Class Numbers of Orders in Quartic Fields
Dissertation
der Fakult at fur Mathematik und Physik der Eberhard-Karls-Universit at Tubingen zur Erlangung des Grades eines Doktors der Naturwissenschaften
vorgelegt von Mark Pavey aus Cheltenham Spa, UK
2006 Mundliche Prufung: 11.05.2006
Dekan: Prof.Dr.P.Schmid 1.Berichterstatter: Prof.Dr.A.Deitmar 2.Berichterstatter: Prof.Dr.J.Hausen Dedicated to my parents:
Deryk and Frances Pavey, who have always supported me.
Acknowledgments
First of all thanks must go to my doctoral supervisor Professor Anton Deit- mar, without whose enthusiasm and unfailing generosity with his time and expertise this project could not have been completed. The rst two years of research for this project were undertaken at Exeter University, UK, with funding from the Engineering and Physical Sciences Re- search Council of Great Britain (EPSRC). I would like to thank the members of the Exeter Maths Department and EPSRC for their support. Likewise I thank the members of the Mathematisches Institut der Universit at Tubingen, where the work was completed. Finally, I wish to thank all the family and friends whose friendship and encouragement have been invaluable to me over the last four years. I have to mention particularly the Palmers, the Haywards and my brother Phill in Seaton, and those who studied and drank alongside me: Jon, Pete, Dave, Ralph, Thomas and the rest. Special thanks to Paul Smith for the Online Chess Club, without which the whole thing might have got done more quickly, but it wouldn’t have been as much fun.
5
Contents
Introduction iii
1 Euler Characteristics and In nitesimal Characters 1 1.1 The unitary duals of SL2(R) and SL4(R) ...... 1 1.2 Euler characteristics ...... 5 1.3 Euler-Poincar e functions ...... 9 1.4 Euler characteristics in the case of SL4(R) ...... 15 1.5 The unitary dual of KM ...... 19 1.6 In nitesimal characters ...... 21
2 Analysis of the Ruelle Zeta Function 28 2.1 The Selberg trace formula ...... 30 2.2 The Selberg zeta function ...... 32 2.3 A functional equation for ZP, (s) ...... 37 2.4 The Ruelle zeta function ...... 43 2.5 Spectral sequences ...... 47 2.6 The Hochschild-Serre spectral sequence ...... 48 2.7 Contribution of the trivial representation ...... 51 2.8 Contribution of the other representations ...... 53
3 A Prime Geodesic Theorem for SL4(R) 56 3.1 Analytic properties of R , (s) ...... 57 3.2 Estimating (x) and ˜(x) ...... 61 3.3 The Wiener-Ikehara Theorem ...... 66 3.4 The Dirichlet series ...... 71 3.5 Estimating n(x) ...... 76 3.6 Estimating 1(x) ...... 79 3.7 Estimating (x), ˜(x) and 1(x) ...... 81
i ii
4 Division Algebras of Degree Four 85 4.1 Central simple algebras and orders ...... 85 4.2 Division algebras of degree four ...... 87 4.3 Sub elds of M(Q) generated by ...... 89 4.4 Field and order embeddings ...... 90 4.5 Counting order embeddings ...... 96
5 Comparing Geodesics and Orders 108 5.1 Regular geodesics ...... 108 5.2 Class numbers of orders in totally complex quartic elds . . . 117 Introduction
In this thesis we present two main results. The rst is a Prime Geodesic Theorem for compact symmetric spaces formed as a quotient of the Lie group SL4(R). The second is an application of the Prime Geodesic Theorem to prove an asymptotic formula for class numbers of orders in totally complex quartic elds with no real quadratic sub eld. Before stating our results we give some background. Let D be the set of all natural numbers D 0, 1 mod 4 with D not a square. Then D is the set of all discriminants of orders in real quadratic elds. For D D the set ∈ x + y√D D = : x yD mod 2 O ( 2 ) is an order in the real quadratic eld Q(√D) with discriminant D. As D varies, runs through the set of all orders of real quadratic elds. For OD D D let h( D) denote the class number and R( D) the regulator of the order∈ . ItOwas conjectured by Gauss ([22]) andOproved by Siegel ([55]) OD that, as x tends to in nity 2x3/2 h( )R( ) = + O(x log x), D D 18 (3) D∈D O O XD x where is the Riemann zeta function. For a long time it was believed to be impossible to separate the class number and the regulator in the summation. However, in [51], Theorem 3.1, Sarnak showed, using the Selberg trace formula, that as x we have → ∞ e2x h( ) . D 2x D∈D O R(XOD) x
iii Introduction iv
More sharply, 3x/2 2 h( D) = L(2x) + O e x D∈D O R(XOD) x ¡ ¢ as x , where L(x) is the function → ∞ x et L(x) = dt. t Z1 Sarnak established this result by identifying the regulators with lengths of closed geodesics of the modular curve SL2(Z) H, where H denotes the upper half-plane, and using the Prime Geodesic Theorem\ for this Riemannian sur- face. Actually Sarnak proved not this result but the analogue where h( ) is replaced by the class number in the narrower sense and R( ) by a “regula-O O tor in the narrower sense”. But in Sarnak’s proof the group SL2(Z) can be replaced by PGL2(Z) giving the above result. The Prime Geodesic Theorem in this context gives an asymptotic formula for the number of closed geodesics on the surface SL2(Z) H with length less than or equal to x > 0. This formula is analogous to the asymptotic\ formula for the number of primes less than x given in the Prime Number Theorem. The Selberg zeta function (see [52]) is used in the proof of the Prime Geodesic Theorem in a way analogous to the way the Riemann zeta function is used in the proof of the Prime Number Thoerem (see [9]). The required properties of the Selberg zeta function are deduced from the Selberg trace formula ([52]). It seems that following Sarnak’s result no asymptotic results for class numbers in elds of degree greater than two were proven until in [15], Theo- rem 1.1, Deitmar proved an asymptotic formula for class numbers of orders in complex cubic elds, that is, cubic elds with one real embedding and one pair of complex conjugate embeddings. Deitmar’s result can be stated as follows. Let S be a nite set of prime numbers containing at least two elements and let C(S) be the set of all complex cubic elds F such that all primes p S are non-decomposed in F . For F C(S) let OF (S) be the set of all∈isomorphism classes of orders in F whic∈h are maximal at all p S, ie. ∈ are such that the completion = Z is the maximal order of the eld Op O p Fp = F Qp for all p S. Let O(S) be the union of all OF (S), where F ranges ov er C(S). For a∈ eld F C(S) and an order O (S) de ne ∈ O ∈ F ( ) = (F ) = f (F ), S O S p p∈S Y Introduction v
where fp(F ) is the inertia degree of p in F . Let R( ) denote the regulator and h( ) the class number of the order . O ForOx > 0 we de ne O
(x) = ( )h( ). S S O O O∈O(S) RX(O) x Then as x we have → ∞ e3x (x) . S 3x More sharply, e9x/4 (x) = L(3x) + O S x µ ¶ as x . This→ ∞result was again proved by means of a Prime Geodesic Theorem, this time for symmetric spaces formed as a compact quotient of the group SL3(R). There exists a Selberg trace formula for such spaces ([59]) by means of which the required properties of a generalised Selberg zeta function can be deduced ([14]) in order to prove the Prime Geodesic Theorem. The class number formula is then deduced by means of a correspondence between primitive closed geodesics and orders in complex cubic number elds, under which the lengths of the geodesics correspond to the regulators of the number elds. In this thesis we follow the methods of [15]. In the rst three chapters we prove a Prime Geodesic Theorem for compact quotients of SL4(R). In the nal two chapters we give an arithmetic interpretation in terms of class numbers. Our main results can be stated as follows. Let G = SL4(R) and let K be the maximal compact subgroup SO(4). Let G be discrete and cocompact. We then have a one to one correspondence bet ween conjugacy classes in and free homotopy classes of closed geodesics on the symmetric space X = G/K. Let \ a a A = : 0 < a < 1 a 1 a 1 and SO(2) B = . SO(2) µ ¶ Introduction vi
Let E ( ) be the set of primitive conjugacy classes [ ] in such that is conjugate in G to an element a b of A B. For we write a also ∈ for the top left entry in the matrix a and de ne the length l of to be 8 log a . Let G , be the centralisers of in G and respectively and let K = K G . For x > 0 de ne the function ∩
(x) = 1( ), [ ]∈E ( ) eXl x where 1( ) is the rst higher Euler characteristic of the symmetric space X = G /K . \ Theorem 0.0.1 (Prime Geodesic Theorem) For x we have → ∞ 2x (x) . log x More sharply, x3/4 (x) = 2 li(x) + O log x µ ¶ as x , where li(x) = x 1 dt is the integral logarithm. → ∞ 2 log t We now state our resultR on class numbers. Let S be a nite, non-empty set of prime numbers containing an even number of elements. We de ne the sets C(S) and O(S) and the constants S( ) as above, replacing complex cubic elds with totally complex quartic eldsO in the de nitions. A totally complex quartic eld has at most one real quadratic sub eld, as can be seen by comparing numbers of fundamental units. Let C c(S) C(S) be the subset of elds with no real quadratic sub eld and let Oc(S) O(S) be the subset of isomorphy classes of orders in elds in C c(S). Let R ( ) and h( ) once again denote respectively the regulator and class numberOof the orderO . O Theorem 0.0.2 (Main Theorem) For x > 0 we de ne
(x) = ( )h( ). S S O O O∈Oc(S) RX(O) x Then as x we have → ∞ e4x (x) . S 8x Introduction vii
Transfering the methods of [15] to the case of totally complex quartic elds does not proceed entirely smoothly, there are various technical di cul- ties to be overcome. These have been successfully overcome to the point of proving the Prime Geodesic Theorem. However, the correspondence between primitive closed geodesics and orders is considerably more complicated than in either the real quadratic or the complex cubic case. As a result of this ex- tra complexity our result is weaker than that obtained in the complex cubic case. We have had to impose the extra condition on the nite set S of prime numbers that it must contain an even number of elements and we have been unable to provide an error term in our nal asymptotic. Also we have had to restrict ourselves to counting the orders in elds without a real quadratic sub eld, as the elds with a real quadratic sub eld cannot be counted using our method. Indeed, any real quadratic eld which occurs as a sub eld of a totally complex quartic eld may in fact occur as a sub eld of in nitely many such elds. The fundamental units in the quadratic eld are powers of fundamental units in the quartic elds, so the (possibly in nitely many) quartic elds all have regulator less than or equal to that of the quadratic eld. Hence an asymptotic formula for a sum of class numbers of orders bounded by the regulators is not even possible in the case of totally complex quartic elds with a real sub eld. In comparison with the real quadratic and complex cubic cases we might have expected to get the asymptotic e4x (x) . S 4x In the result that we do in fact get there is an extra factor of one half. The fact that we have had to restrict the elds over which we are counting gives an explanation for this discrepancy. We also prove the following asymptotic in which we introduce an extra factor into the summands. For an order O(S) this extra factor is de ned in terms of the arguments of the fundamenO ∈ tal units of under the O embeddings of F into C as follows. If ε is a fundamental unit in then ε 1, ε and ε 1 are also fundamental units in , where is a root ofOunity contained in . Let O O 1 (ε) ( ) = 1 , O 2 (ε) O ε X Y µ | |¶ where is the number of roots of unity in , the sum is over the 2 O O O Introduction viii di erent fundamental units of and the product is over the embeddings of O into C. We have: O Theorem 0.0.3 For x > 0 we de ne
˜ (x) = ( ) ( )h( ). S O S O O O∈Oc(S) RX(O) x Then as x we have → ∞ e4x ˜ (x) . S 2x An element in is called regular if it centraliser in G is a torus and non-regular otherwise. The factor ( ) was originally introduced in order to separate the contribution of non-regularO elements from that of the regular elements in the Prime Geodesic Theorem. As it turned out, the complexity of the correspondence between geodesics and orders meant that we did not really gain anything from this approach. However, the factor ( ) contains O information about the arguments of the fundamental units in the order which is interesting in its own right. In comparison with Theorem 0.0.2 wOe can see that “on average” the value of ( ) as R( ) goes to in nity is 4. If ε is a fundamental unit in with argumenO ts , O , , under the four O embeddings of into C then a simple calculation shows that O (ε) 1 = 4(1 cos 2 )(1 cos 2 ). (ε) Y µ | |¶ Since this takes “on average” the value 4 we can see that there is a sense in which we can say that the arguments of the fundamental units in the orders are evenly distributed about /2 as R( ) goes to in nity. O In what follows we give a chapter by chapterO summary of the arguments and techniques used in the proof of our main results. In particular we shall point out the di culties that arise in applying the methods of [15] to our situation. Chapter 1 provides some necessary background and preliminary results. In Chapter 2 we introduce the zeta functions we shall be studying and prove some of their analytic properties. The results of Chapter 1 are made use of here in the de nitions of the zeta functions and proofs of their analytic properties. In Chapter 3 we apply standard techniques from analytic number theory to the results of Chapter 2 in order to prove the Prime Geodesic Introduction ix
Theorem required in our context. In Chapter 4 we show that the elds we are interested in can be obtained as maximal sub elds of a suitably chosen division algebra over Q and count the number of embeddings of the elds into the division algebra. Finally, in Chapter 5 we apply the results of Chapter 4 to give the arithmetic interpretation of the Prime Geodesic Theorem in terms of class numbers. The correspondence between closed primitive geodesics in the Prime Geo- desic Theorem and orders in totally complex quartic elds goes via the choice of a division algebra M(Q) of degree four over Q whose maximal sub elds include the ones we are interested in, ie. the elds in the set C c(S). We are in fact able to choose M(Q) so that as well as having the properties described above it also satis es M(Q) R = Mat (R). The map M(Q) R R 4 → induced from the reduced norm on M(Q) agrees with the determinant map on Mat4(R). We de ne
G (R) = X M(Q) R : det X = 1 = SL (R). { ∈ } 4 If we let M(Z) denote the maximal order of M(Q) ([49]) then
= (M(Z) 1) G (R) ∩ is a discrete, cocompact subgroup of SL4(R) ([3]). Note that is not neces- sarily torsion free. The group forms the link between geodesics and class numbers in the following sense. Firstly, there is a one-to-one correspondence between con- jugacy classes in and free homotopy classes of closed geodesics on the symmetric space X = SL4(R)/SO(4). Under this correspondence primi- tive elements of corresp\ond to primitive geodesics. Secondly, the primitive elements of viewed as a subset of the maximal order M(Z) in M(Q) cor- respond to fundamental units of orders of sub elds of M(Q). We shall be interested in the primitive conjugacy classes in which correspond to orders in totally complex quartic elds with no real quadratic sub eld. Our strategy is to use a suitably de ned generalised Selberg zeta function to get information about the distribution of the primitive, closed geodesics c on X which correspond to orders in O (S). The de nition of such a function is given as a product over the relevant conjugacy classes in . We are able to study our chosen zeta function by choosing a suitable test function for the Selberg trace formula on X so that the geometric side gives a higher Introduction x derivative of the zeta function. We do this using the theory developed in [14]. It is at this point that we encounter the rst obstacle. We say that an element g G is weakly neat if the adjoint Ad(g) has no non-trivial roots of unity as eigen∈ values. A subgroup of G is weakly neat if every element is. The results of [14] use the assumptions that the group G has trivial centre and the group is weakly neat. Note that, if G has trivial centre then weakly neat implies torsion free, since any non weakly neat torsion elements must be central. To generalise the results of [14] to the case where is not weakly neat requires two things. Firstly, we need to modify the de nition of the zeta function slightly from that given in [14]. Secondly, the de nition of the zeta function includes the rst higher Euler characteristic of the spaces X for E ( ). In [14] these are only de ned for torsion free so we need to broaden∈ the de nition. In Chapter 1 we give a de nition of rst higher Euler characteristics which is general enough for our application. We further prove that its value is always positive in the cases we consider. This fact is needed in the proof of the Prime Geodesic Theorem. In [14] it is shown that the position of the poles and zeros of the generalised Selberg zeta function depend on the Lie algebra cohomology of the irreducible, unitary representations of SL4(R). Using a result of Hecht and Schmid ([26]) it su ces to look at the in nitesimal characters of the irreducible, unitary representations of SL4(R). For this purpose, in Chapter 1 we also describe the unitary dual of SL4(R) using a result of Speh ([56]) and give a result about the in nitesimal characters of certain elements of this set. In Chapter 2 we de ne the generalised Selberg zeta function and use the trace formula to deduce its analytic properties. In particular, we give a for- mula for its vanishing order at a given point and prove a functional equation, from which we can deduce that it is of nite order. We then de ne the gener- alised Ruelle zeta function and prove that it is a nite quotient of generalised Selberg zeta functions. In particular, in the cases we are interested in it has a zero at s = 1 and all other poles and zeros in the half plane Res 3 , 4 and is furthermore of nite order. We introduce the Ruelle zeta function as its logarithmic derivative is a Dirichlet series from whose properties we can prove the Prime Geodesic Theorem. The de nitions of the generalised Selberg and Ruelle zeta functions de- pend on the choice of a nite dimensional virtual representation of a particu- lar closed subgroup M of SL4(R). The trace of this representation at elements Introduction xi of appears in the Dirichlet series arising as the logarithmic derivative of the Ruelle zeta function. We can choose such a representation ˜ so that its trace is zero for all non-regular elements of . The factors ( ) in Theorem 0.0.3 come from the traces of the representation ˜. O In Chapter 3 we apply the methods of [47] together with standard tech- niques of analytic number theory to prove a Prime Geodesic Theorem. In carrying out the application to class numbers it is necessary to show that cer- tain subsets of the summands contribute negligibly to the asymptotic. This is done using a “sandwiching” argument. To carry out this “sandwiching” argument we de ne a sequence of Dirichlet series, which are not connected to any zeta function, but whose analytic properties can also be deduced from the Selberg trace formula. We use a version of the Wiener-Ikehara theorem to prove an asymptotic result for an increasing sequence of functions derived from these Dirichlet series, which we use to “sandwich” the product over the elements we are interested in against the sum over all elements which comes from the Ruelle zeta function. Unfortunately the asymptotics we are able to derive from the Wiener-Ikehara theorem do not provide an error term like those we can deduce from the Ruelle zeta function using the methods of [47]. This is why we lose the error term in Theorem 0.0.2. In Chapter 4 we use a classi cation of the set of equivalence classes of division algebras over Q by means of a description of the Brauer group of Q ([45], Theorem 18.5) to show the existence of a division algebra M(Q) whose maximal sub elds include the elds in the set C c(S) and such that M(Q) R = Mat (R). In fact the maximal sub elds of M(Q) are all quartic 4 extensions of Q and the set of totally complex maximal sub elds of M(Q) coincides with C(S). We further show that for an order in O(S) the O number of embeddings of into M(Q), up to conjugation by the unit group O of the maximal order M(Z), is S( )h( ). The restriction on the set S of prime numbers that it has to conOtainOan even number of elements is a consequence of the classi cation of division algebras over Q. In Chapter 5 we prove a correspondence between the primitive, closed geodesics in the Prime Geodesic Theorem and the orders of totally complex quartic elds. Under this correspondence the lengths of the geodesics cor- respond to the regulators of the orders. It turns out that it is the regular, weakly neat elements of E ( ) which correspond to orders in totally complex quartic elds with no real quadratic sub eld. We use the above mentioned “sandwiching” argument to isolate these elements in the Prime Geodesic Theorem. Introduction xii
We nish this introduction with a few remarks about the limitations of the method used here in terms of further applications and mention a couple of other recent results in the same direction. In order to be able to make use of the trace formula for compact spaces we have had to limit our sum over class numbers by means of the choice of a nite set of primes, as described above. In [18] Deitmar and Ho mann have been able to use a di erent trace formula to prove that as x → ∞ e3x h( ) , O 3x R(XO) x where the sum is over all isomorphism classes of orders in complex cubic elds. In order to get the error term in the Prime Geodesic Theorem we have made use of the classi cation of the unitary dual of SL4(R). At present the unitary dual is not known for any higher dimensional groups so a Prime Geodesic Theorem with error term is not possible using our methods. Finally we mention that the correspondence between geodesics and orders actually works by identifying primitive geodesics with fundamental units in orders. By Dirichlet’s unit theorem, an order in a number eld F has a unique fundamental unit (up to inversion and multiplication by a root of unity) only if F is real quadratic, complex cubic or totally complex quartic. Hence an asymptotic of our form can be proven only in these three cases. In [17] Deitmar has proved a Prime Geodesic Theorem for higher rank spaces, from which he deduces an asymptotic formula for class numbers of orders in totally real number elds of prime degree. Chapter 1
Euler Characteristics and In nitesimal Characters
In this chapter we introduce some concepts and prove some results which will be needed for our consideration of the zeta functions in the next chapter.
1.1 The unitary duals of SL2(R) and SL4(R) For a locally compact group G (or more generally a topological group) a representation ( , V ) of G on a complex Hilbert space V = 0 is a homomor- 6 phism of G into the group of bounded linear operatots on V with bounded inverses, such that the resulting map of G V into V is continuous. A representation will also be denoted simply as . An invariant subspace for such a ( , V ) is a vector subspace U V such that (g)U U for all g G. The representation is irreducible if it has no closed invarian t subspaces∈ other than 0 and V . The representation is unitary if (g) is unitary for all g G. Two representations ( , V ) and ∈ ( , V ) are equivalent if there is a bounded linear map E : V V with a bounded inverse such that (g)E = E (g) for all g G. If →and are ∈ unitary, they are unitarily equivalent if they are equivalent via an operator E that is unitary. The unitary dual Gˆ of G is the set of all equivalence classes of irreducible unitary representations of G. If is an irreducible, unitary representation of G then we shall write, by slight abuse of notation, Gˆ. ∈ For a real Lie Algebra gR a Lie algebra representation ( , V ) of gR on
1 1. Euler Characteristics and In nitesimal Characters 2
a complex vector space V = 0 is a homomorphism of gR into the Lie 6 algebra of all automorphisms of V . Invariant subspaces, irreducibility and equivalence are de ned in an analogous way as for group representations. Let G1 = SL2(R). We de ne the following subgroups: let K1 be the maximal compact subgroup SO(2); let M = I ; 1 { 2} a A = : a > 0 ; 1 a 1 ½µ ¶ ¾ and 1 x N = : x R . 1 1 ∈ ½µ ¶ ¾ Let g1, k1, a1, n1 be the respective complexi ed Lie algebras. Let 1 a1 be de ned by ∈ a = a, 1 a µ ¶ and let P1 = M1A1N1 be the parabolic subgroup of G1 with split torus A1 and unipotent radical N1. D + D For n 2 we de ne the discrete series representations n and n of G1. D + The Hilbert space for n is the space of analytic functions f on the upper half plane such that ∞ ∞ f 2 = f(x + iy) 2yn 2 dx dy < . k k | | ∞ Z0 Z ∞ The inner product is given by ∞ ∞ f, g = f(x + iy)g(x + iy)yn 2 dx dy h i Z0 Z ∞ and the group action is a b az c D + f(z) = ( bz + d) nf . n c d bz + d µ ¶ µ ¶ D The representations n are de ned analogously for functions on the lower half plane. We also de ne the two limit of discrete series representations D + D 1 and 1 on the spaces of analytic functions f on the upper (respectively, lower) half plane such that ∞ f 2 = sup f(x + iy) 2 dx < . k k | | ∞ y>0 Z ∞ 1. Euler Characteristics and In nitesimal Characters 3
The inner product is given by ∞ f, g = sup f(x + iy)g(x + iy) dx h i y>0 Z ∞ and the group action is as for the discrete series representations. The repre- D + D sentations n and n are irreducible unitary representations of G1 for all n 1. Given an irreducible, unitary representation of M1 and an element a G1 of we de ne the induced representation Ind P1 ( ) of G1 = SL2(R). Consider the space f : G V f continuous, f(xman) = e ( + 1) log a (m) 1f(x) { 1 → | } with the inner product
f, g = f(k), g(k) dk. h i h i ZK1 G1 1 G acts on the completion of this space via Ind P1 ( )f(x) = f(g x), to give a representation, which is irreducible. + Let be the trivial representation on M1 and the representation de ned by ( I ) = 1. For a we de ne 2 ∈ 1 P , = Ind G1 ( ). P1 For x R de ne the principal series representations ∈ P ,ix = P ,ix 1 . Then the representations P ,ix are unitary and are all irreducible except for P ,0 D + D P , = 1 1 . The only equivalences among the representations are that P+,ix and P ,ix are equivalent to P +, ix and P , ix respectively for all x R. We also∈ de ne the complementary series representations C x = P+,x 1 , for 0 < x < 1. Instead of the usual inner product we give C x the inner product: ∞ ∞ f(x), g(y) + f, g = h i dx dy. h i x y 1 x Z ∞ Z ∞ | | With this inner product the representations C x are unitary for all 0 < x < 1. We have the following classi cation theorem: 1. Euler Characteristics and In nitesimal Characters 4
Theorem 1.1.1 The unitary dual of SL2(R) consists of a) the trivial representation; b) the principal series representations P +,ix for x R and P ,ix for ∈ x R r 0 ; ∈c) the{complementary} series C x for 0 < x < 1; D + D d) the discrete series n and n for n 2 and limits of discrete series D + D 1 and 1 . P ,ix P , ix The only equivalences among these representations are = for all x R. ∈ Proof: [39], Theorem 16.3. ¤
Now let G = SL4(R) and K = SO(4), so K is a maximal compact sub- group of G. Let P 0 = M 0A0N 0 be a parabolic subgroup of G with split component A0 and unipotent radical N 0. Let g and a0 be the complexi ed Lie algebras of G and A0 and let a0 be the complex dual of a0. Let 0 be the half sum of the positive roots of the system (g0, a0). Then we can de- ne induced representations in an entirely analogous way to that used for 0 for G1 = SL2(R) above. For an irreducible, unitary representation of M and for a0 we write the corresponding induced representation of G as G ∈ Ind P 0 ( ). 0 G If = ix for some x R then the induced representation Ind P 0 ( ) is unitary with respect to ∈the inner product
f, g = f(k), g(k) dk. h i h i ZK G In this case we call Ind P 0 ( ) a principal series representation. For other 0 G choices is a it may be possible to make Ind P 0 ( ) unitary with respect ∈ G to a di erent inner product, in which case we call Ind P 0 ( ) a complemen- tary series representation. There are also certain irreducible, unitary sub- representations of induced representations (see [56], p121) which are called limit of complementary series representations. These limit of complementary series representations can often be realised as induced representations from a parabolic subgroup P 00 P 0. We de ne the following subgroups of G. Let SL (R) M = S 2 SL (R) µ 2 ¶ det (x), det (y) = 1 = (x, y) Mat2(R) Mat2(R) , ∈ | det (x)det (y) = 1 ½ ¾ 1. Euler Characteristics and In nitesimal Characters 5
a a A = : a > 0 , a 1 a 1 and I Mat (R) N = 2 2 . 0 I µ 2 ¶ Let P be the parabolic subgroup of G with Langlands decomposition P =
MAN. For m1, m2 N, we denote by m1,m2 the representation of M induced ∈ D + D + from the representation m1 m2 of SL2(R) SL2(R). For m N, let G 1 ∈ m = m,m and let Im = Ind P m 2 P . The representations Im each have a unique irreducible quotient kno wn as the Langlands quotient (see ¡ ¢ [39], Theorem 7.24). We denote the Langlands quotient of Im by m. We have the following classi cation theorem:
Theorem 1.1.2 The unitary dual of SL4(R) consists of a) the trivial representation; b) principal series representations; G c) complementary series representations Ind P ( m t P ), for m N and 1 ∈ 0 < t < 2 ; d) complementary series representations induced from parabolics other than P = MAN; e) limit of complementary series representations, which are irreducible unitary subrepresentations of I , for m N; m ∈ f) the family of representations , indexed by m N. m ∈ Proof: This follows from [56], Theorem 5.1, where the unitary dual of SL (R) = X Mat (R) : det X = 1 is given. ¤ 4 { ∈ 4 } 1.2 Euler characteristics
Let G be a real reductive group and suppose that there is a nite subgroup E of the centre of G and a reductive and Zariski-connected linear group G such that G/E is isomorphic to a subgroup of G (R) of nite index. Note these conditions are satis ed whenever G is a Levi component of a connected semisimple group with nite centre. Let K be a maximal compact subgroup of G and let be a discrete, cocompact subgroup of G. Let X be the locally 1. Euler Characteristics and In nitesimal Characters 6 symmetric space G/K. If is torsion-free we de ne the rst higher Euler characteristic of \to be
dim X (X ) = ( ) = ( 1)j+1jhj(X ), 1 1 j=0 X j where h (X ) is the jth Betti number of X . We want to generalise this de nition to cases where the group is not necessarily torsion free. We prove below a proposition which allows us broaden the de nition to all cases required by our applications. Let be the Cartan involution xing K pointwise. There exists a - stable Cartan subgroup H = AB of G, where A is a connected split torus and B K is a Cartan of K. We assume that A is central in G. Let C denote the centre of G. Then C H and we write BC , C for B C and C respectively. Let G1 be the derived group of G and let 1 = G∩1 C. ∩ ∩ We note in particular that, since G = G1C, we have 1C 1AB. Let A = A C BC be the projection of C to A. Then A is discrete and cocompact in∩A (see [62], Lemma 3.3). Let gR be the real Lie algebra Lie(G) with polar decomposition gR = kR pR. Let b be a xed nondegenerate invariant bilinear form on the Lie algebra gR which is negative de nite on kR and positive de nite on pR. Then the form b(X, (Y )) is positive de nite and thus de nes a left invariant metric on G. Unless otherwise stated all Haar measures will be normalised according to the Harish-Chandra normalisation given in [25], 7. Note that this normalisation depends on the choice of the bilinear form b§ on g. On the space G/H we have a pseudo-Riemannian structure given by the form b. The Gauss-Bonnet construction generalises to pseudo-Riemannian structures to give an Euler-Poincar e measure on G/H. De ne a (signed) Haar measure on G by
= (Haar measure on H). EP The Weyl group W = W (G, H) is de ned to be the quotient of the normaliser of H in G by the centraliser. It is a nite group generated by elements of order two. We de ne the generic Euler characteristic by
( G) ( ) = (X ) = EP \ . gen gen W | | 1. Euler Characteristics and In nitesimal Characters 7
Proposition 1.2.1 Assume is torsion free, A is central in G of dimension one and 0 is of nite index in . Then A/ acts freely on X and A gen(X ) = 1( )vol(A/ A). It follows that
[ : 0 ] ( ) = ( 0) A A . 1 1 [ : 0]
Proof: The group A = A/ acts on G/B by multiplication from the A \ right. We claim that this action is free, i.e., that it de nes a bre bundle
A G/B G/H. → \ → \ To see this let xaB = xB for some a A and x G. Then a = x 1 xb 1 ∈ 1 ∈ 1 for some and b B. Since C AB we can write as a b , 1 ∈1 ∈ with and a A and b BC = B C. It follows that a A. ∈ ∈ ∈ 1 ∩ 1 1 1 1 1∈ 1 Since A is central in G, we can write = aa xb x b . Since G 1 1 ∈ and aa A C, we must have aa = 1, so a = a A, which implies the claim.∈ ∈ In the same way we see that we get a bre bundle
A G/K A G/K. (1.1) → \ → \ Let be the usual Euler characteristic. From [31] we take the equation (K/B) = W . Using multiplicativity of Euler numbers in the bre bundle | | K/B A G/B A G/K → \ → \ we get
( G/H) (X ) = vol(A/ ) \ gen A W | | ( G/H) = vol(A/ ) \ A (K/B) (A G/B) = vol(A/ ) \ A (K/B)
= vol(A/ ) (A G/K) A \ = vol(A/ ) (A X ). A \ 1. Euler Characteristics and In nitesimal Characters 8
It remains to show that (A X ) = 1( ). 1 \ 1 Let aR and gR be the Lie algebras of A and G respectively. Then we can write 1 gR = aR g zR, (1.2) R where zR is central in gR. Let X be the bi-invariant vector eld on G/K \ generating the A action. Then we can consider X as an element of aR under the decomposition (1.2). Considering the dual of (1.2), we can identify aR with a subspace of gR. Since a is central in g, a non-zero element of aR gives us an A -invariant, closed 1-form ω on G/K such that for every \ gK G/K we have ω( gK)(X) = 0. Since A = R/Z is connected and compact,∈ \the cohomology of the de Rham6 complex G/K coincides with the \ A cohomology of the subcomplex of A -invariants ( G/K) . Using local triviality of the bundle one sees that \
( G/K)A = (A G/K) (A G/K) ω, \ \ \ ∧ where denotes the projection map. Set C = (A G/K) and C = 0 \ 1 C C ω = ( G/K)A . Then 0 0 ∧ \ Hp(C ) = Hp(C ) Hp 1(C ) 1 0 0 and so
(C ) = ( 1)p+1p dim Hp(C ) 1 1 1 p 0 X = ( 1)p+1p dim Hp(C ) + dim Hp 1(C ) 0 0 p 0 X ¡ ¢ = ( 1)p+1(p (p + 1)) dim Hp(C ) 0 p 0 X = ( 1)p dim Hp(C ) 0 p 0 X = (C0). (1.3) This gives the required result. ¤ Let be a discrete, cocompact subgroup of G. It is known that every arithmetic subgroup of G has a torsion free subgroup of nite index ([2], Proposition 17.6). Let 0 be such a subgroup. Suppose further that the 0 torus A is central in G and of dimension one. De ne A and A as above. 1. Euler Characteristics and In nitesimal Characters 9
We de ne the rst higher Euler characteristic of as [ : 0 ] (X ) = ( ) = ( 0) A A . (1.4) 1 1 1 [ : 0] Propostition 1.2.1 shows that this is well-de ned. We note that in the case that itself is torsion free, this de nition of rst higher Euler characteristic agrees with the one given above.
1.3 Euler-Poincar e functions
For the next de nition we assume G to be a semisimple real reductive group of inner type and we x a maximal compact subgroup K. We further assume that G contains a compact Cartan subgroup. Let gR be the real Lie algebra Lie(G) with polar decomposition gR = kR pR, and write g = k p for its complexi cation. Recall that we are using Harish-Chandra’s Haar measure normalisation as given in [25], 7 and this normalisation depends on the choice of an invariant bilinear form§ b on g, which for our purposes in this chapter we shall leave arbitrary. A virtual representation of a group is a formal di erence of two repre- sentations = + , which is called nite dimensional if both + and are. Two virtual represen tations = + and = + of a group + + are said to be isomorphic if there is an isomorphism = . The trace and determinant of a virtual representation = + are de ned by tr = tr + tr and det = det +/det . The dimension of is de ned as dim = dim + dim . If V is a representation space with Z-grading then we shall consider it + naturally as a virtual representation space by V = Veven and V = Vodd. In particular, if V is a subspace of g we shall always consider the exterior product V as a virtual representation V = even V odd V with respect to the adjoint representation. We consider symmetric powers and cohomologyV spaces similarly. V V V For a smooth function f on G of compact support and an irreducible unitary representation ( , V ) Gˆ de ne the operator ∈ (f) = (g)f(g)dg ZG on V . Since f is smooth and has compact support, (f) is of trace class. 1. Euler Characteristics and In nitesimal Characters 10
Let ( , V ) be a nite dimensional virtual representation of G. An Euler- Poincare function f for is a compactly supported, smooth function on G such that f (kxk 1) = f (x) for all k K and for every irreducible unitary ∈ representation ( , V ) of G the following identity holds:
dim(p) p K tr (f ) = ( 1)p dim V p V , (1.5) p=0 X ³ ^ ´ where the superscript K denotes the subspace of K invariants. By [14], Theorem 1.1 such functions exist. We note that the value of such functions depends on the choice of Haar measure. We shall assume all Euler-Poincar e functions are given with respect to the Harish-Chandra normalisation. In the following lemmas we prove some of their properties.
Lemma 1.3.1 Let G denote a semisimple real reductive group of inner type, with connected component G0, maximal compact subgroup K and compact Cartan subgroup T K. Let G+ = T G0. Further let be a nite dimen- + sional representation of G, = + and f an Euler-Poincare function |G for on G. + + Then f + is an Euler-Poincare function for on G . |G Proof: This is Lemma 1.5 of [14]. ¤
Lemma 1.3.2 Let H, H1, H2 be real reductive groups of inner type such that H = H H . Let be an irreducible representation of H. There exist 1 2 irreducible representations , of H , H respectively such that = 1 2 1 2 1 2 and let f i be an Euler-Poincare function for i on Hi.
Then f (h1, h2) = f 1 (h1)f 2 (h2) is an Euler-Poincare function for on H.
Proof: Let Hˆ . Then = 1 2 for some i Hˆi. Let Ki be a maximal compact∈ subgroup of H and let K = K ∈K be a maximal i 1 2 compact subgroup of H. Let h R = k R p R be the polar decomposition of i, i, i, the real Lie algebra hi,R of Hi and write hi = ki pi for its complexi cation. Let p = p p . We note in particular that 1 2 p q p = p p . 1 2 p,q ^ M ^ ^ 1. Euler Characteristics and In nitesimal Characters 11
Then
tr (f ) = tr f (h )f (h ) ( (h ) (h )) dh dh 1 1 2 2 1 1 2 2 1 2 µZH1 ZH2 ¶ = tr f (h ) (h )dh f (h ) (h )dh 1 1 1 1 1 2 2 2 2 2 ·µZH1 ¶ µZH2 ¶¸ = (tr 1(f 1 )) (tr 2(f 2 ))
dim pi p Ki = ( 1)p dim V p V i i i i=1,2 p=0 Y X ³ ^ ´ dim p1 dim p2 p q K = ( 1)p+q dim V p p V 1 2 p=0 q=0 X X ³ ³^ ^ ´ ´ dim p j K = ( 1)j dim V p V , j=0 X ³ ^ ´ which is what was required to show. ¤
Lemma 1.3.3 Let G denote a semisimple real reductive group of inner type, with maximal compact subgroup K and compact Cartan subgroup T K. Let g G be central. Let be a nite dimensional representation of G and ∈ let f , h be Euler-Poincare functions for on G. Then f (g) = h (g).
Proof: Since g is central, the orbital integral can be written as
(f ) = f (x 1gx) dx = f (g), Og ZGg\G where Gg denotes the centraliser of g in G (which in this case, since g is central, is the whole of G). Similarly (h ) = h (g). By [14], Proposition Og 1.4, the value of the orbital integral g(f ) of an Euler-Poincar e function for on G depends only on g, not on theO particular Euler-Poincar e function chosen. Hence, f (g) = (f ) = (h ) = h (g), as claimed. ¤ Og Og
Lemma 1.3.4 Let g1 be an Euler-Poincare function for the trivial represen- tation on SL (R). Then g (1) = g ( 1) R. 2 1 1 ∈ Proof: For v R let P +,iv and P ,iv be principal series representations on ∈ SL (R). For n N, n 2 let D + and D be discrete series representations 2 ∈ n n 1. Euler Characteristics and In nitesimal Characters 12
D D + D on SL2(R) and write n for n n . By [39], Theorem 11.6 there is a constant M > 0 such that for any compactly supported, smooth function f on SL2(R) we have
∞ f(1) = M (n 1)tr D (f) n n=2 X M ∞ v v + tr P+,iv(f)vtanh + tr P ,iv(f)vcoth dv. 4 ∞ 2 2 Z ³ ´ ³ ´ Lemma 1.3 of [14] tells us that tr P ,iv(g ) = 0 for all v R, so we have 1 ∈ ∞ g (1) = M (n 1)tr D (g ). (1.6) 1 n 1 n=2 X By the de nition of an Euler-Poincar e function, for all n 2 2 p SO(2) tr D (g ) = ( 1)p dim D p , n 1 n p=0 X ³ ^ ´ where p is the complex Lie algebra
a b p = : a, b C . b a ∈ ½µ ¶ ¾
Hence g1(1) R. ∈ D We want to know for which values of n the trace tr n (g1) is non-zero. D For this we need to know the SO(2)-types of n and p.
Lemma 1.3.5 For l Z, de ne the one dimensional representation εl of SO(2) by ∈ il εlR( ) = e , where cos sin R( ) = SO(2). sin cos ∈ µ ¶ Note that ε0 is the trivial representation, which we shall denote by triv. The unitary dual of SO(2) is the set ε : l Z . { l ∈ } 1. Euler Characteristics and In nitesimal Characters 13
Proof: Since SO(2) is abelian all its irreducible representations are one dimensional by Schur’s Lemma ([39], Proposition 1.5). By unitarity and the fact that I = R(0) = R(2 ) we have that SO(2)\ = ε : l Z . ¤ { l ∈ } By computing the adjoint action of SO(2) on p we get: Lemma 1.3.6 We have the following isomorphisms of SO(2)-modules:
0 p = triv 1 ^ p = ε ε 2 2 2 ^ p = triv. ¤ ^ Lemma 1.3.7 For n N we have the following isomorphism of SO(2)- modules: ∈ D n = εj. |j| n j Mn mod2
Proof: Let n be the unique n-dimensional representation of SL2(R) (see [39], Chapter II 1). It follows from the de nition of n that the following isomorphism of SO§ (2)-modules holds:
n = εj. |j| n 1 j n M1 mod(2)
Let P1 = M1A1N1 be the minimal parabolic of SL2(R). Then the unitary dual of M1 = 1, 1 consists of two one dimensional representations, which we denote by {1 = triv} and 1. We denote by the character of A 1 1 a = a, 1 a 1 µ ¶ and write for Ind 1 (n 1) . Then we have the following exact 1,n 1 P P sequences of SL (R)-modules (see [39], Chapter II 5): 2 § 0 D 0, n N, n even → n → 1,n 1 → n 1 → ∈ and 0 D 0, n N, n odd, n = 1 → n → 1,n 1 → n 1 → ∈ 6 1. Euler Characteristics and In nitesimal Characters 14
and the isomorphism of SL2(R)-modules
D 1 = 1,0. From the so-called compact picture of induced representations given in [39], Chapter VII 1, it is fairly straightforward to compute the following isomor- phisms of SO§(2)-modules, where the sums are over all integers with the same parity:
1,n 1 = εj j even M 1,n 1 = εj. jModd The lemma then follows easily by combining the various elements of the proof. ¤
D From the previous two lemmas we can see that tr n (g1) is non-zero only if n = 2 and in that case tr D (g ) = 2, so g (1) = 2M. 2 1 1 It remains to show that g1( 1) = g1(1). Let Rz be the right multiplica- ∞ tion operator of SL2(R) on the space Cc (SL2(R)) of smooth, compactly sup- ported functions on SL (R). That is, R g(x) = g(xz) for all x, z SL (R) 2 z ∈ 2 and g C∞ (SL (R)). Let be an irreducible, unitary representation of ∈ c 2 SL (R). The matrix 1 = I is central in SL (R) so ( 1) commutes with 2 2 2 (x) for all x SL2(R) and hence, by Schur’s Lemma ([39], Proposition 1.5) is scalar. This∈ means that for any irreducible, unitary representation on SL (R) we have tr (R g ) = ( 1)tr (g ). Thus we get 2 1 1 1
g1( 1) = R 1g1(1) ∞ = M (n 1)D ( 1)tr D (g ) n n 1 n=2 X = D ( 1)g (1), 2 1 D D + where 2 ( 1) is a scalar, which we now compute. For g 2 and z C we have the action ∈ ∈ a b az c D + g(z) = ( bz + d) 2g . 2 c d bz + d µ ¶ µ ¶ 1. Euler Characteristics and In nitesimal Characters 15
Hence 1 z D + g(z) = ( 1) 2g = g(z). 2 1 1 µ ¶ µ ¶ Similarly we get 1 D g(z) = g(z) 2 1 µ ¶ so D ( 1) = 1 and the lemma is proved. ¤ 2
1.4 Euler characteristics in the case of SL4(R)
We now consider the particular case G = SL4(R). Let K = SO(4), a maximal compact subgroup of G and let be a discrete, cocompact subgroup of G. Let A be the rank one torus a a A = : a > 0 , a 1 a 1 and let B be the compact group SO(2) B = . SO(2) µ ¶ B is a compact Cartan subgroup of SL (R) M = S 2 SL (R) µ 2 ¶ det (x), det (y) = 1 = (x, y) Mat2(R) Mat2(R) . ∈ | det (x)det (y) = 1 ½ ¾ Let I Mat (R) N = 2 2 0 I µ 2 ¶ and let P = MAN be a parabolic subgroup of G with Levi component MA and unipotent radical N. Let a a A = : 0 < a < 1 , a 1 a 1 1. Euler Characteristics and In nitesimal Characters 16 be the negative Weyl chamber in A with respect to the root system given by the choice of parabolic. Let EP ( ) be the set of -conjugacy classes of elements which are conjugate in G to an element of A B. We shall use this notation∈ for the rest of this chapter. We say g G is regular if its centraliser is a torus and non-regular (or singular) otherwise.∈ Clearly, for regularity is a property of the - ∈ conjugacy class [ ]. Let [ ] EP ( ) and write G for the centraliser of in ∈ G. The element is conjugate in G to an element a b A B and we de ne 1/2 ∈ the length l of to be l = b(log a , log a ) . It follows that if is regular then G = AB. In the rst case K = B is a maximal compact subgroup of G , the group B is then clearly a Cartan subgroup of K , the product AB is a - stable Cartan subgroup of G and A is central in G . If we let = G ∩ denote the centraliser of in then is discrete and cocompact in G . Let 0 0 0 be a torsion free subgroup of nite index in and let = G . Then 0 ∩ 0 is a torsion free subgroup of nite index in . We de ne ,A, ,A as in Section 1.2. The rst higher Euler characteristic 1( ) of is then de ned as in (1.4) as: 0 ,A : ( ) = ,A ( 0 ). (1.7) 1 0 1 £ : ¤
In the second case K = S(O(2)£ O(2))¤ is a maximal compact subgroup of G . Furthermore, B is a Cartan subgroup of K , the product AB is a -stable Cartan subgroup of G and A is central in G . The de nition of 1( ) then proceeds exactly as above.
Lemma 1.4.1 For E ( ) we have that ∈ P
C vol( G ) 1( ) = \ , l 0 where C is an explicit constant depending only on , which is equal to one when is regular, and 0 is the primitive geodesic underlying .
0 Proof: Since is torsion free we may take from the second proposition in section 2.4 of [11] the equation
C vol( 0 G ) 0 \ 1( ) = 0 , 1. Euler Characteristics and In nitesimal Characters 17 where 0 denotes the volume of the maximal compact at in 0 G/K con- \ taining , and C is an explicit constant depending only on . We note that in [11] there is an extra factor in the equation which does not show up here. The reason is that in [11] a di erential form, not a measure, was constructed. The value of the constant C is given in [11] in terms of the root system of G with respect to a Cartan subgroup. In the case that is regular, as noted above we have that G = AB and hence G is a Cartan subgroup of itself. It is then easy to see that value of C in this case is one. We denote by the volume of the maximal compact at in G/K 0 \ containing . The values of and are given by the volumes of the images 0 of ,A A and ,A A respectively under their embeddings into G/K and 0 \ \ \ G/K respectively. Since A is one-dimensional, = l 0 . Furthermore, \ 0 from the de nition of the groups ,A and ,A it follows that
0 0 0 ,A : ,A = / = /l 0 . Since £ ¤ vol( 0 G ) = : 0 vol( G ), \ \ the lemma follows from (1.7). £ ¤ ¤
Theorem 1.4.2 For all EP ( ) the Euler characteristic 1( ) is posi- tive. If is regular then ∈
0 ,A : ( ) = ,A . 1 0 £ : ¤ £ ¤ 0 Proof: If is regular then G = AB, K = B and is a complete lattice 0 1 in G so X 0 = G /K = S , the unit circle in C. The Betti numbers \ hj(S1) are equal to zero for j = 0, 1 and one for j = 0, 1, hence ( 0 ) = 1. 6 1 The claimed value for 1( ) then follows immediately from (1.7). Suppose is not regular. Since EP ( ), we have that is conjugate in G to a b A B AM. Let ( , V )∈be a nite dimensional representation ∈ of M and let KM be the maximal compact subgroup S(O(2) O(2)) of M, which contains the compact Cartan B of M. We saw above that there exist Euler-Poincar e functions of on M. Let f be one such. We denote by M the centraliser of b in M and by M (f ) the orbital integral Ob
1 f xb x dx. ZM/M ¡ ¢ 1. Euler Characteristics and In nitesimal Characters 18
From Proposition 1.4 of [14] we get the equation M (f ) = C tr (b ), Ob where C is the constant from Lemma 1.4.1. Together with Lemma 1.4.1 this gives us M b (f )vol( G ) 1( ) = O \ . l 0 tr (b ) Choosing = 1, the trivial representation of M, this simpli es to M b (f1)vol( G ) 1( ) = O \ . (1.8) l 0 To complete the proof of the theorem we shall show that the orbital integral M (f ) is positive. In the case we are considering Ob 1 1 b = , 1 µ ¶ where 1 denotes the identity matrix in SL2(R), hence it is central and M = M so we have simply that M (f ) = f (b ). (1.9) Ob 1 1 The group M = SL2(R) SL2(R) is the connected component of M = S(SL (R) SL (R)). M has a maximal compact subgroup K = O(2) O(2) 2 2 M and compact Cartan T = SO(2) SO(2). We have that T M = M . M M Hence by Lemma 1.3.1, since f1 is an Euler-Poincar e function for the trivial representation on M we have that f1 = f1 M is an Euler-Poincar e function for the trivial representation on M . | Let g1, h1 be Euler-Poincar e functions of the trivial representation on SL2(R). By Lemma 1.3.2 the function f˜1 = g1h1 is an Euler-Poincar e function of the trivial representation on M . We recall that 1 b = , 1 µ ¶ which is central in M and deduce from Lemma 1.3.3 that f (b ) = f (b ) = f˜ (b ) = g ( 1)h ( 1). 1 1 1 1 1 From Lemmas 1.3.3 and 1.3.4 it follows that g (1) = g ( 1) = h (1) = h ( 1) R 1 1 1 1 ∈ and so f1(b ) is positive. ¤ 1. Euler Characteristics and In nitesimal Characters 19
1.5 The unitary dual of KM
Let K = K M = S(O(2) O(2)). We shall need in the proof of later M ∩ results to know the unitary dual KM of KM , which is given in the following proposition. First we de ne the following done dimensional representations of SO(2) and SO(2) SO(2). ε R( ) = eil , for all l Z, l ∈ R( ) ε = ei(l +k ), for all l, k Z, l,k R( ) ∈ µ ¶ where cos sin R( ) = SO(2). sin cos ∈ µ ¶ Note that ε0 and ε0,0 are the trivial representation of their respective groups.
Proposition 1.5.1 For l, k Z not both zero there are unique representa- 2 ∈ tions l,k of KM on C with = ε ε , l,k|SO(2) SO(2) l,k l, k and 1 1 (z , z ) = (z , z ). l,k 1 1 2 2 1 1 We can also de ne the representation of KM on C by (X, Y )(z) = (det X)z = (det Y )z.
We have that K = triv, , : l, k Z not both zero . M { l,k ∈ } \ Proof: By Lemmad 1.3.5, we have that SO(2) = εl : l Z . In general, for two locally compact groups H and K, the map{ ( , )∈ } de nes 7→ an isomorphism Hˆ Kˆ = H\K. Thus the map from SO\(2) SO\(2) to SO(2)\SO(2) given by ( , ) is an isomorphism. Hence we have 1 2 7→ 1 2 that SO(2)\SO(2) is the set ε ε l, k Z = ε l, k Z . { l k| ∈ } { l,k| ∈ } 1. Euler Characteristics and In nitesimal Characters 20
Let 1 1 T = 1 1 and R( ) R( , ) = . R( ) µ ¶ For l, k Z we note that ∈ 2 l,k(T ) l,k(T )(z1, z2) = (z1, z2) = l,k(T )(z1, z2) and
i(l +k ) i(l +k ) l,k(T ) l,k(R( , )) l,k(T )(z1, z2) = (e z1, e z2) = (R( , ))(z , z ) l 1 2 = l(T R( , )T )(z1, z2), so l,k is indeed a representation. We shall show that the representations given in the proposition are in fact the only irreducible unitary representations of KM . Let KM . Then by [39], Theorem 1.12(d) the representation re- stricted to∈SO(2) SO(2) is a direct sum of irreducible representations, that is d = ε ε ε , |SO(2) SO(2) l1,k1 l2,k2 ln,kn for some l , . . . , l , k , . . . , k Z. Let v be an (SO(2) SO(2))- 1 n 1 n ∈ ∈ eigenvector with eigenvalue ei(l1 +k1 ). Then the equation T R( , ) = R( , )T tells us that (T )v is also an SO(2)-eigenvector, with eigenvalue e i(l1 +k1 ). 2 If (T )v is a scalar multiple of v then l1 = k1 = 0 and the equation T = I tells us that = triv or . Otherwise = ε ε and |SO(2) SO(2) l1,k1 l1, k1 = l1,k1 . For l = k = 0 we have = triv . For all other values of l and k 0,0 the representation is irreducible. Also, for l, k Z we have that is l,k ∈ l,k unitarily equivalent to l, k. There are no other equivalences between the representations l,k. The proposition follows. ¤ 1. Euler Characteristics and In nitesimal Characters 21 1.6 In nitesimal characters
Let G be a connected reductive group with maximal compact subgroup K, real Lie algebra gR and complexi ed Lie algebra g. Let be a represen- tation of G. By [39], Theorem 1.12(d), the restriction of to K splits into an orthogonal sum of irreducible representations of K. The irreducible K-representations occuring in this decomposition are called the K types oc- curing in . By [39], Theorem 1.12(e), the K types of occur in the above decomposition with well de nied multiplicity, which is either a non-negative integer or + . We say that is admissible if (K) acts by unitary operators ∞ and if each K type occurs with nite multiplicity in K . In particular is admissible for all Gˆ ([39], Chapter VIII, Theorem| 8.1). We say a vec- tor in the representation∈ space is K- nite if its K-translates span a nite dimensional space. Let K denote the space of K- nite vectors in . For X gR and v de ne ∈ ∈ K (exp tX)v v (X)v = lim . t→0 t By [39], Propositions 3.9 and 8.5 the limit exists and de nes a linear map . Thus we get a representation of gR on , which extends to a K → K K representation of the universal enveloping algebra U(g) of g on K . This representation we also denote by . Let h be a Cartan subalgebra of g. The Weyl group W = W (g, h) is a nite group generated by the root re ections in the root system (g, h). The group W acts on h and hence on the dual space h . Suppose that is an admissible representation of G such that (Z) acts as a scalar on K for all Z in the centre zG of the universal enveloping algebra of g. In particular this condition is satis ed when is irreducible admissible ([39], Chapter VIII, Corollary 8.14). In this situation the representation gives us a character of zG. Via the Harish-Chandra homomorphism (see [39], Chapter VIII, 5,6) the set of characters of zG can be identi ed with the set of Weyl group§§orbits in h . If h corresponds under this identi cation ∈ to the character of zG given by we say that has in nitesimal character and we write = . The in nitesimal character is de ned up to the action of the Weyl group. It follows from the de nition of the in nitesimal character that if is a sub- or quotient representation of then = . Up until now we have assumed that G is connected, however if we merely assume that G is a Lie group and is an irreducible representation of G, then 1. Euler Characteristics and In nitesimal Characters 22
by [39], Corollary 3.12, we have that (Z) commutes with (g) for all Z zG and all g G. Then by Schur’s Lemma ([39], Proposition 1.5) we see ∈that ∈ (Z) is a scalar operator for all Z zG, so we can de ne the in nitesimal character of in this case also. ∈ Let G = SL4(R) and the subgroups K and P = MAN be as above. Let h be the diagonal subalgebra of g. In this case the Weyl group W (g, h) acts on h by interchanging elements of the leading diagonal. We shall see in the next chapter that the analytic properties of the zeta functions considered there are related to the in nitesimal characters of elements of Gˆ. The following proposition gives us information about these in nitesimal characters which will be required in the following chapter. Let P be the half-sum of the positive roots of the system (g, a), where a is the complexi ed Lie algebra of A, so that a a = 4a. P a a Let be a nite dimensional virtual representation of M, whose KM types are all contained in the set triv, , : l, k 0, 2 . Let Mˆ be the subset { l,k ∈ { }} of all Mˆ such that tr (f ) = 0 for all Euler-Poincar e functions f for ∈ on M. (Note that the value of tr (f ) = 0 depends only on and and not on the choice of Euler-Poincar e function f .) Let Gˆ be the set of all elements of Gˆ except for: the trivial representation; representations induced from parabolic subgroups other than P = MAN and representations induced from Mˆ . We de ne an order on the real dual space of a by > if and only if ∈( ) = t for some t > 0. P Proposition 1.6.1 Let be a nite dimensional virtual representation of M and let Gˆ . Then the in nitesimal character of satis es ∈ 1 Re(w ) a or Re(w ) a | 2 P P | for all w W (g, h). ∈ Proof: We shall prove the proposition by considering di erent cases in turn. By Theorem 1.1.2 we have the following cases to consider. The case when is a principal series representation induced from P = MAN; the case when 1. Euler Characteristics and In nitesimal Characters 23 is a complementary series representation induced from P = MAN; the case when is a limit of complementary series representation and the case when is one of the representations for m N. m ∈ First we consider the principal series representations. Let = , = G Ind P ( ) be induced from P , where is an irreducible, unitary represen- tation of M = S(SL2 (R) SL2 (R)) and a is imaginary (ie. iaR, ∈ ∈ where aR is the real dual space of a). 0 0 Let be an irreducible subspace of SL2(R) SL2(R). Then has in nites- | 0 imal character 0 and = 0 . We have that = where and 1 2 1 2 are irreducible unitary representations of SL2(R). To limit the possibilities for 1 and 2 that we need to consider we use the double induction formula (see [39], (7.4)).
Lemma 1.6.2 (Double induction formula) Let M A N be a parabolic subgroup of M, so that M (A A)(N N) is a parabolic subgroup of G. If is a unitary representation of M and a = ∈ (LieCA ) , a , then there is a canonical equivalence of representations ∈ Ind G Ind M ( ) = Ind G ( ( + )) . MAN M A N M (A A)(N N) ¤ ¡ ¢
We may assume that neither 1 nor 2 are induced since then, by the double induction formula, we would have the case that was induced from a parabolic other than P = MAN, which was excluded. By Theorem 1.1.1, the remaining possibilities for 1 and 2 are the trivial representation and the discrete series and limit of discrete series representations.
Let be the in nitesimal character of and i be the in nitesimal character of i, then = 1 + 2 . Recall that h is the diagonal subalgebra of g. We lift and to h by de ning to be zero on a and to be zero on the diagonal elements of m, so that = + ([39], Proposition 8.22). The Weyl group W = W (g, h) acts on . Let w W , we will show that ∈ either tr (f ) = 0 or 1 Re(w a) | 2 P or Re(w a). P | 1. Euler Characteristics and In nitesimal Characters 24
First we take 1 and 2 to be the trivial representation. This gives us that s s = s + t. t t If a a = a, iR, a ∈ a then a b c a b c a b 2 b a 2 a + b = a+b + (1.10) c + 2 2 c a+b 2 a b a + b a + b = + c + + 2 2 2 µ ¶ = a + c + (a + b). 2
If we let w = 1 then from above we see that Re( a) = 0. If we take w to be the transposition interchanging b and c we get | a b w = a + b + (a + c), c 2 a b c 1 the real part of which when restricted to a gives 2 P . All other Weyl group 1 1 elements are dealt with similarly and we see that 2 P Re( a) 2 P for all w W . | ∈ It remains to consider the case when either or both of 1 and 2 are a discrete series representation or limit of discrete series representation with 1. Euler Characteristics and In nitesimal Characters 25
D + D D + D parameter mi. We set 0 = 0 = triv and let i = mi or i = mi , where m 0, and m and m are not both zero. From (1.5) we get i 1 2
dim pM p KM tr (f ) = ( 1)p dim V p V . (1.11) M p=0 X ³ ^ ´
We shall use Proposition 1.5.1 to examine the KM types to limit the possi- bilities for m , m for which tr (f ) = 0. 1 2 6
Lemma 1.6.3 For m1, m2 both non-zero we have the following isomorphism of KM -modules: V = . j,k j,k j m1,j m1mod2 k m2,kM m2mod2
If m1 = 0 we have V = 0,k, k m2,kM m2mod2 and for m2 = 0 we have
V = j,0. j m1,jM m1mod2 Proof: This follows from Lemma 1.3.7. ¤
Lemma 1.6.4 We have the following isomorphisms of KM -modules:
0 pM = triv 1 ^ p = M 2,0 0,2 2 ^ p = M 2,2 2, 2 3 ^ p = M 2,0 0,2 ^4 pM = triv. ^ Proof: KM acts on pM by the adjoint representation and we can compute
p = . M 2,0 0,2 1. Euler Characteristics and In nitesimal Characters 26
The other isomorphisms follow straightforwardly from this. ¤
Let v = v v v V p V , where the v ’s all lie in a single 1 2 3 ∈ M i KM type of their respective representation spaces. Lemma 1.6.4 tells us that V v2 is in one of the following KM types: triv, , 2,0, 0,2, 2,2, 2, 2 by our assumption on , the possibilities for the KM type containing v3 are also the same. It follows that tr (f ) is non-zero only if m1, m2 0, 2, 4 . It follows from the exact sequences in the proof of∈Lemma{ }1.3.7 that D + D P ,(m 1) 1 P m m , where the index on is + if m is even and if ,(m 1) 1 m is odd. The in nitesimal character of P is simply (m 1) 1 (see [39], Proposition 8.22), hence it follows that the in nitesimal characters of D + and D are both equal to (m 1) . This gives us in the cases where m m 1 tr (f ) is non-zero:
s s = (m1 1)s + (m2 1)t, m1, m2 0, 2, 4 . (1.12) t ∈ { } t By computing the action of the di erent Weyl group elements we can see 1 that in all cases either w a or w a . Since = + | 2 P | P and is imaginary we see that Re( a) = a, so the claim follows. The complementary series represen| tations|are dealt with similarly. We recall from Theorem 1.1.2 that the complementary series induced from P are G 1 = Ind P ( m t P ), for m N and 0 < t < 2 . We may argue as above to nd the possibilities for m suc∈h that tr (f ) = 0. In this way we see that m 6 there are two possibilities for m for which tr m(f ) = 0, namely m = 2 and 1 6 m = 4. In the rst case, w a 2 P , for all w W (g, h). In the second case we have | ∈ a b = (3 + 2t)a + 2tb + 3c. c a b c If w W (g, h) is the element which swaps the rst and fourth diagonal ∈ 3 1 entries then w a = 2 P . In all other cases we have w a 2 P . The limit of complemen| tary series representations are closely| related to the family of representations , m N, so we shall deal with them together. m ∈ 1. Euler Characteristics and In nitesimal Characters 27
For m N, we denote by m the representation of M induced from ∈ D + D + the representation m m of SL2(R) SL2(R). For m N we have the limit of complementary series representation given as an irreducible∈ unitary subrepresentation of I = Ind G( 1 ), which we will denote by c . The m P m 2 m representations m are the Langlands quotients of the Im. c Let m be the in nitesimal character of m and m the in nitesimal c 1 c character of m. Clearly m = m = m + 2 P , so we need only consider
m . The value of a b m c a b c is equal to a b a+b 2 2 b a a+b 2 1 2 + P m a+b+2c 2 a+b 2 2 a+b 2c a+b 2 2 a b a + b = (m 1) + (m 1)(c + ) + (a + b) 2 2 = ma + b + (m 1)c. Restricting to a we get a a 1 = 2a = P . m a 2 a The Weyl group W acts on m . Now for all elements w W except one we 1 ∈ get w m a 2 P for all m 1. The exception is w1 which swaps the rst and fourth| diagonal entries. F or this we have a a w1 = 2(m 1)a. m a a 1 Thus w1 m a 2 P if m = 1, 2 and if m 3 then w1 m a P . This completes| the proof of the proposition. | ¤ Chapter 2
Analysis of the Ruelle Zeta Function
Let G = SL (R) and G be discrete and cocompact. Let gR = sl (R) 4 4 and g = sl4(C) be respectively the Lie algebra and complexi ed Lie algebra of G. As in the previous chapter, all Haar measures will be normalised as in [25], 7. Recall that this normalisation depends on the choice of an invariant bilinear§ form b on g. Let b be the following multiple of the trace form on g: b(X, Y ) = 16tr (XY ). (2.1) We choose this normalisation in order to get the rst zero of the Ruelle zeta function at the point s = 1 in Theorem 2.4.3 below. Let K G be the maximal compact subgroup SO(4). Let kR gR be its Lie algebra and let pR gR be the orthogonal space of kR with respect to the form b. Then b is positive de nite and Ad(K)-invariant on pR and thus de nes a G-invariant metric on X = G/K, the symmetric space attached to G. Let a a A = : a > 0 , a 1 a 1 SO(2) B = . SO(2) µ ¶ B is a compact Cartan subgroup of SL (R) M = S 2 SL (R) µ 2 ¶ 28 2. Analysis of the Ruelle Zeta Function 29
det (x), det (y) = 1 = (x, y) Mat2(R) Mat2(R) . ∈ | det (x)det (y) = 1 ½ ¾ We also de ne I Mat (R) I 0 N = 2 2 and N = 2 , 0 I Mat (R) I µ 2 ¶ µ 2 2 ¶ and set KM = K M. Let m denote the∩ complexi ed Lie algebra of M and let m = k p be M M its polar decomposition, where kM is the complexi ed Lie algebra of KM = K M. Let h be the Cartan subalgebra of g consisting of all diagonal matrices,∩ and let a, n and n be the complexi ed Lie algebras of A, N and N respectively. Let P denote the parabolic with Langlands decomposition P = MAN and P the opposite parabolic with Langlands decomposition P = MAN . Let P be the half-sum of the positive roots of the system (g, a), so that a a = 4a. P a a Let 1 1 1 H1 = aR. 8 1 ∈ 1 1 Then it follows that b(H1) = b(H1, H1) = 1 and P (H1) = 2 . Let A = exp(tH ) : t > 0 be the negative Weyl chamber in A. Let E ( ) be the set { 1 } P of all conjugacy classes [ ] in such that is conjugate in G to an element a b of A B. An element is called primitive if for and n N the equation n = implies that∈ n = 1. Clearly primitivity is∈invariant ∈under conjugacy, E p E so we may view it as a property of conjugacy classes. Let P ( ) P ( ) be the subset of primitive classes. 1/2 For [ ] EP ( ) we de ne the length of to be l = b(log a , log a ) . Let G be the∈ centraliser of in G, let = G be the centraliser of ∩ in and let 1( ) be the rst higher Euler characteristic as in the previous chapter. 2. Analysis of the Ruelle Zeta Function 30 2.1 The Selberg trace formula
Let Gˆ be the unitary dual of G. The group G acts on the Hilbert space L2( G) by translations from the right. Since G is compact this represen- tation\ decomposes discretely: \
L2( G) = N ( ) \ M ∈Gˆ with nite multiplicities N ( ), (see [23]). Recall that for we denote by G and the centraliser of in ∈ G and respectively. For a function f on G, denote by (f) the orbital integral O (f) = f(x x 1) dx. O ZG/G The Selberg trace formula (see [59], Theorem 2.1) says that for suitable functions f on G the following identity holds:
N ( )tr (f) = vol( G ) (f), (2.2) \ O X∈Gˆ X[ ] where the sum on the right is over all conjugacy classes in . The set of suitable functions includes, but is not limited to, all dim G + 1 times con- tinuously di erentiable functions of compact support on G. In fact, we shall need to extend the set of test functions for which the trace formula is valid.
Lemma 2.1.1 Let d N, d 16, that is d = 2d0 for some d0 > dim G/2. Let f be a d-times di er∈ entiable function on G such that Df L1(G) for all left invariant di erentiable operators D on G with complex c∈oe cients and of degree d. Then the trace formula is valid for f.
Proof: This is Lemma 1.3 of [17] in the case G = SL4(R). ¤
For g G and V any complex vector space on which g acts linearly let ∈ E(g V ) R be de ned by | + E(g V ) = : is an eigenvalue of g on V . | {| | } Let (g V ) = min(E(g V )) and (g V ) = max(E(g V )). min | | max | | 2. Analysis of the Ruelle Zeta Function 31
For am AM de ne ∈ (a n) (am) = min | , (m g)2 max | where we are considering the adjoint action of G on n and g resepectively. De ne the set (AM) = am AM : (am) < 1 . { ∈ } An element of G is said to be elliptic if it is conjugate to an element of a compact torus. Let Mell denote the set of elliptic elements in M.
Lemma 2.1.2 The set (AM) has the following properties: (1) A M (AM) ; ell (2) am (AM) a A ; (3) am,∈a0m0 (AM⇒) ,∈g G with a0m0 = gamg 1 a = a0, g AM. ∈ ∈ ⇒ ∈ Proof: See [14], Lemma 2.4. ¤
For the construction of our test function we shall need, for given s C and ∈ j N, a smooth, conjugation invariant, j-times continuously di erentiable function∈ on AM, with support in (AM) . Further, we require that at each j+1 sla point ab A B the function takes the value la e . In [14] a function j ∈ gs is constructed with these properties, with the one di erence that there the positive Weyl chamber A+ is used instead of the negative Weyl chamber A . With only very minor modi cation the construction in [14] will yield a j function with our required properties, which we shall also call gs. Let : N R be a smooth, non-negative function of compact support, → which is invariant under conjugation by elements of KM and satis es
(n) dn = 1. ZN Let f : M C be a smooth, compactly supported function, invariant under → conjugation by KM . Suppose further that the orbital integrals of f on M satisfy M (f) = f(xmx 1) dx = 0 Om ZM/Mm whenever m is not conjugate to an element of B, where Mm denotes the centraliser of m in M. The group AM acts on n according to the adjoint representation. 2. Analysis of the Ruelle Zeta Function 32
Given these data we de ne = : G C by f,j,s → gj(am) (knam(kn) 1) = (n)f(m) s , (2.3) det (1 (am) 1 n ) | for k K, n N, am AM. To∈see that∈ is w∈ell de ned we recall that, by the decomposition G = KNAM, every g G which is conjugate to an element of (AM) can be written in the form∈ knam(kn) 1. By the properties of (AM) we see that two of these representations can only di er by an element of KM . The com- ponents of the function are all invariant under conjugation by KM , and we note that det (1 (am) 1 n) = 0 for all am (AM) , so we can see that is well-de ned. | 6 ∈ Proposition 2.1.3 The function is (j 14)-times continuously di eren- tiable. For j and Re(s) large enough it go es into the trace formula and we have: lj+1e sla N ( )tr ( ) = vol( G ) M (f) a . (2.4) b 1 n E \ O det (1 (a b ) ) X∈Gˆ [ ]∈XP ( ) | Proof: Noting that 2 dim n+dim k = 14 we see that this follows from Propo- sition 2.5 of [14]. This proposition was proved in the case that the function f is an Euler-Poincar e function for some nite dimensional representation of M. However the only properties of Euler-Poincar e functions used in the proof are those given above for f, namely that it is smooth, of compact sup- port, invariant under conjugation by K, and the orbital integrals satisfy the given condition. ¤
2.2 The Selberg zeta function
An element g G is said to be weakly neat if the adjoint Ad(g) has no non-trivial roots∈ of unity as eigenvalues. A subgroup H of G is said to be E p weakly neat if every element of H is weakly neat. Let [ ] P ( ) so that ∈ is conjugate in G to a b A B. We want to know which roots of unity can occur as eigenvalues of Ad(∈ ). These are the same as the roots of unity which occur as eigenvalues of Ad(a b ). If R( ) b = , R( ) µ ¶ 2. Analysis of the Ruelle Zeta Function 33 where cos sin R( ) = , sin cos µ ¶ 2i 2i then the eigenvalues of Ad(a b ) are e and e . De ne the sets
R = n N : n Z and R = n N : n Z { ∈ ∈ } { ∈ ∈ } and R = min R , min R . { } Then R contains either 0, 1 or 2 elements. We can see that weakly neat E n is equivalent to R = ∅. For P ( ), where = 0 for 0 primitive, the value of ( ) depends on whether∈ or not n R . 1 ∈ 0 For I R with I = ∅ de ne n to be the least common multiple of the 6 I elements of I and set n∅ = 1. Further, de ne
|I| ( 1) |J| I ( ) = ( 1) 1( nJ ). nI J I X Let z C r 0 and q Q. We de ne zq to be equal to eq log z, where we take the branc∈ h {of}the logarithm∈ with imaginary part in the interval ( , ]. For any nite-dimensional virtual representation of M we de ne, for Re(s) 0, the generalised Selberg zeta function ( ) snI l nI n I ZP, (s) = det 1 e V S (n) , (2.5) p | [ ]∈E ( ) n 0 I R YP Y Y ¡ ¢ where Sn(n) denotes the nth symmetric power of the space n and acts on V Sn(n) via (b ) Adn(a b ). In the case that is weakly neat this simpli es to
( ) Z (s) = det 1 e sl V Sn(n) 1 , P, | [ ]∈E p ( ) n 0 YP Y ¡ ¢ (see [14]). Let Gˆ. We recall that a vector in the representation space is said ∈ to be K- nite if its K-translates span a nite dimensional space and let K denote the space of K- nite vectors in . We say that a real or complex 2. Analysis of the Ruelle Zeta Function 34 vector space V is a (g,K)-module if it is a g-module and a locally nite and semi-simple K-module. Further, the actions of g and K must satisfy, k X v = Adk(X) k v, for v V, k K and X U(g), where U(g) is the universal enveloping algebra∈ of g, and∈ we must ha∈ve that the action of K is di erentiable on every K-stable nite dimensional subspace of V and has k as its di erential. | The space K is a (g, K)-module (see [4], I, 2.2). The Lie algebra n acts on K and we denote by H (n, K ) (resp. H (n, K )) the corresponding Lie algebra cohomology (resp. homology) (see [4],[8]). We have the following isomorphism of AM-modules (see [26], p57): 4 H (n, ) = H4 p(n, ) n. (2.6) p K K For a let ^ ∈ p KM m ( ) = ( 1)p+q dim Hq(n, ) p V , (2.7) K M p,q 0 X ³ ^ ´ where for an a-module V and a , V denotes the generalised -eigenspace ∈ v V n N such that (a (a))nv = 0 a a and the superscript K { ∈ | ∃ ∈ ∀ ∈ } M denotes the subspace of KM invariants. We say that an admissible representation of a linear connected reduc- 0 G0 tive group G has a global character = if for all smooth, compactly 0 G0 supported functions f on G the operator (f) is of trace class and is a locally L2 function on G0 satisfying
G0 tr (f) = (g)f(g) dg 0 ZG for each such f. By [39], Theorem 10.2, every irreducible admissible, and in particular every irreducible unitary, representation of G0 has a global char- acter. Theorem 2.2.1 Let be a nite dimensional virtual representation of M. The function ZP, (s) extends to a meromorphic function on the whole of C. For a , the vanishing order of Z (s) at the point s = (H ) is ∈ P, 1
N ( )m ( ). (2.8) X∈Gˆ Further, all the poles and zeros of Z (s) lie in R ( 1 + iR). P, ∪ 2 2. Analysis of the Ruelle Zeta Function 35
Proof: The analogue of this theorem in the case that G has trivial centre and is weakly neat is proved in [14], Theorem 2.1. If G has trivial centre then weakly neat implies torsion free, since non weakly neat torsion elements must be central. With a few mod cations the proof of [14], Theorem 2.1 becomes valid in our case also. In fact the assumption that is weakly neat is used in two places. We sketch the proof here, pointing out the necessary modi cations for it to be valid in our case. The theorem is proved by setting f = f in the test function = j,s = ,j,s de ned in (2.3), where f is an Euler-Poincar e function for on M. We then get that for j N and Re(s) su ciently large, the right hand side of (2.4) is equal to ∈
lj+1e sl vol( G ) M (f ) . (2.9) b 1 n E \ O det (1 (a b ) ) [ ]∈XP ( ) | In [14] the following equation from [11], proved under the assumption that is torsion free, is used:
C vol( G ) 1( ) = \ , (2.10) l 0 where C is an explicit constant depending only on , and 0 denotes the primitive geodesic underlying . We have shown in Lemma 1.4.1 that (2.10) holds for all EP ( ) in our case also. From [14], Proposition 1.4 we take the equation ∈ M (f ) = C tr (b ), Ob which, together with (2.10) gives us
M (f )vol( G ) = l tr (b ) ( ). Ob \ 0 1 Substituting this into (2.9) we get
lj+1e sl l tr (b ) ( ) . 0 1 1 n E det (1 (a b ) ) [ ]∈XP ( ) | We claim that this is equal to
∂ j+2 ( 1)j+1 log Z (s). ∂s P, µ ¶ 2. Analysis of the Ruelle Zeta Function 36
Indeed ∞ sml e m m n log ZP, (s) = I ( ) tr (b ) tr ((a b ) S (n)) . p m | [ ]∈E ( ) m=1 I R n 0 XP X X X We note that tr ((a b )m Sn(n)) = det (1 (a b )m n) 1 | | n 0 X 1 = det 1 (a b ) m n | and the claimed equality follows. ¡ ¢ In [14], since is weakly neat it follows that G n = G for all and ∈ n N. In [20] it is shown that X = G /K and so it follows that for ∈ \ all n N we have X n = X and hence ( n ) = ( ). Thus the above ∈ 1 1 equality involving the logarithmic derivative of ZP, (s) is derived with the simpler Euler product expansion for ZP, (s) given above. In our case we may have an element [ ] E p( ) with a non-trivial root ∈ P of unity as an eigenvalue. For such a we have 1( n ) = 1( ) for n R . For this reason we have had to introduce the more complicated6 Euler pro∈duct expansion for ZP, (s) so that the above equality involving the logarithmic derivative of ZP, (s) still holds. On page 909 of [14] it is shown that
MA j tr ( s) = f (m) H (n, K )(ma)dm gs(a)da. (2.11) ZMA Using the property (1.5) of f we get
KM j+1 sla tr ( s) = tr a H (n, K ) pM V la e da ZA µ ¯ ¶ ∞ ¯³ ^ ´ ¯ ( (H1) s)t j+1 = m¯ ( ) e t dt 0 a Z X∈ ∂ j+1 1 = ( 1)j+1 m ( ) . (2.12) ∂s s (H ) a 1 µ ¶ X∈ Proposition 2.1.3 then gives us that ∂ j+2 ∂ j+1 1 log Z (s) = N ( ) m ( ) , (2.13) ∂s P, ∂s s (H ) a 1 µ ¶ X∈Gˆ µ ¶ X∈ 2. Analysis of the Ruelle Zeta Function 37
from which it follows that the vanishing-order of ZP, (s) at s = (H1) is
N ( )m ( ). X∈Gˆ Two further comments are in order. First, in [14] the positive Weyl + chamber A = exp(tH1) : t < 0 in A is considered, where we have instead considered the negativ{ e chamber}A . For this reason we have interchanged the positions of the Lie algebras n and n from the way they are used in [14]. This is easily seen to give a precisely equivalent result. Secondly, the results of [14] are stated for a nite dimensional represen- tation of M. By linearity the results extend in a straightforward way to the case where is a virtual representation, which we use here. ¤
2.3 A functional equation for ZP, (s) Proposition 2.3.1 For a let be the norm given by the form b in ∈ k k (2.1). There are m1 N and C > 0 such that for every Gˆ and every a we have ∈ ∈ ∈ 4 q q m1 ( 1) dim(H (n, K ) ) C(1 + ) . ¯ ¯ k k ¯ q=0 ¯ ¯X ¯ ¯ ¯ Further, let S denote¯ the setof all pairs ( ,¯ ) Gˆ a such that ∈ 4 ( 1)q dim Hq(n, ) = 0. K 6 q=0 X ¡ ¢ Then there is m N such that 2 ∈ N ( ) < . (1 + )m2 ∞ ( X, )∈S k k Proof: These results follow from [16], Proposition 2.4 and Lemma 2.7. ¤
An entire function f is said to be of nite order if there is a positive constant a and an r > 0 such that f(z) < exp ( z a) for z > r. If f is of nite order then the order of f is de ned| | to be the| |in mum| |of such a’s. 2. Analysis of the Ruelle Zeta Function 38
It is well known that the classical Selberg zeta function is an entire func- tion of order two (see [52]). Our next lemma gives an analogous result for the generalised Selberg zeta function we are considering here. Let be a nite dimensional virtual representation of M. Theorem 2.2.1 tells us that ZP, (s) is meromorphic and hence it may be written as the quotient of two entire functions:
Z1(s) ZP, (s) = , Z2(s) where the zeros of Z1(s) correspond to the zeros of ZP, (S) and the zeros of Z2(s) correspond to the poles of ZP, (s). The orders of the zeros of Z1(s) (resp. Z2(s)) equal the orders of the corresponding zeros (resp. poles) of ZP, (s). The functions Z1(s) and Z2(s) are not uniquely determined, but clearly their respective sets of zeros, together with the orders of the zeros, are. For i = 1, 2 let Ri denote the set of zeros of Zi(s) counted with multiplicity.
Lemma 2.3.2 There exist Zi(s), for i = 1, 2, with the above properties, which are, in addition, of nite order.
Proof: Let Z1(s), Z2(s) be as above. We shall show that we may take them to be of nite order. For Gˆ let ( ) be the set of all a with = 0 and m ( ) = 0. ∈ ∈ 6 6 Let Gˆ( ) be the set of Gˆ such that N ( ) = 0 and let S denote the set of all pairs ( , ) such that∈ Gˆ( ) and 6 ( ). For a let be the norm given by the form b in∈ (2.1). ∈ ∈ k k The expression (2.8) tells us that s = 0 is a zero or pole of ZP (s) if and 6 only if s = (H1) for some a , for which there exists Gˆ( ) such that ( , ) S. ∈ ∈ ∈ Since the function ZP, (s) is meromorphic and non-zero, it follows that there exists c > 0 such that
(H ) c (1 + ) (2.14) | 1 | k k for all such that ( , ) S for some . ∈ By the de nition of m ( ) we deduce immediately from Proposition 2.3.1 that there exist m1 N and C > 0 such that for every Gˆ and every a we have ∈ ∈ ∈ m ( ) C (1 + )m1 , (2.15) | | k k 2. Analysis of the Ruelle Zeta Function 39 and that there exists m N such that 2 ∈ N ( ) < . (2.16) (1 + )m2 ∞ ( X, )∈S k k Let k = m + m . Then, for ( , ) S, by (2.14) and (2.15) we have, 1 2 ∈ m ( ) 1 m ( ) k k | | k (H1) c (1 + ) ¯ ¯ k k ¯ ¯ C 1 ¯ ¯ . ¯ ¯ ck (1 + )m2 k k It then follows from (2.8) and (2.16) that, for i = 1, 2,
1 N ( ) m ( ) k | k | < . (2.17) (H1) ∞ r ∈RXi {0} | | ( X, )∈S | |
We say that Z1(s) and Z2(s) are of nite rank. By the Weierstrass Factorisation Theorem ([10], VII.5.14) and (2.17), there exist entire functions g1(s), g2(s) such that s s s2 sk Z (s) = sni egi(s) 1 exp + + , (2.18) i 2 2 k k r ∈RYi {0} µ ¶ µ ¶ where ni is the order of the zero of Zi(s) at s = 0. From (2.13) we recall the equation
j j 1 d d m ( ) log ZP, (s) = N ( ) , (2.19) ds ds s (H1) µ ¶ ∈XGˆ( ) µ ¶ ∈ (X )∪{0} where j N is su ciently large. Let J = max(j, k). It follows from (2.18) and (2.19)∈ that
d J 1 ( 1)J 1(J 1)! ( 1)J 1(J 1)! (g0 (s) g0 (s)) + ds 1 2 (s )J (s )J ∈R ∈R µ ¶ X1 X2 is equal to ( 1)J 1(J 1)! N ( )m ( ) J . (s (H1)) ∈XGˆ( ) ∈ (X )∪{0} 2. Analysis of the Ruelle Zeta Function 40
Remembering that the zeros of Zi(s) are included with multiplicity in Ri(s) and bearing in mind the expression (2.8) for the vanishing order of ZP, (s), we see that this implies that
d J 1 (g0 (s) g0 (s)) = 0. ds 1 2 µ ¶ It follows that g (s) g (s) is a polynomial of degree at most J. Without loss 1 2 of generality we may take g2(s) to be zero, so that g1(s) is itself a polynomial of degree at most J. Finally, Theorem XI.2.6 of [10] tells us that since Z1(s) and Z2(s) are of nite rank and g1(s) and g2(s) are both polynomials of degree at most J, it follows that Z1(s) and Z2(s) are both of order at most J. ¤
Before we give the next lemma we make a couple of de nitions. Let G0 be a linear connected reductive group with maximal compact subgroup K 0 and let be a representation of G0. Let g0 be the complexi ed Lie algebra of G0, let h0 g0 be the diagonal subalgebra and let 0 (h0) be the half sum of the positiv e roots of the system (g0, h0). We say that∈ is tempered if for all 0 0 K - nite vectors u, v 0 there exist constants c such that for all g G ∈ K u,v ∈
0(H(g 1k)) (g)u, v cu,v e dk, |h i| 0 ZK where H(g 1k) denotes the logarithm of the split part of g 1k under the Iwasawa decomposition and , is the inner product on V . An admissible representation of G0 is calledh standar i d if it is induced from an irreducible tempered representation of M 0 G0, where P 0 = M 0A0N 0 is a parabolic subgroup of G0. The Weyl group W (G, A) has two elements, let w be the nontrivial ele- ment therein. Then w acts on M by conjugation and for a representation of M we can de ne the representation w by w (m) = (wmw 1).
w Lemma 2.3.3 Suppose that = as KM -modules. Then ZP, (s) and Z (1 s) have the same poles and zeros with multiplicity. P, Proof: By Theorem 2.2.1 a , the vanishing order of Z (s) at the point ∈ P, s = (H1) is equal to ∈Gˆ N ( )m ( ). We shall show that P m ( ) = m 2 P ( ) (2.20) 2. Analysis of the Ruelle Zeta Function 41
for all Gˆ. Since 2 P (H1) = 1 the lemma will follow. Let ∈ Gˆ. Recall from (2.7) that ∈ p KM m ( ) = ( 1)p+q dim Hq(n, ) p V . K M p,q 0 X ³ ^ ´ G The global character of on G can be written as a linear combination with integer coe cients of characters of standard representations ([39], Chapter G X, 10.2). We are interested in the values taken by on MA . By [39], Prop§ osition 10.19, the only characters which are non-zero on MA are those characters of representations induced from P = MAN. So there exist n N ∈ and for all 1 i n integers ci, tempered representations i of M and i a such that ∈ n G G = ci i , (2.21) i=1 X i G where = Ind P ( i i). From [26], Theorem 3.6 and (2.6) it follows that for all regular ma MA we have ∈ 4 G MA det a n (ma) = (ma) | , H (n, K ) det (1 ma n) ¡ V |¢ and the same holds for all i. Together with (2.21) this implies that
n MA MA H (n, )(ma) = i (ma) K H (n, K ) i=1 X for all regular ma MA . Substituting this into (2.11) and proceeding as in (2.12) we see that∈ it su ces to show (2.20) for the representations i. The Weyl group element w acts on the group MA by conjugation, which has the e ect of swapping the two components. For a we therefore have ∈ w = . Recall that for a representation of M we let w denote the rep- resentation w (m) = (wmw 1). By [39], Theorem 8.38, the representations i w i G w and = Ind P ( i ( i)) are equivalent. From the de nition of the induced group action we can see that
q i q w i 2 P H (n, K ) = H (n, K ) . Thus we see that
p KM dim Hq(n, i ) p V K M ³ ^ ´ 2. Analysis of the Ruelle Zeta Function 42 is equal to p KM dim Hq(n,w i ) 2 P p V . K M Using the notation³ of Proposition 1.5.1 ^we note that´the KM -types satisfy wtriv = triv , wdet = det and w = . Lemma 1.6.4 KM KM KM KM l,m m,l = m, l and the isomorphism of KM -modules n = 2,2 2, 2 p tell us that the KM -modules pM and n are invariant under the action of w w 2 and we have assumed that = as KM -modules. Since w is the identity element of W (G, A) we can concludeV that
p KM dim Hq(n, i ) p V K M ³ ´ is equal to ^ p KM dim Hq(n, i ) 2 P p V . K M i ³ i ^ ´ ¤ Hence m ( ) = m 2 P ( ) and the lemma follows.
w Theorem 2.3.4 Suppose that = as KM -modules. Then there exists a polynomial G(s) such that ZP, (s) satis es the functional equation Z (s) = eG(s)Z (1 s). P, P,
Proof: Let Ri, ni be as above for i = 1, 2. We saw in the proof of Lemma 2.3.2 that there exist entire functions Z1(s) and Z2(s) of nite order such that Z1(s) ZP, (s) = Z2(s) and polynomials g1(s) and g2(s) such that s s s2 sk Z (s) = sni egi(s) 1 exp + + . i 2 2 k k r ∈RYi {0} µ ¶ µ ¶ Lemma 2.3.3 tells us that Z (s) and Z (1 s) have the same poles and ze- P, P, ros. Hence we can in the same way conclude that there exist entire functions W1(s) and W2(s) of nite order such that
W1(s) ZP, (1 s) = W2(s) 2. Analysis of the Ruelle Zeta Function 43
and polynomials h1(s) and h2(s) such that
s s s2 sk W (s) = sni ehi(s) 1 exp + + . i 2 2 k k r ∈RYi {0} µ ¶ µ ¶ Setting G(s) = g (s) + h (s) g (s) h (s) 1 2 2 1 we can see that the claimed functional equation does indeed hold. ¤
2.4 The Ruelle zeta function
For any nite-dimensional virtual representation of M we de ne, for Re(s) 0, the generalised Ruelle zeta function
( ) snI l nI I R , (s) = det 1 e V . p | [ ]∈E ( ) I R YP Y ¡ ¢ We have the following theorem giving a relationship between the generalised Selberg zeta function and the generalised Ruelle zeta function.
Theorem 2.4.1 Let be a nite dimensional virtual representation of M. The function R , (s) extends to a meromorphic function on C. More pre- cisely 4 q ( 1)q R (s) = Z q s + . , P,( V n ) 4 q=0 Y ³ ´ E ( ) Proof: Let P ( ) let ( ) N be the least such that = 0 for some E p( ). W∈e compute at rst∈ 0 ∈ P
snI l nI log R , (s) = I ( )tr log 1 e V p | [ ]∈E ( ) I R XP X ∞ ¡ ¢ 1 smnI l nI = I ( ) e tr (b ) p m [ ]∈E ( ) I R m=1 XP X X e snI l = ( ) tr (bnI ). I ( ) E I R [ ]∈XP ( ) X 2. Analysis of the Ruelle Zeta Function 44
Similarly, for = q n we have q V ∞ snI l nI n log ZP, q (s) = I ( ) tr log (1 e V q S (n) p | [ ]∈E ( ) I R n=0 XP X X ∞ ∞ ¡ ¢ 1 smnI l mnI n = I ( ) e tr V q S (n) p m | [ ]∈E ( ) I R n=0 m=1 XP X X X ¡ q ¢ e snI l tr (bnI n ) nI = I ( ) tr (b ) | nI n E ( ) det (1 (a b ) ) [ ]∈ P ( ) I R V | X X q snI l nI e tr (b n ) nI | = I ( ) tr (b ) n . ( ) tr ((a b ) I n ) [ ]∈E ( ) I R V XP X | Since n is an M-module de ned over the reals we conclude that Vthe trace 1 tr ((a b ) n ) is a real number. Therefore it equals its complex conju- | 1 gate, which is tr a b n . Summing over q we get V | ¡ V ¢ 4 q log R (s) = ( 1)q log Z s + . , P, q 4 q=0 X ³ ´ The theorem follows. ¤
We shall be interested in the zeta function RP, (s) in the case where = triv = 1 is the trivial representation of M and in the case where is the following virtual representation of M. De ne
0 2 4 ˜+ = 15 m + 6 m + m and ^ ^ ^ 1 3 ˜ = 10 m + 3 m and let ˜ = ˜+ ˜ , where M acts^on n m^according to Adn. The reason for this choice of ˜ is that it allows us to separate the contribution of the V non-regular elements of EP ( ) from the regular ones, as made clear in the following lemma. Lemma 2.4.2 Let E ( ) with ∈ P R( ) b = , R( ) µ ¶ 2. Analysis of the Ruelle Zeta Function 45 where cos sin R( ) = . sin cos µ ¶ Then tr ˜(b ) = det (1 b : m/b) = 4(1 cos 2 )(1 cos 2 ), where b is the complexi ed Lie algebra of the group B. In particular we have tr ˜(b ) 0 for all EP ( ), and tr ˜(b ) = 0 if and only if is non-regular. ∈
Proof: As B-modules, we have the isomorphism m = b (m/b). The group B is abelian and so acts trivially on the 2-dimensional Lie algebra b. Hence, for 0 n 6, we have the following isomorphism of B-modules n 2 q m = m/b. (2.22) p p+q=n ^ M µ ¶ ^
We set a0 = 1, a1 = 3, a2 = 6, a3 = 10, a4 = 15 and note that, for k = 0, ..., 4, these satisfy
k 2 a = ( 1)k. (2.23) k m m m=0 X µ ¶ The B-module isomorphism
4 n (m/b) = a4 n m = ˜ n=0 ^ X ^ then follows from (2.22) and (2.23). We can then see that
tr ˜(b ) = tr b : m/b = det (1 b : m/b). ³ ^ ´ The adjoint action of B on m/b can easily be computed to give the claimed value. Finally we note that tr ˜(b ) = 0 if and only if , Z and recall from the proof of Lemma 4.3.1 that this occurs precisely ∈when is non- regular. ¤
The main result of this chapter is the following theorem. 2. Analysis of the Ruelle Zeta Function 46
Theorem 2.4.3 The function R ,1(s) has a double zero at s = 1. The function R , ˜(s) has a zero of order eight at s = 1. Apart from that, for 1, ˜ , all poles and zeros of R (s) are contained in the union of the interval∈ { } , 1, 3 with the ve vertical lines given by k + iR for k = 2, 1, 0, 1, 2. 4 4 £ The¤ theorem will follow from the following proposition, which we prove in the remainder of the chapter.
Proposition 2.4.4 The function ZP,1(s) has a double zero at s = 1. The function ZP, ˜(s) has a zero of order eight at s = 1. Apart from that, for 1, ˜ , all poles and zeros of ZP, (s) lie in the half-plane Re(s) 3 .∈ F{urther,} for being the representation of M on q n (q {1, . . . , 4 ) 4 } q ∈ { } obtained from the adjoint representation and q, q ˜ , all poles and zeros of Z (s) lie in the half-plane Re(s) 1∈. { V } P, { } Let us assume for a moment that the proposition has been proved. The representations of M considered in the proposition all satisfy the isomor- w phism of KM -modules = , where w is the non-trivial element of the Weyl group W (G, A). Hence we can apply the functional equation given in Theorem 2.3.4 to see that the poles and zeros of the functions ZP, (s) all lie in the region 0 Re(s) 1. We can then apply Theorem 2.2.1 to see that 1 the poles and zeros in fact lie in [0, 1] ( 2 + iR). Finally, an application of Theorem 2.4.1 completes the proof of Theorem∪ 2.4.3. The proof of the proposition will take up the rest of the chapter. We see from (2.8) that a representation Gˆ makes a contribution to the vanishing order of Z (s) only if m ( ) = 0∈for some a . From (2.12) it follows P, 6 ∈ that, if tr ( s, ) = 0 for some Gˆ and nite dimensional representation ∈ of M, then m ( ) = 0 for all a . Thus, if we can show that tr ( s, ) = 0 for some and , then we will∈ know that makes no contribution to the vanishing order of ZP, . We de ne a partial order on the real dual space aR by setting > if 1 and only if = t P for some t > 0. Since P (H1) = 2 , we shall be able to prove Prop osition 2.4.4 with the following steps. Firstly , irrespective of the choice of , all eigenvalues of a on H (n, K ) satisfy
Re( ) 2 , (2.24) P with equality only if = triv. Secondly, in the case 1, ˜ , if is the ∈ { } trivial representation on G, then H 4(n, ) 2 P = 0. If = 1 then this gives K 6 2. Analysis of the Ruelle Zeta Function 47 a double zero at the point s = 1 and if = ˜ this gives a zero of order eight at s = 1. Apart from this, for all Gˆ and all a , we have either tr ( ) = 0, or that H (n, ) = 0 implies∈ ∈ K 6 3 Re( ) . (2.25) 2 P The rst step will be proven in the following two sections using the Hochschild-Serre spectral sequence. Then the second step will be proven us- ing a result of Hecht and Schmid which relates the action of a on H (n, K ) to the in nitesimal character of .
2.5 Spectral sequences
In what follows we shall need to use the notion of a spectral sequence. We give here a brief de nition, for more details see [41], Chapter XX, 9. By a complex we mean a sequence (Ki, di), for i 0, of ob§jects and homomorphisms in a given abelian category satisfying
d0 d1 d2 K0 / K1 / K2 / and such that di+1 di = 0. We say that a complex L = (Li, ei) is a subcomplex i i i i i i of K = (K , d ), and we write L K, if L K and e = d Li for all i 1. For a complex K = (Ki, di), a ltr ation F K of K is a decreasing| sequence of subcomplexes
K = F 0K F 1K F 2K F nK F n+1K = 0 . { } The quotient of two complexes can be formed in the obvious way. For a complex K = (Ki, di) we de ne the i-th cohomology group of K as Hi(K) = ker di/Im di 1. The associated graded object
H(K) = Hi(K) i 0 M is called the cohomology of K. To a ltered complex F K is associated the graded complex
Gr F K = Gr K = Gr iK = F iK/F i+1K. i 0 i 0 M M 2. Analysis of the Ruelle Zeta Function 48
A ltration F iK on K induces a ltration F iH(K) on the cohomology by
j i j ker(d F iK ) F H (K) = j |1 . Im (d i ) |F K The associated graded object is F iHj(K) Gr H(K) = Gr iHj(K) = . F i+1Hj(K) i,j i,j M M A spectral sequence is a sequence (E , d ), for r 0, of bigraded objects r r p,q Er = Er p,q 0 M together with homomorphisms (known as di erentials) d : Ep,q Ep+r,q r+1 satisfying d2 = 0, r r → r r and such that the cohomology of Er is Er+1, that is
H(Er) = ker dr/Im dr = Er+1. (2.26) If we have E = E = for some r , then this limit object is called E r0 r0+1 0 ∞ and we say that the spectral sequence abuts to E∞. If there exist m, n N p,q ∈ such that for all r we have Er = 0 only if p m and q n, then it m,n 6 m,n m,n follows from (2.26) that E2 = E3 = = E∞ . In [41], Chapter XX, Proposition 9.3, it is shown that given a ltered complex F K one can derive a spectral sequence (Er), which in particular has
p,q p p+q E∞ = Gr (H (K)).
2.6 The Hochschild-Serre spectral sequence
We shall need the following proposition. Proposition 2.6.1 Let P 0 = M 0A0N 0 be a parabolic subgroup of G and let a0,n0 be the complexi ed Lie algebras of A0,N 0 respectively. De ne a partial 0 0 order >a0 on the dual space a of a by > if and only if is a non-zero linear combination with positive integral coe cients of positive roots of (g, a). 0 p Let a and 0 p < dim n be such that H (n, K ) = 0. Then there ∈ 0 dim n 6 exists a with >a0 such that H (n, ) = 0. ∈ K 6 2. Analysis of the Ruelle Zeta Function 49
Proof: This follows from [26], Proposition 2.32 and the isomorphism (2.6). ¤
Let M G be the subgroup of diagonal matrices with each diagonal 0 entry equal to 1, let A0 G be the subgroup of diagonal matrices with positive entries and let N G be the subgroup of upper triangular matrices 0 with ones on the diagonal. Then P0 = M0A0N0 is the minimal parabolic subgroup of G. Let a0 and n0 be the complexi ed Lie algebras of A0 and N0 respectively, and let a be the dual of a . Let a be de ned as follows: 0 0 0 ∈ 0 a b = 3a + 2b + c. 0 c d Then 0 is the half-sum of the positive roots of the system (g, a0). Let a0,R be the real Lie algebra of A0, which we may consider as a subalgebra of a0. Let a be the set of all X a R such that (X) < 0 for all positive roots 0,R ∈ 0, of the system (g, a ). Let a ,+ be the set of all a such that (X) < 0 0 0,R ∈ 0,R for all X a . Then for all a ,+ we have = , where the sum ∈ 0,R ∈ 0,R is over the positive roots of (g, a0) and every > 0. P For Gˆ a matrix-coe cient of is any function G C of the form ∈ → f (g) = (g)u, v u,v h i for some u, v . ∈ ˆ Lemma 2.6.2 Let G and let a0 be such that H (n0, K ) = 0. Then ,+ ∈ ∈ 6 2 + a , except in the case = triv, when H 6(n , ) 2 0 = 0 and ∈ 0 0,R 0 K 6 other than this H (n , triv) = 0 implies 2 + a ,+. 0 6 ∈ 0 0,R Proof: According to [26], Theorem 4.16, there exists a countable set E ( ) a1 and a collection of polynomial functions pu,v indexed by u, v K and E ( ) such that if f , for u, v , is a matrix coe cient of ∈ then: ∈ u,v ∈ K
( + 0)(X) fu,v(exp X) = pu,v(X)e , (2.27) E ∈X( ) E for all X a0,R. We assume that ( ) is minimal and hence that each ∈ polynomial pu,v is non-zero. 2. Analysis of the Ruelle Zeta Function 50
If = triv then every matrix coe cient of is a constant function so in E 0 (2.27) we have ( ) = 0 and pu,v = Cu,v, where Cu,v is a constant for all u, v . { } ∈ If Gˆ r triv then is in nite dimensional so by [32], Theorem 5.1 all the matrix∈ co{ e cien} ts of vanish at in nity. Hence, for all E ( ) and ∈ E for all X a0,R, we have ( + 0)(X) < 0. In other words, for all ( ) ∈ ,+ ∈ we have 0 + a0,R . ∈ We have assumed that H (n0, K ) = 0. Let >a0 be the partial order 6 de ned on a by >a if and only if is a linear combination with 0 0 positive integral coe cients of positive roots of (g, a0). By Proposition 2.6.1 6 there exists a0 such that H (n0, K ) = 0 and a0 . ∈ 6 6 min Let = + 0 a0 : H (n0, K ) = 0 and let be the set of { ∈ 6 } E min elements of which are minimal with respect to >a0 . Let ( ) be the set E of ( ) which are minimal with respect to >a0 . Theorem 4.25 of [26] says that∈ min = E ( )min. If = triv then min = and the claims of the proposition follow { 0} from the de nition of and [26], Proposition 2.32 as quoted above. If Gˆ r triv then min = E ( )min implies that there exists E ( ) ∈ { } ,+ ∈ such that a0 0. We saw above that 0 + a0,R so the claim follows. ∈ ¤
Proposition 2.6.3 Let Gˆ and let a be such that H (n, K ) = 0. Then Re( ) 2 , with ∈equality only if∈ = triv. 6 P Proof: Let n = n m and a = a m. Then n = n n and M 0 ∩ M 0 ∩ 0 M a0 = a aM . In [29, 30] a ltered complex is constructed so that the spectral sequence derived from it has
p,q p q E2 = H (nM , H (n, K )) and p,q p p+q n E∞ = Gr H ( 0, K ), p+q where H (n0, K ) is appropriately ltered. This is the so-called Hochschild- Serre spectral sequence. By Proposition 2.6.1 it su ces to prove the proposition in the case
H4(n, ) = 0. K 6 2. Analysis of the Ruelle Zeta Function 51
In this case it then follows that
H2(n , H4(n, ) ) = 0 M K 6 and so there exists a such that M ∈ M H2(n , H4(n, ) ) M = 0. M K 6
Since A acts trivially on nM , this equals
2 4 + M 2,4 + M H (nM , H (n, K )) = (E2 ) ,
where we consider + M as an element of a0 = a aM . For all r we p,q have that Er = 0 only if 0 p 2 and 0 q 4, so it follows that 2,4 2,4 6 E2 = E∞ . Since the action of A0 commutes with the di erentials of the spectral sequence it follows that
(E2,4) + M = 0 ∞ 6 and hence that H6(n , ) + M = 0. 0 K 6 The proposition then follows from Lemma 2.6.2 by projection of + M onto the a component. ¤
2.7 Contribution of the trivial representation
For a KM -module let 2 denote the module . We shall need the following:
Lemma 2.7.1 We have the following isomorphisms of KM -modules:
0 pM = triv 1 ^ p = M 2,0 0,2 2 ^ p = 2 M 2,2 2, 2 3 ^ p = M 2,0 0,2 ^4 pM = triv. ^ 2. Analysis of the Ruelle Zeta Function 52 and 0 m = triv 1 ^ m = 2 2,0 0,2 2 ^ m = triv 2 2( ) 2,0 0,2 2,2 2, 2 3 ^ m = 2triv 2triv 2( ) 2,0 0,2 2,2 2, 2 4 ^ m = triv 2 2( ) 2,0 0,2 2,2 2, 2 Proof: The^isomorphisms for pM were given in Lemma 1.6.4. For m note that KM acts on m by the adjoint representation and we can compute m = 2 . 2,0 0,2 The other isomorphisms follow straightforwardly from this. ¤ Proposition 2.7.2 Let be the trivial representation on G. For the trivial representation on M, the representation contributes a double zero of ZP, (s) at the point s = 1. For = ˜ the representation contributes a zero of ZP, (s) of order eight at the point s = 1. In both cases, all other poles and zeros contributed by are in Re(s) 3 . 4 Proof: The space H0(n, K ) = K /n K is one ©dimensionalªwith trivial a- 1 action. The action of a on n is given by 2 P , so the isomorphism (2.6) tells 4 us that a acts on the one dimensional space H (n, K ) according to 2 P . q Proposition 2.6.1 tells us that for q = 0, 1, 2, 3 and a , H (n, K ) = 0 implies 3 . Since 2 (H ) = 1 this gives a ∈pole or zero at s =6 1 2 P P 1 and evaluation of the relevant characters at H1 also gives the other poles and zeros contributed by in Re(s) 3 . { 4 } It remains to compute the vanishing order of ZP, (s) for 1, ˜ at the 4 2 P ∈ { } point s = 1. Since dim H (n, K ) = 1, we get from (2.8) the following expression for the vanishing order:
p KM N ( ) ( 1)p dim p V . M p 0 X ³^ ´ For = 1 this is equal to
p KM N ( ) ( 1)p dim p M p 0 X ³^ ´ 2. Analysis of the Ruelle Zeta Function 53 and for = ˜ to
p q KM N ( ) ( 1)p+q dim p a m , M q p,q 0 X ³^ ^ ´ where a = 15, a = 10, a = 6, a = 3, a = 1 and a = 0 for q 5. The 0 1 2 3 4 q only functions on L2( G) invariant under the action of G are the constant \ functions, hence N ( ) = 1. We can then use Lemma 2.7.1 to see that the above expressions take the claimed values. ¤
2.8 Contribution of the other representations
The following proposition gives a relationship between the action of a on H (n, K ) and the in nitesimal character of .
Proposition 2.8.1 Let Gˆ. ∈ Suppose H (n, K ) = 0 for some a . Then = w a P , where w W (g, h) and is 6a representative∈of the in nitesimal char| acter of . ∈ Moreover, for p Z we have ∈
p p w P H (n, K ) = H (n, K ) . w∈MW (g,h) Proof: This follows from [26], Corollary 3.32 and the isomorphism (2.6). ¤
In light of this proposition, in order to complete the proof of Proposition 2.4.4, and hence of Theorem 2.4.3, it will be su cient to show that for all Gˆ r triv and for 1, ˜ either tr ( ) = 0 or the in nitesimal ∈ { } ∈ { } character of satis es 1 Re(w ) a or Re(w ) a | 2 P P | for all w W (g, h). We rst∈ consider the elements of Gˆ induced from parabolic subgroups other that P = MAN. We say that two parabolics P 0 = M 0A0N 0 and P 00 = M 00A00N 00, where A0, A00 A = diag(a, b, c, (abc) 1) a, b, c > 0 are associate if there exists a 0 { | } 2. Analysis of the Ruelle Zeta Function 54 member w of the Weyl group W (g, h) such that w 1MAw = M 0A0. Repre- sentations of G induced from associate parabolics are equivalent. Up to association G has four parabolic subgroups: P de ned above; the minimal parabolic P0 = M0A0N0 where M0 = 1 and N0 is the group of real, upper triangular matrices with ones on the{ diagonal;} the parabolic 0 0 0 0 0 0 P with Langlands decomposition P = M A N where M = SL2 (R) 1 , A0 = diag(a, a, b, a 2b 1) a, b > 0 and N 0 is the group of real, upper triangular{ } matrices{ with ones on| the diagonal} and otherwise with zeros in the second column; and the parabolic P 00 with Langlands decomposition 00 00 00 00 00 00 3 P = M A N where M = SL3 (R), A = diag(a, a, a, a ) a > 0 and N 00 is the group of real, upper triangular matrices{ with ones on the| diagonal} whose only non-zero entries above the diagonal are in the rightmost column.
Proposition 2.8.2 Let be a principal series or complementary series rep- 0 00 resentation induced from the parabolic P = M A N , where P = P0, P or P and let be a nite dimensional representation on M. Then tr ( ) = 0 so contributes no zeros or poles to ZP, (s).
Proof: Let be the global character of the irreducible unitary representa- tion , so that
tr ( ) = (x) (x) dx. ZG The Weyl integration formula can be applied (see [14], p908) to give us
1 1 tr ( ) = f mlm dm (al)d(al) dl, W (L) + L | | ZA L ZM/L X ¡ ¢ where the sum is over conjugacy classes of Cartan subgroups L of M, and we denote by W (L) = W (L, M) the Weyl group of L in M, by f an Euler- Poincar e function for and d(al) is an explicitly given function on A+L. Proposition 1.4 of [14] tells us that for L = B 6
1 f mlm dm = 0, ZM/L ¡ ¢ hence
1 1 tr ( ) = f mbm dm (ab)d(ab) db. W (B) + | | ZA B ZM/B ¡ ¢ 2. Analysis of the Ruelle Zeta Function 55
The character of is non-zero only on Cartan subgroups of G that are G-conjugate to Cartan subgroups of M A (see [39], Proposition 10.19). The subgroup BA is not G-conjugate to any Cartan subgroup of M A , so it follows that tr ( ) = 0. ¤
G ˆ Now let = Ind P ( ) for some M and a . For KM let P : V V ( ) be the pro jection onto the∈ -isotype. ∈For any function∈ f on → G which is su ciently smooth and of su cient decay the operator (fd) is of trace class. Its trace is
+ P 1 a f(k mank) tr P (m)P dman dk. K MAN ∈K Z Z XdM
Plugging in the test function f = , where triv, ˜ , this gives us, as in [15]: ∈ { }
tr ( ) = C(a) tr (f ) da, + ZA where C(a) depends only on a and f is an Euler-Poincar e function on M attached to the representation . We can see that tr ( ) is non-zero only if tr (f ) is also. Theorem 2.4.3 now follows from Proposition 1.6.1. Chapter 3
A Prime Geodesic Theorem for SL4(R)
We continue using the notation de ned in the previous chapters, in particular we take G = SL (R) and take G to be a discrete, cocompact subgroup. 4 For let N( ) = el and de ne for x > 0 ∈
(x) = 1( ), and ˜(x) = 1( )tr ˜(b ), p p,reg [ ]∈E ( ) [ ]∈E ( ) P P NX( ) x NX( ) x where is the centraliser of in , we denote by 1( ) the rst higher Euler characteristic and ˜ is the virtual representation de ned in Section 2.4. We recall from Theorem 1.4.2 that the rst higher Euler characteristics are all positive and from Lemma 2.4.2 that the traces tr ˜(b ) are also all positive, so both (x) and ˜(x) are monotonically increasing functions.
Theorem 3.0.3 (Prime Geodesic Theorem) For x we have → ∞ 2x 8x (x) and ˜(x) . log x log x More sharply,
x3/4 x3/4 (x) = 2 li(x) + O and ˜(x) = 8 li(x) + O log x log x µ ¶ µ ¶ as x , where li(x) = x 1 dt is the integral logarithm. → ∞ 2 log t R 56 3. A Prime Geodesic Theorem for SL4(R) 57
We also prove the following Prime Geodesic Theorem, which will be needed for our application to class numbers. Let B0 be a closed subset of the compact group B with the following properties: it is of measure zero; it is invariant under the map b b 1 and 7→ contains all xed points of this map; and its complement B1 = BrB0 in B is homeomorphic to an open subset of Euclidean space each of whose connected components is contractible. The assumption that B0 contains all xed points of the map b b 1 is equivalent to the assumption that B0 contains all non- 7→ E p,1 E p regular elements of B. Let P ( ) be the subset of all [ ] P ( ) such that is conjugate in G to an element of A B1. We de ne for∈x > 0
1 (x) = 1( ). p,1 [ ]∈E ( ) P NX( ) x
Then as in the case of (x) above, 1(x) is a monotonically increasing func- tion.
Theorem 3.0.4 For x we have → ∞ 2x 1(x) . log x
The proof of these two theorems will occupy the rest of the chapter.
3.1 Analytic properties of R , (s) This section and the following proceed according to the methods of [47] and [48], in which the Selberg zeta function for quotients of the hyperbolic plane are considered. In this section the analogs of a series of lemmas are proved in our context. The next section translates the main theorem of [47] into our context. Recall from Theorem 2.3.4 that for a nite dimensional virtual represen- tation of M we have the functional equation
Z (1 s) = e G(s)Z (s), (3.1) P, P, where G(s) is a polynomial. Let D be the degree of the polynomial G(x). 3. A Prime Geodesic Theorem for SL4(R) 58
Lemma 3.1.1 Let H be a half-plane of the form Re(s) < (1 + ε) for some ε > 0 and let be a nite dimensional virtual{ representation of} M. Then there exists a constant C > 0 such that for s H we have ∈ R0 (s)/R (s) C s D 1. | , , | | | Proof: From Theorem 2.4.1 we get the identity
4 q ( 1)q R (s) = Z q s + , , P,( V n V ) 4 q=0 Y ³ ´ which implies 0 4 0 q Z q s + R , (s) q P,( V n V ) 4 = ( 1) q . (3.2) R (s) q , q=0 ZP,( V n V ) ¡s + 4 ¢ X Considering this identity, it will su ce to prove that¡ when¢ K is a half-plane of the form Re(s) < ε for some ε > 0, there exists a constant C > 0 such { } that for s K we have ∈ 0 D 1 Z q (s)/Z q (s) C s | P,( V n V ) P,( V n V ) | | | for all q = 0, . . . , 4. Since the proof does not depend on the value of q or on we shall abbreviate our notation for the zeta function to ZP (s), which notation we shall use for the rest of the section. It follows that Z0 (1 s) Z0 (s) P = P G0(s). Z (1 s) Z (s) P P From the de nition (2.5) of ZP (s) and Proposition 2.4.4, we can see that ZP (s) is both bounded above and bounded away from zero on the half plane 0 0 0 K = Re(s) > 1 + ε . It follows that ZP (s)/ZP (s) is bounded on K . This implies{ the lemma. } ¤
For t > 0, let N(t), denote the number of poles and zeros of ZP (s) at 1 points s = 2 + x, where 0 < x < t. Lemma 3.1.2 N(t) = O(tD)
2 G(s) Proof: De ne (s) = (ZP (s)) e , where G(s) is the polynomial in the functional equation (3.1) for ZP (s). We note that, in light of its role in the 3. A Prime Geodesic Theorem for SL4(R) 59 functional equation, the polynomial G(s) must satisfy G(s) = G(1 s). It then follows that (1 s) = (s). Note that (s) is real on the real axis and so (s ) = (s). Fix a real number 1 < a < 5/4 and let t > 0. Let R be the rectangle de ned by the inequalities 1 a Res a and t Im a t. We assume that t has been chosen so that no zero or pole occurs on the boundary of R. Then 1 1 0(s) 1 1 0(s) N(t) = ds N = Im ds N , 4 2 i (s) 0 4 2 (s) 0 Z∂R µZ∂R ¶ where N0 is the number of poles and zeros of ZP (s) on the real line. It then follows from the functional equation for (s) and from the fact that (s ) = (s), that we have
1 0(s) N(t) = Im ds N , 2 (s) 0 µZC ¶ where C is the portion of ∂R consisting of the vertical segment from a to 1 a + it plus the horizontal segment from a + it to 2 + it. Now the de nition of (s) gives us
0(s) Z0 (s) = 2 P G0(s), (s) ZP (s) so that 0(s) Z0 (s) Im ds = 2.Im P ds Im G0(s)ds (s) Z (s) µZC ¶ µZC P ¶ µZC ¶ Z0 (s) 1 = 2.Im P ds Im G + it + G(a) Z (s) 2 µZC P ¶ µ µ ¶ ¶ Z0 (s) = 2.Im P ds + O(tD). Z (s) µZC P ¶ It thus remains to show that Z0 (s) S(t) = Im P ds = O(tD). Z (s) µZC P ¶ 3. A Prime Geodesic Theorem for SL4(R) 60
Note that S(t) is the variation of the argument of ZP (s) along C. We may 1 extend the de nition of S(t) to those values of t for which 2 + it is a pole or zero of Z (s) by de ning it to be lim 1 (S(t + ε) + S(t ε)). P ε→0 2 From the de nition of ZP (s) we can see that S(t) = h(t) + O(1), where h(t) is the variation of the argument of ZP (s) along the segment from a + it 1 to 2 + it. The value of h(t) is bounded by a multiple of the number of zeros of Re(ZP (s)) on this segment, since the point ZP (s) cannot move between the right and left half-planes without crossing the imaginary axis. On the segment, the real part of ZP (s) coincides with 1 f (w) = (Z (w + it) + Z (w it)), P 2 P P
1 where w runs from 2 to a on the real axis. Since we have assumed that 1 2 + it is not a zero or pole of ZP (s), the function fP (w) is holomorphic in 1 a neighbourhood of the closed disc S, centred at a, of radius a 2 . As this 1 disc contains the interval from 2 to a, we may apply Jensen’s Formula ([10], XI.1.2) to conclude that
h(t) = O log f (w) dw | P | µZ∂S ¶ = O log Z (w + it) + Z (w it) dw | P P | µZ∂S ¶ = O log Z (w + it) + log Z (w it) dw | P | | P | µZ∂S ¶ = O log Z (w + it) dw + log Z (w it) dw . | i | | i | Ãi=1,2 ∂S ∂S ! X Z Z There remains one nal step in the proof of the lemma:
D Z (x + it) = eO(|t| ) (3.3) | i | uniformly in the strip 1 a x a for i = 1, 2. We know that the function ZP (s) is bounded and bounded away from zero on the line Res = a. Hence, by Lemma 2.3.2 and the functional equation (3.1), we can apply the Phragm en Lindel of Theorem ([10], Theorem VI.4.1) to give (3.3). ¤ 3. A Prime Geodesic Theorem for SL4(R) 61
Lemma 3.1.3 Given a < b R, there exists a sequence (yn) tending to in nity such that ∈ R0 (x + iy ) n = O(y2D) R (x + iy ) n ¯ n ¯ ¯ ¯ for a < x < b. ¯ ¯ ¯ ¯ Proof: As in Lemma 3.1.1 it will su ce to prove the result for ZP (s). Using the notation of Lemma 2.3.2 we have that
0 ZP (s) 1 0 0 i 1 k k 1 = (n1 n2) + g1(s) g2(s) + ( 1) s (s ) . ZP (s) s i=1,2 r X ∈RXi {0} 1 Let t0 > 2 be xed and consider the segment of the line Res = 2 given by 1 2 + it for t0 1 < t t0 + 1. Let N(t) be as above, then by Lemma 3.1.2 we know that N (t) = O (tD). It follows immediately that the number of roots D on the segment is O(t0 ). 1 By the Dirichlet principle, there exists a 2 + iy in the segment whose distance from any pole or zero is greater that C/T D, for some xed C > 0. i 1 k k 1 We conclude that the portion of the sum i=1,2( 1) s (s ) 2D corresponding to poles and zeros in the segment for sx = x + iy is O(y ), k k P P since sx = O(1) for these , when a < x < b. | | 1 To deal with the segments 2 +it for 0 < t t0 1 and t0 +1 < t < , we i 1 k k ∞ 1 proceed as follows. The portions of the sum i=1,2( 1) s (s ) corresponding to the rst and second segments resp ectively, can be written P P tp 1 1 k 1 1 sk + it s it dN(t) x 2 x 2 Z0 µ ¶ µ ¶ and ∞ 1 k 1 1 sk + it s it dN(t). x 2 x 2 Ztp+1 µ ¶ µ ¶ Recalling that N(t) = O(tD), we easily conclude that both of these expres- sions are O(T 2D). ¤
3.2 Estimating (x) and ˜(x)
To simplify notation, in what follows we write for an element of EP ( ) and for a primitive element and recall that E ( ) implies that is 0 ∈ P 3. A Prime Geodesic Theorem for SL4(R) 62
conjugate in to an element a b A B. Unless otherwise speci ed, sums ∈ E E p involving or 0 will be taken over conjugacy classes in P ( ) and P ( ) respectively. If and 0 occur in the same formula it is understood that 0 E reg wil be the primitive element underlying . We denote by P ( ) the subset of regular elements in EP ( ). For x > 0 let
(x) = 1( )l 0 E [ ]∈ P ( ) NX( ) x and ˜ (x) = 1( 0 )tr ˜(b )l 0 . reg [ ]∈E ( ) P NX( ) x Let be a nite dimensional virtual representation of M. We have for Re(s) > 1: R0 (s) , = ( )tr (b )l e sl . (3.4) R (s) 1 0 , X The following propositions are analogs of Theorem 2 of [47], from which, in the next section, we prove the Prime Geodesic Theorem using standard techniques of analytic number theory.
Proposition 3.2.1 (x) = 2x + O x3/4
Proof: Let D be the degree of the¡ polynomial¢ G(s), as in the previous section, and suppose k 2D is an integer and x, c > 1. Then, by (3.4) and [24], Theorem 40,
1 c+i∞ R0 (s) ,1 s 1(s + 1) 1 (s + k) 1xsds 2 i R (s) Zc i∞ ,1 1 c+i∞ = ( )l e sl s 1(s + 1) 1 (s + k) 1xsds 2 i 1 0 c i∞ Ã ! Z X 1 N( ) k = ( )l 1 . (3.5) k! 1 0 x NX( ) x µ ¶ 0 Theorem 2.4.3 tells us that all poles of R (s)/R (s) lie in the strip 1 Re(s) 1. By virtue of Lemmas 3.1.1 and 3.1.3 it is permissible to shift 3. A Prime Geodesic Theorem for SL4(R) 63 the line of integration in (3.5) into the half plane Re(s) < 1, taking into account the residues of the poles of R0 (s)/R (s). Hence, for c0 < 1 1 N( ) k ( )l 1 (3.6) k! 1 0 x NX( ) x µ ¶ c0+i∞ 0 1 R ,1(s) 1 1 1 s = ck( )x + s (s + 1) (s + k) x ds, 2 i c0 i∞ R ,1(s) ∈Sk Z ReX( )>c0 where S denotes the set of poles of (R0 (s)/R (s))s 1(s+1) 1 (s+k) 1 k ,1 ,1 and ck( ) denotes the residue at . For x > 1, Lemma 3.1.1 implies the integral in (3.6) tends to zero as c0 . If we de ne → ∞ x (x) = (x), (x) = (t)dt, j N, 0 j j 1 ∈ Z0 it is well known that 1 (x) = ( )l (x N( ))j j j! 1 0 NX( ) x and we deduce from (3.6) that
k+ k(x) = ck( )x . (3.7) ∈S Xk Let d > 0. For a function f : R R de ne the operator by setting → 2D 2D f(x) = ( 1)i f(x + (2D i)d). i i=0 X µ ¶ It follows from (3.7) and Theorem 2.4.3, setting k = 2D, that
2 2 2D+1 2D+ 2D+ 2D(x) = x + c2D( )x + c2D( )x , (2D + 1)! R ∈S p= 2 ∈Sp/4 X2D X X2D (3.8) R where S2D = S2D (R r 1 ), the real elements of S2D not including = 1, q ∩ { } and S = S (q + i(R r 0 )), the non-real elements of S on the line 2D 2D ∩ { } 2D 3. A Prime Geodesic Theorem for SL4(R) 64
Re(s) = q. The coe cient ((2D + 1)!) 1 on the leading term comes from the 0 fact that (R ,1(s)/R ,1(s)) has a double pole at s = 1 and from the factors s 1...(s + 2D) 1. In general if f is at least 2D times di erentiable,
x+d t2D+d t2+d f(x) = f (2D)(t ) dt ...dt . (3.9) 1 1 2D Zx Zt2D Zt2 By applying the Mean Value Theorem we get xr = d2Dr(r 1)...(r (2D 1))x˜r 2D, (3.10) where x˜ [x, x+2Dd]. In particular we notice that (x2D+1) = O(x), hence ∈ 2 x2D+1 = ax + b (2D + 1)! µ ¶ for some a, b R. Computations show that ∈ 2D 1 a = 2 ( 1)j ((2D j)d)2D = 2d2D. (3.11) j!(2D j)! j=0 X (2D) By de nition 0(x) = 2D (s) so from (3.9) we have
x+d t2D+d t2+d (x) = (t ) dt ...dt . (3.12) 2D 0 1 1 2D Zx Zt2D Zt2 By Theorem 1.4.2, the Euler characteristics 1( ) are all positive. Hence 0(x) is non-decreasing and it follows from (3.12) that (x) d 2D (x) (x + 2Dd). (3.13) 0 2D 0 It also follows from (3.10) and (3.11) that
2 d 2D x2D+1 + c ( )x2D+ = 2x + O(x3/4). (2D + 1)! 2D ∈SR X2D Thus it remains to show that for p 2, 1, 0, 1, 2 ∈ { }
2D 2D+ 3/4 d c2D( )x = O(x ) ∈Sp/4 X2D 3. A Prime Geodesic Theorem for SL4(R) 65 and the proposition follows by (3.13). Let p 2, 1, 0, 1, 2 . In order to estimate ∈ { }
2D 2D+ d c2D( )x ∈Sp/4 X2D we need two estimates for c ( )x2D+ , where Sp/4. The residues 2D ∈ 2D at the poles of R0 (s)/R (s) are O(1), so for Sp/4 ,1 ,1 ¡ ¢ ∈ 2D d 2D c ( )x2D+ = O d 2D (2D+1)x2D+p/4 . 2D | | On the other hand,¡it follows from¢ (3.10)¡ that ¢
d 2D c ( )x2D+ = O 1xp/4 . 2D | | ¡ ¢ ¡0 ¢ De ne Np(t) to be the number of poles of R ,1(s)/R ,1(s) on the interval p D 4 + i(0, t]. From Lemma 3.1.2 we have that Np(t) = O t . Thus ¡ ¢
2D 2D+ d c2D( )x ∈Sp/4 X2D KD ∞ = O xp/4 t 1dN(t) + d 2Dx2D+p/4 t (2D+1)dN(t) D Ã Z1 ZK ! = O KD 1xp/4 + K (D+2)d 2Dx2+p/4 (3.14)
If we choose K¡ and d appropriately, then we can¢ conclude from (3.14) that 2 2+ 3/4 d p/4 c2( )x = O(x ), as required. ¤ ∈S2 ³P ´ Proposition 3.2.2 ˜(x) = 8x + O x3/4
Proof: The proof follows exactly as¡ for ¢the previous proposition, replacing R ,1(s) with R , ˜(s) throughout. It follows in particular from the fact (The- orem 2.4.3) that R , ˜(s) has a zero of order eight at the point s = 1 and from E p the fact (Lemma 2.4.2) that tr ˜(b ) 0 for all P ( ) and tr ˜(b ) = 0 if and only if is non-regular. ∈ ¤ 3. A Prime Geodesic Theorem for SL4(R) 66 3.3 The Wiener-Ikehara Theorem
We shall use the following version of the Wiener-Ikehara theorem (see also [9], Chapter XI, Theorem 2, and [17], Theorem 3.2). Let R (s), k N be a sequence of rational functions on C. for an open set k ∈ U C let N(U) be the set of natural numbers k such that the pole divisor of Rk does not intersect U. We say that the series
Rk(s) N Xk∈ converges weakly locally uniformly on C if for every open U C the series
Rk(s) N k∈X(U) converges locally uniformly on U.
Theorem 3.3.1 (Wiener-Ikehara) Let A(x) 0 be a monotonic measurable function on R+. Suppose that the integral
∞ L(s) = A(x)e sx dx Z0 converges for s C with Re(s) > 1. Suppose further that there are j N, ∈ ∈ r R and a countable set I, and for each i I there is C with Re( ) < 1 ∈ ∈ i ∈ i and c Z, such that the function i ∈ ∂ j+1 r ∂ j+1 1 L(s) ci ∂s s 1 ∂s s i i∈I µ ¶ X µ ¶ extends to a holomorphic function on the half-plane Re(s) 1. Here we assume the sum converges weakly locally uniformly absolutely on C. Then
lim A(x)x (j+1)e x = r. x→∞
Proof: Let B(x) = A(x)x (j+1)e x. Since
r ∞ = r e (s 1)x dx s 1 Z0 3. A Prime Geodesic Theorem for SL4(R) 67 we get ∂ j+1 r ∞ = r xj+1e (s 1)x dx, ∂s s 1 µ ¶ Z0 and similarly ∂ j+1 1 ∞ = xj+1e (s i)x dx. ∂s s µ ¶ i Z0 Let f be a smooth function of compact support on R which is real-valued and even. Then its Fourier transform fˆ will also be real valued. We further assume f to be of the form f = f f for some f . Then fˆ = (fˆ )2 is positive 1 1 1 1 on the reals. Let I(f) be the set of i I such that Im ( i) suppf. Then the function ∈ ∈
∂ j+1 r ∂ j+1 1 g(s) = L(s) ci . ∂s s 1 ∂s s i µ ¶ i∈XI(f) µ ¶ extends to a holomorphic function on the set
s C : Re(s) 1, Im (s) suppf . { ∈ ∈ } We have that ∞ ∞ j+1 (s 1)x j+1 (s i)x g(s) = (B(x) r)x e dx ci x e dx. 0 0 Z i∈XI(f) Z For ε > 0 let g (t) = g(1 + ε + it). We then have for y R ε ∈ ∞ iyt iyt j+1 (ε+it)x f(t)e gε(t)dt = f(t)e (B(x) r)x e dx dt R R 0 Z Z µZ ∞ ¶ iyt j+1 (1+ε+it i)x ci f(t)e x e dx dt. R 0 i∈XI(f) Z Z We want to change the order of integration. The only potential problem is with the summand involving B(x). Since A(x) is non-negative and non- decreasing, we have for real s and x > 0
∞ ∞ A(x)e xs L(s) = A(u)e usdu A(x) e usdu = , s Z0 Zx 3. A Prime Geodesic Theorem for SL4(R) 68 that is A(x) sL(s)exs. Since L(s) is holomorphic for Re(s) > 1, it follows that A(x) = O(exs) for every s > 1, which implies that A(x) = o(exs) for every s > 1. Hence B(x)e x = A(x)e (1+ )x = o(1) for every > 0. This implies that the integral
∞ (B(x) r)xj+1e (ε+it)xdx Z0 converges uniformly in the interval 2 t 2 . Hence we can interchange the order of integration to obtain
∞ iyt j+1 εx i(y x)t f(t)e gε(t)dt = (B(x) r)x e e f(t)dt dx R 0 R Z Z ∞ µZ ¶ j+1 (1+ε i)x i(y x)t ci x e e f(t)dt dx. 0 R i∈XI(f) Z Z ∞ = (B(x) r)xj+1e εxfˆ(y x) dx 0 Z ∞ j+1 (1+ε i)x ci x e fˆ(y x) dx 0 i∈XI(f) Z We want to take the limit of both sides as ε 0 and show that the limit passes under the integration signs. → Since g(s) is holomorphic for Re(s) 0 it follows that gε(t) g(1 + it) uniformly on the support of f, as ε 0. Thus the limit of the left→hand side exists and → iyt iyt lim f(t)e gε(t) dt = f(t)e g(1 + it) dt. ε→0 ZR ZR We note that since fˆ is rapidly decreasing
∞ ∞ lim xj+1e εxfˆ(y x) dx = xj+1fˆ(y x) dx, ε→0 Z0 Z0 and since B(x) is non-negative and non-decreasing it follows that
∞ ∞ lim (B(x) r)xj+1e εxfˆ(y x) dx = (B(x) r)xj+1fˆ(y x) dx. ε→0 Z0 Z0 Considering the sum over I(f), we note that the imaginary parts of the i for i I lie in a compact set and hence the convergence of the sum is uniform ∈ 3. A Prime Geodesic Theorem for SL4(R) 69 in ε and the limit may be drawn under the summation. Finally, we note once more that the integrand is non-negative and monotonically increasing as ε 0, so the limit may once more be taken under the integral. We conclude→ that ∞ f(t)eiytg(1 + it) dt = (B(x) r)xj+1fˆ(y x) dx R 0 Z Z ∞ j+1 (1 i)x ci x e fˆ(y x) dx. 0 i∈XI(f) Z By the Riemann-Lebesgue lemma the left hand side tends to zero as y . → ∞ For y 0 and C with Re( ) < 1 we estimate ∈ ∞ ∞ xj+1e( 1)xfˆ(y x) dx xj+1e( 1)xfˆ(y x) dx 0 ∞ Z Z ∞ = (x + y)j+1e( 1)(x+y)fˆ( x) dx Z ∞ Cyj+1e( 1)y, for some constant C. It follows that the sum over I(f) tends to zero as y . We therefore have → ∞ ∞ ∞ lim B(x)xj+1fˆ(y x) dx = r lim xj+1fˆ(y x) dx. (3.15) y→∞ y→∞ Z0 Z0 Lemma 3.3.2 For every non-negative integer k 1 ∞ lim xkfˆ(y x)dx = 2 f(0). y→∞ yk Z0 Proof: We prove the lemma using induction. First, let k = 0. Then
∞ ∞ lim fˆ(y x)dx = lim fˆ(y x)dx y→∞ 0 y→∞ ∞ Z ∞ Z = fˆ( x)dx Z ∞ = 2 f(0). 3. A Prime Geodesic Theorem for SL4(R) 70
We now assume the lemma has been proved for a xed value of k. 1 ∞ 1 ∞ lim xk+1fˆ(y x)dx = lim xk+1fˆ(y x)dx y→∞ yk+1 y→∞ yk+1 Z0 Z ∞ 1 ∞ = lim (x + y)k+1fˆ( x)dx y→∞ yk+1 Z ∞ 1 ∞ = lim xk+1fˆ( x)dx y→∞ yk+1 Z ∞ 1 ∞ + lim (x + y)kfˆ( x)dx y→∞ yk Z ∞ 1 ∞ = lim xkfˆ(y x)dx y→∞ yk Z0 = 2 f(0). ¤
The lemma together with (3.15) implies that
∞ x j+1 lim B(x) fˆ(y x)dx = r2 f(0). y→∞ y Z0 µ ¶ Let S > 0. Since A(x) is monotonic we have A(y S) A(x) A(y + S) whenever y S x y + S. For x in that range w e then ha ve by the de nition of B(x)
B(y s)(y S)j+1ey S B(x)xj+1ex B(y + S)(y + S)j+1ey+S. The rst inequality implies
B(x)xj+1 B(y S)(y S)j+1ey x S B(y S)(y S)j+1e 2S. So for y S, (y S)j+1 y+S y+S xj+1 e 2SB(y S) fˆ(y x)dx B(x) fˆ(y x)dx yj+1 yj+1 Zy S Zy S ∞ xj+1 B(x) fˆ(y x)dx, yj+1 Z0 3. A Prime Geodesic Theorem for SL4(R) 71 which implies 2S 2 f(0) lim sup B(y) re S . y→∞ ˆ S f(x)dx We vary f so that fˆ is small outside [ S,RS]. In this way we get lim sup B(y) re2S. y→∞
Since this holds for any S > 0 it follows that
lim sup B(y) r. y→∞
The inequality lim inf B(y) r. y→∞ is obtained in a similar fashion. The theorem is proven. ¤
3.4 The Dirichlet series
Let B0 be a closed subset of the compact group B with the following prop- erties: it is of measure zero; it is invariant under the map b b 1 and 7→ contains all xed points of this map; and its complement B1 = B r B0 in B is homeomorphic to an open subset of Euclidean space each of whose connected components is contractible. The Weyl group W = W (M, B) contains two elements and the non- trivial element acts on B by b b 1, so the invariance condition above says that both B0 and B1 are invarian7→ t under the action of W . The xed points under the action of the Weyl group are precisely the non-regular elements of B, so the assumption that B0 contains all these xed points is equivalent to Bnreg B0, where Bnreg denotes the subset of nonregular elements of B. The action of W on B1 permutes the connected components and the assumption Bnreg B0 implies that the quotient space B1/W is also homeomorphic to an open subset of Euclidean space each of whose connected components is simply connected. For subsets S and T of G we denote by S.T the subset
S.T = sts 1 : s S, t T { ∈ ∈ } 3. A Prime Geodesic Theorem for SL4(R) 72
1 1 E 0 E 1 of G. Let Mell = M.B M. Let P ( ) and P ( ) be the subsets of 0 1 EP ( ) consisting of all conjugacy classes [ ] such that b B or b B respectively. The assumption that B0 is of measure zero is∈not required∈ for the following lemma, but will be needed later on.
Lemma 3.4.1 There exist a set M M 1 with compact closure in M and a c ell monotonically increasing sequence (gn) of smooth functions on M, supported on Mc, which are invariant under conjugation by elements of KM , and whose orbital integrals satisfy
M (g ) = g (xb x 1) dx 1 as n Ob n n → → ∞ ZM/B for all E 1( ). ∈ P 1 1 Proof: We view Mell as a bre bundle with base space B /W and bres homeomorphic to M/B. Let d( , ) denote the metric on B given by the form b in (2.1). For n N let B˜ B1 be the set B˜ = b B : d(b, B0) 1/n and let ∈ n n { ∈ } B = B˜ /W . Then, by [60], Corollary 1.11, for each n N there exists a n n ∈ function hn : B/W R such that 0 hn(b) 1 for all b B/W , for all b B we have h (b→) = 1 and h is supp orted on B1/W . W∈e may assume ∈ n n n that the hn’s form an increasing series. Let U be a compact neighbourhood of a point in M/B homeomorphic to a subset of Euclidean space. Let k : M/B R be a smooth positive function → supported on U and satisfying M/B k(m)dm = 1. We now de ne, for n N, functions g˜ : M R as follows. On each R n connected component V of∈B1/W we x a trivialisation→ of the restriction to 1 1 V of the bundle M/B Mell B /W . Then for v V and m M/B 1 → → ∈ ∈ de ne g˜n(mvm ) = hn(v)k(m). Since V is simply connected there are no problems with global agreement of this de nition. For m M r M 1 de ne ∈ ell g˜ (m) = 0. Finally we de ne the functions g : M R by n n → 1 g (m) = g˜ (kmk 1) dk n 2 n ZKM for all m M. Then the functions g form an increasing sequence of smooth ∈ n KM -invariant functions supported on the compact set Mc, which we de ne to 3. A Prime Geodesic Theorem for SL4(R) 73
1 be the closure in M of KM .(U.B ). We will show that their orbital integrals have the required properties. Let b B1. ∈ M (g ) = g (mbm 1) dm Ob n n ZM/B 1 = g˜ (kmbm 1k 1) dk dm 2 n ZM/B ZKM 1 = g˜ (mkbk 1m 1) dk dm 2 n ZM/B ZKM 1 = g˜ (mbm 1) + g˜ (mb 1m 1) dk dm 2 n n ZM/B ZB 1 = h (b) + h (b 1) k(m) dm 2 n n ZM/B = hn¡(b). ¢
The last equality holds since bW = b 1W in B1/W . There exists N N such that b B for all n N, hence M (g ) 1 as n , by∈the ∈ n Ob n → → ∞ de nition of the functions hn. ¤
For n N let ∈ j+1 sl j l e Ln(s) = 1( 0 ) b (gn)l 0 1 , O det (1 (a b ) n) [ ]∈E 1 ( ) XP | let
rn = gn(x)dx, ZM and let 4 m = ( 1)qr dim Hq(n, ) . n, n K q=0 X Proposition 3.4.2 For all n N and for j N large enough the series j ∈ ∈ Ln(s) converges locally uniformly in the set s C : Re(s) > 1 . j { ∈ } The function Ln(s) can be written as a Mittag-Le er series
∂ j+1 r ∂ j+1 1 Lj (s) = n + N ( ) m , n ∂s s 1 n, ∂s s (H ) a 1 µ ¶ X∈Gˆ X∈ µ ¶ 3. A Prime Geodesic Theorem for SL4(R) 74 where the summand of the double series corresponding to = triv, = 2 P is excluded. The double series converges weakly locally uniformly on C. For Gˆ, a such that ( , ) = (triv, 2 P ) we have mn, = 0 only if Re∈ ( ) > ∈ 2 . Thus, in particular,6 the double series converges6 locally P uniformly on s C : Re(s) > 1 . { ∈ }
Proof: We can put f = gn in (2.3) to de ne the function gn,j,s, which by Proposition 2.1.3, for j and Re(s) large enough, goes into the Selberg trace formula to give the equation
lj+1e sl N ( )tr ( ) = vol( G ) (g ) . gn b n 1 n E \ O det (1 (a b ) ) X∈Gˆ [ ]∈XP ( ) |
By the de nition of the functions gn, the orbital integrals b (gn) are equal E 0 E 1 O to zero when P ( ). Also, for P ( ), since we have assumed that Bnreg B0 we ha∈ve that is regular and∈ we get from Lemma 1.4.1 that vol( G ) = ( )l . \ 1 0 Thus we can see that the geometric side of the trace formula is equal to j j Ln(s). It follows that for j and Re(s) large enough the Dirichlet series Ln(s) converges absolutely.
Lemma 3.4.3
∂ j+1 1 tr ( ) = ( 1)j+1 r dim H (n, ) . gn ∂s n K s (H ) a 1 µ ¶ X∈
Proof: Replacing f with gn in (2.11) we get
MA j tr ( gn ) = gn(m) H (n, K )(ma)dm gs(a)da ZMA j+1 sla = gn(m)dm tr (a H (n, K )) la e da. A M | Z Z∞ ( (H1) s)t j+1 = rn dim H (n, K ) e t dt 0 a Z X∈ ∂ j+1 1 = ( 1)j+1 r dim H (n, ) . ∂s n K s (H ) a 1 µ ¶ X∈ 3. A Prime Geodesic Theorem for SL4(R) 75
¤
By the above lemma and Proposition 2.6.3, and using similar arguments to those used in the proof of Proposition 2.7.2, we can see that the spectral side of the trace formula is equal to the Mittag-Le er series given in the propostion. j Since Ln(s) is a Dirichlet series with positive coe cients it will converge locally uniformly for s in some open set. By proving the convergence of the Mittag-Le er series we shall show that the Dirichlet series extends to a holomorphic function on Re(s) > 1 and hence, since it has positive coe cients, converges locally{uniformly there.} We recall that for a we write for the norm given by the form b ∈ k k in (2.1). It then follows from Proposition 2.3.1 that there exist m1 N and C > 0 such that for every Gˆ and every a we have ∈ ∈ ∈ m C(1 + )m1 . | n, | k k If S denotes the set of all pairs ( , ) Gˆ a such that m = 0, then ∈ n, 6 there exists m N such that 2 ∈ N ( ) < . (1 + )m2 ∞ ( X, )∈S k k Now, let U C be open and let S(U) be the set of all pairs ( , ) Gˆ a such that m = 0 and (H ) / U. Let V be a compact subset of∈U. We n, 6 1 ∈ have to show that for some j N, which does not depend on U or V , ∈ N ( )mn, sup j+2 < . s∈V (s (H1)) ∞ ( , )∈S(U) ¯ ¯ X ¯ ¯ ¯ ¯ Let m , m be as above and let¯ j m + m¯ 2. Since V U and 1 2 1 2 V is compact there is ε > 0 such that s V and ( , ) S(U) implies ∈ ∈ s (H1) ε. Hence there is c > 0 such that for every s V and every (| , ) S(|U ), ∈ ∈ (s (H ) c(1 + ). | 1 | k k Putting this all together we get
N ( )mn, 1 N ( ) mn, sup j+2 sup j+2 | j+2| s∈V (s (H1)) s∈V c (1 + ) ( , )∈S(U) ¯ ¯ ( , )∈S(U) X ¯ ¯ X k k ¯ ¯ ¯ ¯ 3. A Prime Geodesic Theorem for SL4(R) 76
C N ( ) sup j+2 m s∈V c (1 + ) 2 ( , X)∈S(U) k k < , ∞ which proves the proposition. ¤
3.5 Estimating n(x) Let j+1 j l n(x) = 1( 0 ) b (gn)l 0 1 . O det (1 (a b ) n) [ ]∈E 1 ( ) P | NX( ) x
Lemma 3.5.1 ∞ j x sx j n(e )e dx = Ln(s) Z0 Proof: This follows from Abel’s summation formula ([9], Chapter VII, The- orem 6). ¤
Let 1 n(x) = 1( 0 ) b (gn)l 0 1 . O det (1 (a b ) n) [ ]∈E 1 ( ) P | NX( ) x
Lemma 3.5.2 For each n N we have ∈ (x) r x. n n Proof: By Propostion 3.4.2 and Lemma 3.5.1 we can apply Theorem 3.3.1 j j to the series Ln(s) and the function n(x) to deduce that
j n(x) lim = rn. (3.16) x→∞ x(log x)j+1
Also it is clear that j (x) (x) n , n (log x)j+1 3. A Prime Geodesic Theorem for SL4(R) 77 so it follows that n(x) lim inf rn. x→∞ x Let 0 < < 1. Then
j+1 j l n(x) 1( 0 ) b (gn)l 0 1 O det (1 (a b ) n) [ ]∈E 1 ( ) P | x j+1 j+1 1 (log x) 1( 0 ) b (gn)l 0 1 O det (1 (a b ) n) [ ]∈E 1 ( ) P | x From (3.16) and (3.17) we get (j+1) 1 L rn + L lim sup 1 x→∞ x (j+1) = rn . Since this holds for any value of in the interval 0 < < 1 we get that L r and the lemma follows. ¤ For n N and x > 0 let ∈ (x) = ( ) (g )l . n 1 0 Ob n 0 [ ]∈E 1 ( ) P NX( ) x Proposition 3.5.3 For each n N we have ∈ (x) r x. n n Proof: We note that 0 < det (1 a b n) < 1 for all E reg( ) and that the | ∈ P value of the determinant tends to 1 as l tends to in nity, hence for 0 < ε < 1 E reg there are only nitely many P ( ) such that det (1 a b n) 1 ε. Since E 1( ) E reg( ) the same∈ holds for E 1( ). We x 0 <| ε < 1 and P P ∈ P de ne for each n N the functions n,ε and n,ε to be the same sums as for ∈ E 1 n and n respectively but restricted to those classes [ ] P ( ) such that 1 ε < det (1 a b n) < 1. It then follows that both ∈ | (x) (x) n n,ε 0 (3.18) x → and (x) (x) n n,ε 0 (3.19) x → as x . Now → ∞ 1 ε < det (1 a b n) < 1 | immediately implies 1 ε 1 < 1 < , det (1 a b n) det (1 a b n) | | 3. A Prime Geodesic Theorem for SL4(R) 79 so summing up we get (x) (x) (x) n,ε (1 ε) < n,ε < n,ε . x x x By Lemma 3.5.2 and (3.18), it then follows that n,ε(x) n,ε(x) rn(1 ε) lim inf lim sup rn. x→∞ x x→∞ x It then follows from (3.19) that n(x) n(x) rn(1 ε) lim inf lim sup rn. x→∞ x x→∞ x Since this holds for any value of ε in the interval 0 < ε < 1 the proposition is proven. ¤ 3.6 Estimating 1(x) Lemma 3.6.1 The sequence (rn) is monotonically increasing and rn 2 as n . → → ∞ Proof: That the sequence (rn) is monotonically increasing is clear since the sequence of functions (gn) is monotonically increasing. By Weyl’s integral formula ([39], Proposition 5.27), and since the func- tions gn are only non-zero on elliptic elements of M, we have rn = gn(x) dx ZM 1 = g (xbx 1) D(b) 2 dx db W (B : G) n | | | | ZB ZM/B 1 = M (g ) D(b) 2 db, W (B : G) Ob n | | | | ZB where W (B : G) is the Weyl group and D(b) is the Weyl denominator. Since the sequence of functions gn is monotonically increasing and the functions 3. A Prime Geodesic Theorem for SL4(R) 80 are all supported within a given compact subset of M we can interchange integral and limit to get 1 M 2 lim rn = lim b (gn) D(b) db n→∞ n→∞ W (B : G) O | | | | ZB 1 M 2 = lim b (gn) D(b) db. W (B : G) n→∞ O | | | | ZB M Furthermore, the orbital integrals b (gn) tend to one as n , except for b in a set of measure zero, so it folloOws that → ∞ 1 2 lim rn = D(b) db. n→∞ W (B : G) | | | | ZB The Weyl group W (B : G) consists of the identity element and the ele- ment 1 1 . 1 1 We can also compute the Weyl denominator D(b) for b B. Let ∈ cos sin R( ) = SO(2) sin cos ∈ µ ¶ and let R( ) R( , ) = B. R( ) ∈ µ ¶ Then D(R( , )) = ei( + )(1 e 2i )(1 e 2i ) = (ei( + ) + e i( + )) (ei( ) + e i( )) = 2cos ( + ) 2cos ( ) = 2(cos cos sin sin cos cos sin sin ) = 4 sin sin We have normalised the Haar measure on B so that B db = 1, hence 1 1 2 2 R d d D(b) 2 db = 16 sin2 sin2 W (B : G) | | 2 4 2 | | ZB Z0 Z0 3. A Prime Geodesic Theorem for SL4(R) 81 2 2 2 = sin2 sin2 d d 2 Z0 Z0 2 = . 2 2 = 2. This proves the lemma. ¤ Let 1 (x) = 1( 0 )l 0 . [ ]∈E 1 ( ) P NX( ) x Proposition 3.6.2 1(x) 2x . log x Proof: For all n N and all x > 0 we have ∈ (x) 1(x) (x). n It then follows, using Proposition 3.2.1 and Proposition 3.5.3, that for all n N ∈ 1 1 n(x) (x) (x) (x) rn = lim inf lim inf lim sup lim sup = 2. x→∞ x x→∞ x x→∞ x x→∞ x We can then deduce from Lemma 3.6.1 that 1(x) lim = 2. x→∞ x ¤ 3.7 Estimating (x), ˜(x) and 1(x) To keep the notation less cluttered, in the sums over conjugacy classes in which appear in this section we shall not specify which set of classes the sum is being taken over, since this will be clear from the context. We shall always use 0 to denote primitive elements and where and 0 appear together in the same formula we shall mean that 0 is the primitive element underlying . 3. A Prime Geodesic Theorem for SL4(R) 82 Proposition 3.7.1 (x) (x) lim = lim = 1. x→∞ 2x/ log x x→∞ 2x Proof: We can write (x) = n 0 1( 0 )l 0 , N(X 0) x n 0 where n 0 N is maximal such that N( 0) x. By de nition N( 0) = l ∈ n e 0 , so N( ) x implies nl log x and we can see that 0 0 (x) log(x) (x). (3.20) Next we x a real number 0 < a < 1. By Theorem 1.4.2 the Euler characteristic ( ) > 0 for all E p( ), so for x > 1, 1 0 0 ∈ P (x) 1( 0 )l 0 . a x a (x) a log x 1( 0 ) = a log x( (x) (x )). a x Proposition 3.7.2 ˜(x) ˜(x) lim = lim = 1. x→∞ 8x/ log x x→∞ 8x Proof: Exactly as for the previous proposition, making use of Proposition 3.2.2 and Lemma 2.4.2. ¤ Proposition 3.7.3 x3/4 (x) = 2li(x) + O . log x µ ¶ Proof: We consider the function l 0 S(x) = 1( 0 ) l NX( ) x 1 = ( ) + ( ) . 1 0 1 0 k 1/k N(X 0) x Xk 2 N( X0) x We consider the double sum on the right. Since G is discrete there is a geodesic min of minimum length. For a given x the inner sum contains at least one summand only for k log x/l min . For each such k 2, by Proposition 3.7.1, the inner sum is equal to O(√x/ log x). Therefore we have S(x) = (x) + O √x . (3.21) Now ¡ ¢ x x (t) l 0 dt = 1( ) dt t log2t 0 t log2t Z2 Z2 N( ) t X x l 0 = 1( 0 ) 2 dt N( ) t log t NX( ) x Z 1 1 = 1( 0 )l 0 l log x NX( ) x µ ¶ (x) = S(x) . log x 3. A Prime Geodesic Theorem for SL4(R) 84 Hence x (t) (x) S(x) = dt + . t log2t log x Z2 By Proposition 3.2.1 we have x 2 2x x 1 x3/4 S(x) = dt + + O dt + O log2 t log x t1/4 log2 t log x Z2 µZ2 ¶ µ ¶ 2t x x 2 2x x3/4 = + dt + + O log t log t log x log x · ¸2 Z2 µ ¶ x 2 x3/4 = dt + O . log t log x Z2 µ ¶ Together with (3.21) this proves the proposition. ¤ Proposition 3.7.4 x3/4 ˜(x) = 8li(x) + O . log x µ ¶ Proof: Exactly as for the previous proposition, making use of Proposition 3.2.2 and Proposition 3.7.2. ¤ The proof of Theorem 3.0.4 proceeds exactly as for the proof of Proposi- tion 3.7.1, making use of Proposition 3.6.2. Chapter 4 Division Algebras of Degree Four 4.1 Central simple algebras and orders Let F be a eld. An algebra A over F is a (not necessarily commutative) ring with unity, which is also a vector space over F , such that x(ab) = (xa)b = a(xb) for all x F and a, b A. The eld F can be embedded into the centre of an F -algebra∈A under the∈map x x.1, where 1 is the unity element of A. A ring is said to be simple if it has7→no non-trivial, proper, two-sided ideals. If A is simple and its centre is equal to F then it is known as a central simple F -algebra. If for every non-zero element of A there exists a (two-sided) inverse in A, then A is called a division algebra . The centre of a division algebra is a eld. Every division algebra is simple and hence central simple over its centre. Let (F ) denote the set of isomorphism classes of nite dimensional central simpleC algebras over F and (F ) the set of isomorphism classes of nite dimensional division algebras withD centre F , then (F ) (F ). For convenience we collect together here a number ofDfacts abCout central simple algebras and their orders, which will be used in the sequel. Let F be a eld, K an extension of F and A a nite dimensional algebra over F . Proposition 4.1.1 (a) If n = dimF A, then there exists an injective homo- morphism A , Mat (F ). → n (b) dimK (A K) = dimF A. (c) If A ( F ), then A K (K). ∈ C ∈ C 85 4. Division Algebras of Degree Four 86 (d) If B is a simple subalgebra of A (F ) with centraliser C (B) in A, ∈ C A then (dimF B)(dimF CA(B)) = dimF A. Proof: All the statements of the proposition are proved in [45]. Statement (a) is Corollary 5.5b; (b) is Lemma 9.4; (c) is Proposition 12.4b(ii) and (d) is Theorem 12.7. ¤ The next proposition gives some results about sub elds of algebras. Proposition 4.1.2 (a) If A (F ), then for every x A the set F [x] = f(x) : f F [X] is a sub eld∈ofDA. ∈ { ∈ } 2 (b) If A (F ), then dimF A = m for some m N. The natural number m is called the∈ C degree of A. If E is a sub eld of A ∈then [E : F ] divides m. If K is a sub eld of A containing F then it is said to be strictly maximal if [K : F ] = deg A. (c) A sub eld K of A (F ) is strictly maximal if and only if CA(K) = K. If A (F ) then every∈ Cmaximal sub eld of A is strictly maximal. ∈ D (d) If A (F ) and [K : F ] = deg A = n, then A K = Matn(K) if and only if K is∈isomorphicC as an F -algebra to a strictly maximal sub eld of A. Proof: All the statements of the proposition are proved in [45]. Statement (a) is Lemma 13.1b; (b) is Corollary 13.1a; (c) is Corollary 13.1b and (d) is Corollary 13.3. ¤ Now let F be a number eld and F the ring of integers in F . For any nite dimensional vector space V overOF , a full -lattice in V is a nitely OF generated F -submodule M in V which contains a basis of V . A subring of A whichOis also a full -lattice in A is called an -order, or simply anO OF OF order, in A. A maximal -order in A is an -order which is not properly OF OF contained in any other F -order in A. An element a A is said to be integral over if it is a rootOof a monic polynomial with ∈coe cients in . The OF OF integral closure of F in A is the set of all elements of A which are integral over . O OF Proposition 4.1.3 (a)(Skolem-Noether Theorem) Let A (F ), and B a simple subring of A such that F B A. Then every F -isomorphism∈ C of B onto a subalgebra of A extends to an inner automorphism of A. (b) Every element of an -order is integral in A over . OF O OF 4. Division Algebras of Degree Four 87 (c) The -order is maximal in A if and only if for each prime ideal OF O p of the p-adic completion p is a maximal p-order in Ap. OF O OF, (d) If A (Fp) for some prime ideal p of , then the integral closure ∈ D OF of p in A is the unique maximal p-order in A. OF, OF, Proof: All the statements of the proposition are proved in [49]. Statement (a) is Theorem 7.21; (b) is Theorem 8.6; (c) is Corollary 11.6 and (d) is Theorem 12.8. ¤ 4.2 Division algebras of degree four Let A be a central simple algebra over a eld F . By the Wedderburn Struc- ture Theorem ([45], Theorem 3.5) the algebra A is isomorphic to Matn(D), where n N and D is a nite dimensional division algebra over F , which by [45], Prop∈osition 12.5b is unique up to isomorphism. If B is another central simple algebra over F isomorphic to Matm(E), where m N and E is a nite dimensional division algebra over F , then A and B are∈ called Morita equivalent if and only if D = E. By [45], Proposition 12.5a, the tensor prod- uct over F induces a group structure on the set of equivalence classes. The group de ned in this way is called the Brauer group. By [45], Theorem 17.10, if F is a local, non-archimedean eld, then there is a canonical isomorphism between the Brauer group of F and the group Q/Z. The Brauer invariant Inv A of A is de ned to be the image in Q/Z of the Morita equivalence class of A under this isomorphism. The Schur index Ind A of A is de ned to be the degree of D. By [45], Corollary 17.10a(iii), the order of the Brauer invariant of A in Q/Z equals Ind A. The only (associative) division algebras over R are R itself, C and the quaternions H (see [45], Corollary 13.1c). Since C is itself a eld, any algebra which is central simple over R cannot be isomorphic to a matrix algebra over C. The Brauer group of R is therefore isomorphic to Z/2Z. Let A be a central simple algebra over R. If A = Matn(R) for some n N, then the Brauer invariant of A is de ned to be zero and the Schur index∈of A is de ned to be one. If A = Matn(H) for some n N, then the Brauer invariant of A is one half and the Schur index of A is t∈wo. Let M be a division algebra of degree 4 over Q. Fix a maximal order M(Z) in M. If R is a commutative ring with unit then we denote by M(R) the R-algebra M(Z) Z R. In particular we have M = M(Q) . For any ring 4. Division Algebras of Degree Four 88 R the reduced norm induces a map det : M(R) R. (We note that in the → case that M(R) = Mat4(R) the reduced norm of an element of M(R) equals its determinant, justifying the choice of notation; see [45], Chapter 16.) Let p be a prime. From Proposition 4.1.1(b) we know that deg M(Qp) = 1 3 deg M(Q) = 4. It then follows that if Inv M(Qp) is equal to 4 or 4 then M(Qp) is a division algebra of degree four over Qp, and if Inv M(Qp) = 0 then M( ) Mat ( ). The other possibility, that Inv M( ) = 1 , does Qp = 4 Qp Qp 2 not interest us here. Proposition 4.1.1(b) also tells us that deg M(R) = 4. We know that Inv M(R) equals zero or one half. It follows that M(R) is isomorphic to Mat4(R) or Mat2(H) respectively. Let S be a nite, non-empty set of prime numbers with an even number of elements. We say that M(Q) splits over a prime p if M(Qp) = Mat4(Qp). For all primes p we de ne i as follows. If p S de ne i to be either 1 or 3 p ∈ p 4 4 in such a way that p∈S ip Z. Note that we are able to do this since we have speci ed that S must con∈ tain an even number of elements. For all other P p de ne ip = 0. Then [45], Theorem 18.5 tells us that we may choose M so that Inv M(Qp) = ip for all primes p and Inv M(R) = 0. We can then see that the set of places at which M(Q) does not split coincides with the set S. More particularly, for p S we have that M(Q ) is a division Q -algebra, for ∈ p p p / S we have that M(Qp) = Mat4(Qp) and we also have M(R) = Mat4(R). ∈For a commutative ring R with unity, let G (R) = x M(R) : det (x) = 1 . { ∈ } Then G is a linear algebraic group, de ned over Z. Let = G (Z), then G forms a discrete subgroup of G = (R) = SL4(R). Since M(Q) is a division algebra, it follows that G is anisotropic over Q and so is cocompact in G (see [3], Theorem A). Let P be the parabolic subgroup P = 0 0 0 0 of G. Then P = MAN, where SL (R) M = S 2 SL (R) µ 2 ¶ X = : X, Y Mat (R), det X = det Y = 1 , Y ∈ 2 ½µ ¶ ¾ 4. Division Algebras of Degree Four 89 A = diag(a, a, a 1, a 1) : a > 0 and the elements of N have ones on the diagonal{ and the only other non-zero} entries in the top right two by two square. Let SO(2) B = . SO(2) µ ¶ Then B is a compact subgroup of M. Let A = diag(a, a, a 1, a 1) : a (0, 1) be the negative Weyl chamber { ∈ } of A. Let EP ( ) be the set of conjugacy classes [ ] in such that is E p conjugate in G to an element a b of A B and let P ( ) be the subset of primitive conjugacy classes. We say g G is regular if its centraliser is a torus and non-regular (or singular) otherwise.∈ Clearly, for regularity is a property of the - E p,reg∈ E p conjugacy class [ ], we denote by P ( ) the regular elements of P ( ). By E p E p,reg abuse of notation we sometimes write P ( ) or P ( ) when we mean the conjugacy class of . ∈ ∈ We will call a quartic eld extension F/Q totally complex if it has two pairs of conjugate complex embeddings and no real embeddings into C. 4.3 Sub elds of M(Q) generated by Lemma 4.3.1 Let [ ] E ( ). ∈ P The centraliser M(Q) of in M(Q) is a totally complex quartic eld if and only if is regular. The centraliser M(Q) is a quaternion algebra over the real quadratic eld Q[ ] if and only if is non-regular. Proof: The centraliser M(Q) of in M(Q) is a division subalgebra of M(Q). Suppose that M(Q) = Q. Then since M(Q) we must have ∈ Q, but then is central so M(Q) = M(Q), a contradiction. Suppose ∈ 4 instead that M(Q) = M(Q), ie. is central. Then Q, so det = = 1 and it follows that = 1, which possibility is excluded∈ since = 1 is not primitive. By Proposition 4.1.1(d) we have that dimQ M(Q) divides dimQ M(Q), which is 16, so dimQ M(Q) is equal to 2, 4 or 8. Also dimQ M(Q) = dimR M(R) = dimR M(R)a b . 4. Division Algebras of Degree Four 90 Depending on whether neither, one or both of the SO(2) components of b are I , the dimension dimR M(R) is equal to 4, 6 or 8 respectively. Hence 2 a b dimQ M(Q) is either 4 or 8. If dimQ M(Q) = 4 then b = (R( ), R( )) with , / Z, where ∈ cos sin R( ) = . sin cos µ ¶ If dimQ M(Q) = 8 then b is diagonal. In the rst case G = AB = R+ SO(2) SO(2) so is regular. From Proposition 4.1.1(d) and Proposition 4.1.2(a) it follows that Q[ ] = f( ) : f(x) Q[x] is a sub eld of M(Q) of degree 4. Let f (x) be the { ∈ } minimal polynomial of over Q. Then a b also satis es f (x) = 0. Now a = diag(a, a, a 1, a 1) for some a (0, 1) and b = (R( ), R( )) for some ∈ , / Z, so the complex numbers z = aei and z = a 1ei also sat- ∈ 1 2 isfy f (x) = 0. Neither z1 nor z2 are real, nor do we have z1 = z 2, so Q[ ] = Q[z1, z2], which is a totally complex eld. The division subalgebra M(Q) of M(Q) has dimension four over Q and contains Q[ ], hence is equal to Q[ ]. Suppose instead that b is diagonal. Then G = AM, so is not regular. From Proposition 4.1.1(d) and Proposition 4.1.2 (a) it follows that Q[ ] = f( ) : f(x) Q[x] is a sub eld of M(Q) of degree 2. Let f (x) be the { ∈ } minimal polynomial of over Q. Then a b also satis es f (x) = 0. Now 1 1 a = diag(a, a, a , a ) for some a (0, 1) and b = ( I2, I2). Let us ∈ 1 suppose that b = I4, the other cases are similar. Then f (a) = f (a ) = 0, so f = (x a)(x a 1) and Q[ ] is a real quadratic eld. The division algebra M(Q) is central simple over its centre Z(M(Q) ). Clearly, Q[ ] is contained in Z(M(Q) ). By Proposition 4.1.2(b), the dimen- sion of M(Q) over its centre is m2 for some m N. Hence either M(Q) ∈ is a eld or Z(M(Q) ) = Q[ ]. However, by Proposition 4.1.2(b) again, if F is a sub eld of M(Q) then [F : Q] = 1, 2 or 4. This rules out the pos- sibility of M(Q) being a eld, since dimQ M(Q) = 8. We conclude that Z(M(Q) ) = Q[ ] and by [45], Theorem 13.1, M(Q) is a quaternion algebra over Q[ ]. ¤ 4.4 Field and order embeddings In this section we prove a number of lemmas which will be needed later. 4. Division Algebras of Degree Four 91 Lemma 4.4.1 Let u M(Z). Then u is a unit in M(Z) i det (u) = 1. If ∈ Q[u] is quadratic or is totally complex quartic, or if u = 1, then det (u) = 1. Proof: By Proposition 4.1.2(a) the set F = Q[u] is a sub eld of M(Q) containing u. The set F M(Z) is an order in F . The element u is in ∩ F M(Z) so by Proposition 4.1.3(b) it is in the integral closure F of Z in F .∩ O The eld norm N Q : F Q can be written as a product over all F | → embeddings of F into C: NF |Q(x) = x. Y Hence we can see that it restricts to a group homomorphism F Q and → maps integral elements in F to elements of Z (see [44]). Suppose there exists v such that vu = 1. Then N Q(v), N Q(u) Z and ∈ OF F | F | ∈ NF |Q(v)NF |Q(u) = NF |Q(uv) = NF |Q(1) = 1, so N Q(u) = 1. On the other hand, suppose there exists n Z such that F | ∈ nNF |Q(u) = 1. Then 1 = n u = yu Y for some y . ∈ OF We have shown that u is a unit in F if and only if NF |Q(u) = 1. Proposition 16.2a of [45] tells us that O k det (u) = NF |Q(u) , (4.1) where deg M(Q) = k[F : Q]. The rst statement of the lemma follows. If u = 1 (so that F = Q) or F is quadratic then by (4.1) we have det (u) = 1. Suppose F is a totally complex quartic extension. Let f(X) = 4 3 2 X + a3X + a2X + a1X + a0 be the minimal polynomial of u over Q. Then det (u) = a0. If a0 = det (u) = 1 then f must have a real root, which contradicts the supposition on F . Hence det (u) = 1. ¤ The following lemma holds for any algebraic extension F of Q. Lemma 4.4.2 Let be an order in the number eld F/Q and p a prime O number. Let F = F Q and = Z . Then is the maximal order p p Op O p Op of Fp for all but nitely many primes p. 4. Division Algebras of Degree Four 92 Proof: Let M be the maximal order of F and let M,p = M Zp. We de ne the conductorO of to be the ideal F = O O of . O { ∈ OM | OM O} OM Then for all primes p, if F * p M , then p = M,p. In fact, if F * p M then there is an element F Osuch that O / p O . Every element of O ∈ ∈ OM OM,p can be written as a unit of M,p times a power of p. Since / p M we have , which implies O = . Then, as F it follo∈ wsO that ∈ OM,p OM,p OM,p ∈ = . OM,p OM,p Op It is clear that p M,p. Since F p M for only nitely many primes p, the lemma folloOws. O O ¤ Let F be a number eld. A prime number p is called non-decomposed in F if there is only one place in F lying above p. Lemma 4.4.3 A number eld F embeds into M(Q) if and only if [F : Q] = 1, 2 or 4 and p is non-decomposed in F for all p S. ∈ Proof: The statement about the degree of F is Proposition 4.1.1 (c). The case F = Q is trivial. Let F be a quartic eld and suppose that F embeds into M(Q), then F is a strictly maximal sub eld of M(Q). We say that a sub eld K of M(Q) is a splitting eld for M(Q) (or that K splits M(Q)) if M(Q) K = Mat (K). 4 Proposition 4.1.2(d) tells us that a quartic eld embeds into M(Q) if and only if it splits M(Q). For p a rational prime, the eld Qp is an extension of Q. Hence, by Proposition 4.1.1(c), the Q -algebra M(Q ) is central simple over Q . Let p p p p ∈ S. The Schur index, Ind M(Qp), of M(Qp) is, by [45], Corollary 17.10a(iii), equal to the order of [M(Qp)] in the Brauer group of Qp, where the square brackets denote the Morita equivalence class. By choice of M(Q) we have Ind M(Qp) = 4. Let pi be the ( nitely many) primes in F above p. Then Theorem 32.15 of [49] tells us that, since F splits M(Q), 4 = Ind M(Q ) [Fp : Q ] p | i p for all i. Corollary 8.4 of Chapter II of [44] says that [F : Q] = [Fpi : Qp], (4.2) i X 4. Division Algebras of Degree Four 93 which implies that there is only one prime p in F above p and that [Fp : Qp] = 4. Conversely, suppose that for each p S there is only one prime p in F ∈ above p. From (4.2) we get that [Fp : Qp] = 4, so Ind M(Q ) [Fp : Q ]. (4.3) p | p For p / S, by choice of M(Q), the Schur index, ind M(Qp), is equal to 1, so in this∈case (4.3) also holds for each p in F above p. Then, by [49], Theorem 32.15, the eld F splits M(Q). By Proposition 4.1.2(d) we then deduce that F embeds into M(Q). Now let F be a quadratic eld and suppose that F embeds into M(Q), then we consider F as a sub eld of M(Q). By Proposition 4.1.2(c), the sub eld F of M(Q) is not maximal so there exists a maximal sub eld K of M(Q) properly containing F , which by the same proposition is quartic. It was shown above that every prime in S is non-decomposed in K and hence also in F . Suppose p is non-decomposed in F for all p S. We choose a quaternion ∈ 1 algebra A over F by specifying that the local Brauer invariant at p be 2 for all places p in F over some p S, and that the local Brauer invariant be 0 at all other places of F . Prop∈ osition 4.1.2(c) tells us that there exists a sub eld K of A containing F with [K : F ] = 2. Then by the same argument as above, if p is a place of F over p for some p S, then p is non-decomposed in K. It follows that p is non-decomposed in ∈K for all p S. It was shown ∈ above that the quartic extension K of Q, and thus also the sub eld F of K, embeds into M(Q). The lemma is proven. ¤ Let A n denote the ring of nite adeles over Q, ie. 0 A n = Qp, p Y where p ranges over the rational primes and 0 denotes the restricted prod- uct, that is, almost all components of a given element are integral. Let Q Zˆ = Z A . There is a natural embedding of Q into A , which maps p p n n q Q to the adele every one of whose components is q. From this embedding ∈ Q we derive an embedding of M(Q) in M(A n) and an embedding of G (Q) in G (A n). Lemma 4.4.4 M(A n) = M(Zˆ) M(Q) . 4. Division Algebras of Degree Four 94 Proof: We have the following commutative diagram with exact rows: det 1 / G (Q) / M(Q) / Q / 1 G det 1 / (A n) / M(A n) / A n / 1 O O O det 1 / G (Zˆ) / M(Zˆ) / Zˆ / 1 where the vertical arrows denote the natural embeddings. The commutativ- ity of the diagram and the exactness of the rows are clear, except that the surjectivity of the three determinant maps requires justi cation. The surjec- tivity of the map det : M(Q) Q follows from [49], Theorem 33.15. For → all primes p we have that M(Qp) is a central simple Qp-algebra so it follows from [49], Theorem 33.4 that the map det: M(Q ) Q is surjective. For p → p p S, we have that M(Q ) is a division Q -algebra and so an element x of ∈ p p M(Q ) is in M(Z ) if and only if det x Z ([49], Theorem 12.5). For p / S, p p ∈ p ∈ we have that M(Qp) = Mat4(Qp) and x 1 det = x 1 1 for all x Z . It follows that the map det: M(Zˆ) Zˆ is surjective. From ∈ p → this we can then see that map det: M(A ) A is also surjective. n → n Let v M(A ) and let w = det v A . Since A = Zˆ Q , there ∈ n ∈ n n exist r Zˆ and q Q such that w = rq. By the surjectivity of det , there ∈ ∈ exist r M(Zˆ) and q M(Q) such that det r = r and det q = q. Let ∈ ∈ v = r 1vq 1. Then det v = 1, so v G (A ). By the Strong Approximation ∈ n Theorem (see [40]), we have that G (Q) is dense in G (A n). Hence there exist rˆ G (Zˆ) and qˆ G (Q) such that v = rˆqˆ. Finally, we have ∈ ∈ v = r v q = r (rˆqˆ)q = (r rˆ)(qˆq ), where r rˆ M(Zˆ) and qˆq M(Q) . The lemma follows. ¤ ∈ ∈ 4. Division Algebras of Degree Four 95 Lemma 4.4.5 Let F be a eld that embeds into M(Q). For each embedding 1 : F M(Q) the order = ( (F ) M(Z)) is maximal at all p S. Conversely,→ if is an orOder in the eld F∩ which is maximal at all p ∈ S, O ∈ then there exists an embedding : F M(Q) such that = . → O O Let : F M(Q) be an embedding of the eld F into M(Q). By Lemma 4.4.3 each prime→ p in the set S is non-split in F . Since for p S there is only ∈ one place in F above p, we can note that the p-adic completion Fp = F Qp is again a eld (see [44], Chapter 1, Proposition 8.3). Firstly we need to show that for all p S, the completion = Z is the maximal order ∈ O ,p O p in Fp. By Proposition 4.1.3(c), M(Zp) is the maximal Zp-order in M(Qp). Since M(Qp) is a division algebra, M(Zp) coincides with the integral closure of Zp in M(Qp), by Proposition 4.1.3(d). We can extend to an embedding of F in M(Q ) and then = 1 ( (F ) M(Z )) is the integral closure p p O ,p p ∩ p of Zp in Fp, which is the maximal order of Fp. For the converse, let be an order of F such that the completion O Op is maximal for each p S. Via we consider F to be a sub eld of M(Q). ∈ For any u M(Q) let = F u 1M(Z)u. Let u = u u 1, that is: ∈ Ou ∩ u (x) = u (x)u 1 for all x F . We will show that there is a u M(Q) such that = . The embedding∈ u is then the one required by the∈ lemma. O Ou Let 1 = F M(Z). Since and 1 are orders, by Lemma 4.4.2 they are both maximalO at∩ all but nitelyO manOy places. Let T be the set of primes p such that either or is not maximal at p. Then T is nite and T S = ∅. O O1 ∩ Furthermore, if TF denotes the set of places of F lying over T , we have that for any place p of F with p / T the completions p and p are maximal ∈ F O O1, in Fp and thus equal, by uniqueness of maximal orders. Hence, for p / T we have ∈ = F M(Z ). Op p ∩ p Let p T , then p / S so M(Q ) = Mat (Q ). Considering F as a ∈ ∈ p 4 p p Qp-vector space we see that we can embed Fp into LinQp (Fp, Fp) by sending an element x F to the linear map a ax. We choose a Z -basis of , ∈ p 7→ p Op which is then also a Qp-basis of Fp. This gives us an isomorphism Lin (F , F ) Mat ( ) M( ). Qp p p = 4 Qp = Qp This isomorphism and the above embedding F , LinQ (F , F ) then give p → p p p us an embedding : F , M(Q ) such that = 1 ( (F ) M(Z )). p p → p Op p p p ∩ p The map p is a Qp-isomorphism of Fp (considered as a sub eld of M(Qp) 4. Division Algebras of Degree Four 96 via a suitable extension of ) onto its image in M(Qp), so by the Skolem- Noether Theorem (Proposition 4.1.3(a)), there exists up M(Qp) such that u xu 1 = (x) for all x F . Hence ∈ p p p ∈ p = F u 1M(Z )u . Op p ∩ p p p For p / T set u = 1 and let u˜ = (u ) A . By Lemma 4.4.4 ∈ p p ∈ n M(A n) = M(Zˆ) M(Q) so there exists an element u M(Q) such that for all primes p we have ∈ 1 1 u M(Zp)u = up M(Zp)up. This implies that = F u 1M(Z )u (4.4) Op p ∩ p for all primes p and we deduce that = F u 1M(Z)u = . O ∩ Ou This completes the proof of the lemma. ¤ 4.5 Counting order embeddings Let F/Q be an algebraic number eld and let be an order in F . Let I( ) be the set of all nitely generated -submoO dules of F . According toOthe Jordan-Zassenhaus Theorem ([49], TheoremO 26.4), the set [I( )] of isomorphism classes of elements of I( ) is nite. Let h( ) be the cardinalitO y of the set [I( )], called the class numbO er of . O Let I be Oa non-trivial, nitely generatedO -submodule of F . Then, by [49], Theorem 10.6, I is isomorphic as an O-module to an ideal of so O O also has the property that I Q = F . If J is another nitely generated -submodule of F , such that J = I, then the isomorphism extends to an automorphismO of F considered as a module over itself, and hence J = Ix for some x F . So h( ) = card([I( )]) = I( )/F . ∈ O O | O | For a prime p Z let Fp = F Qp. Let p1, ...pn be the prime ideals of F above p. By ∈[44], Chapter 1, Prop osition 8.3 there exists a canonical 4. Division Algebras of Degree Four 97 Qp-algebra isomorphism de ned by: n F = F Q = Fp (4.5) p p i i=1 Y x ( ( ).x)n , 7→ i i=1 where Fpi denotes the completion of F with respect to the pi-adic absolute value and the embedding of F into Fp . If is an order in F and p Z a i 1 O ∈ prime then let p = Zp. Recall that OS is aO nite, non-empty set of prime numbers with an even number of elements. Let C(S) be the set of all totally complex quartic elds F such that all primes p in S are non-decomposed in F . Note in particular that by the isomorphism (4.5) this means that for each eld F in C(S) and for each p S we have that F = F Q is once again a eld, namely the ∈ p p completion of F at the unique place of F above p. For F C(S) let OF (S) be the set of isomorphism classes of orders in F which are∈ maximal at all O p S, ie. are such that the completion = Z is the maximal order ∈ Op O p of the eld Fp for all p S. Let O(S) be the union of all OF (S), where F ranges over C(S). ∈ Let F C(S) and O (S), then, by Lemma 4.4.3, F embeds into ∈ O ∈ F M(Q). By Lemma 4.4.1, the group M(Z) of units in M(Z) contains as a subgroup. By Lemma 4.4.5 we know that there is an embedding of F 1 into M(Q) such that = = ( (F ) M(Z)). Let u M(Z) and, u O 1 O ∩ ∈ as above, let = u u for embeddings : F M(Q). Then u = , → O O so the group M(Z) acts on the set ( ) of all embeddings with = . O O O This M(Z) action also restricts to an action of on ( ). We de ne O ( ) = (F ) = f (F ), S O S p p∈S Y where fp(F ) is the inertia degree of p in F . Our aim is to prove the following proposition. Proposition 4.5.1 The quotient M(Z) ( ) is nite and has cardinality equal to the product h( ) ( ). \ O O S O We prepare for the proof of the propostition with the following discussion and then prove some lemmas, from which the proposition will follow. 4. Division Algebras of Degree Four 98 Fix an embedding : F , M(Q) with = and consider F as a → O O sub eld of M(Q) such that = F M(Z). For u M(Q) let O ∩ ∈ = F u 1M(Z)u. Ou ∩ Let U be the set of all u M(Q) such that ∈ = F M(Z) = F u 1M(Z)u = . O ∩ ∩ Ou Let a F , then ∈ F a 1u 1M(Z)ua = a 1 F u 1M(Z)u a = F u 1M(Z)u, ∩ ∩ ∩ so F acts on U by multiplication¡ from the right.¢ Let v M(Z) , then ∈ F u 1v 1M(Z)vu = F u 1M(Z)u, ∩ ∩ so M(Z) acts on U by multiplication from the left. Let ( ), so that = . Then by the Skolem-Noether Theorem (Proposition∈4.1.3(a))O there O O exists u M(Q) such that = u = u u 1, ie. (F ) = uF u 1. Then ∈ = F u 1M(Z)u = u 1 uF u 1 M(Z) u Ou ∩ ∩ = u 1 ( (F ) M(Z)) u ¡ ∩ ¢ = O = . O Conversely, it is clear that for all u U we have u ( ), and that u = v if and only if u = vx for some∈x F . Hence ∈ O ∈ M(Z) U/F = M(Z) ( ) , | \ | | \ O | and the proposition will be proved if we can show that the left hand side equals h( ) S( ). For u O U letO ∈ I = F M(Z)u. u ∩ Then I is a nitely generated -module in F . We shall prove that the map u O : M(Z) U/F I( )/F \ → O u I , 7→ u 4. Division Algebras of Degree Four 99 is surjective and S( ) to one. This will be done in the following lemmas through localisation andO strong approximation. Let p Z be a prime and let I( p) be the set of all nitely generated -submo∈dules of F and [I( )] theOset of isomorphism classes. Let I be a Op p Op p non-trivial, nitely generated p-submodule of Fp. Then, by [49], Theorem 10.6, the submodule I is isomorphicO as an -module to an ideal of so p Op Op also has the property that I Q = F . If J is another nitely generated p p p p -submodule of F , such that J = I , then the isomorphism extends to an Op p p p automorphism of Fp considered as a module over itself, and hence Jp = Ipx for some x F . So card([I( )]) = I( )/F . ∈ p Op | Op p | For a prime p and u M(Q ) let p ∈ p = F u 1M(Z )u Op,up p ∩ p p p and recall that = F M(Z ). Op p ∩ p Let U be the set of all u M(Q ) such that p p ∈ p = . Op,up Op As in the global case above, Fp acts on Up from the right and M(Zp) acts on U from the left. For u U let p p ∈ p I = F M(Z )u . up p ∩ p p Then I is a nitely generated -module in F . up Op p Lemma 4.5.2 Let p be a prime. Then I( )/F = 1. | Op p | Proof: Suppose p S. Then p is non-decomposed in F so there exists a unique prime ideal p∈of F above p, so by the isomorphism (4.5) we can see that Fp = Fp and hence Fp is itself a eld. By [44], Chapter II, Proposition 3.9, every ideal of p is principal and hence all non-zero ideals of p are isomorphic. The claimO holds for p S. O Suppose p / S. Let p , ..., p ∈be the prime ideals of F over p. Under ∈ 1 n the isomorphism (4.5) an ideal in gives ideals in p , ..., p , where p Op O 1 O n O 1 denotes the ring of integers in the eld Fpi . As we saw above, these ideals are each unique up to isomorphism and hence the original ideal in p is also unique up to isomorphism and the claim holds for p / S. O ¤ ∈ We now prove the surjectivity of . 4. Division Algebras of Degree Four 100 Lemma 4.5.3 The map is surjective. Proof: Let I be an ideal and for a prime p let I = I Z . By Lemma O p p 4.5.2, for all primes p the -module F M(Z ) is isomorphic to I , so there Op p ∩ p p exists u F such that p ∈ p F M(Z )u = I . p ∩ p p p We have further that F u 1M(Z )u = u 1(F M(Z )u p ∩ p p p p p ∩ p p = F M(Z ). p ∩ p Let u˜ = (u ) A . By Lemma 4.4.4 there exist z˜ M(Zˆ) and u p ∈ n ∈ ∈ M(Q) such that u˜ = z˜u. Then for all primes p we have that F u 1M(Z )u = F M(Z ) p ∩ p p ∩ p and F M(Z )u = I . p ∩ p p We deduce from this that F u 1M(Z)u = F M(Z) ∩ ∩ and F M(Z)u = I. ∩ This is what was required to complete the proof of the lemma. ¤ In the following series of lemmas we prove that is ( ) to one. S O Lemma 4.5.4 Let K be a non-archimedian local eld and its ring of OK integers. For n N let n Matn(K) be the subspace of diagonal matrices and suppose that∈u GL (K ) is such that ∈ n Mat ( ) u 1Mat ( )u. n ∩ n OK n ∩ n OK Then there exist z Matn( K ) and x n such that zu = x. If we suppose further∈ thatO ∈ Mat ( ) = Mat ( )u n ∩ n OK n ∩ n OK then we have x ( Mat ( )) . ∈ n ∩ n OK 4. Division Algebras of Degree Four 101 Proof: The rst claim of the lemma will be proved by induction on n. For n = 1 we take z = 1 and x = u and the claim holds. Assume now that the claim holds for some n 1 N. For any ring R let Upp (R) denote ∈ n the subalgebra of upper triangular matrices in Matn(R). We can choose an 0 element y Matn( k) such that u = yu Uppn(K) is upper triangular. Then u0 has∈ the formO ∈ a b u0 = , c µ ¶ where a K, b Mat (K) and c Upp (K). Then ∈ ∈ 1 (n 1) ∈ n 1 a 1 a 1bc 1 (u0) 1 = . c 1 µ ¶ Since u satis es Mat ( ) u 1Mat ( )u n ∩ n OK n ∩ n OK it follows that u0 satis es Upp ( ) (u0) 1Upp ( )u0. n ∩ n OK n ∩ n OK It follows that there exist K , Mat1 (n 1)( K ) and Uppn 1( K ) such that ∈ O ∈ O ∈ O a 1( b + c bc 1 c) 1 = (u0) 1 u0 = , c 1 c 0 µ ¶ µ ¶ µ ¶ where the matrix on the right is the n by n matrix with a one in the top left corner and all other entries zero. It then follows that = 1, c 1 c = 0 and b = c. Let 1 w = Matn( K ) . In 1 ∈ O µ ¶ Then 1 a c a wu0 = = . I c c µ n 1 ¶ µ ¶ µ ¶ so we have that a wyu = . c µ ¶ It is then clear that c satis es Mat ( ) c 1Mat ( )c n 1 ∩ n 1 OK n 1 ∩ n 1 OK 4. Division Algebras of Degree Four 102 0 0 so by the inductive hypothesis there exist z Matn 1( K ) and x n 1 such that z0c = x0. Setting ∈ O ∈ 1 a z = wy Mat ( ) and x = z0 ∈ n OK x0 ∈ n µ ¶ µ ¶ we get that zu = x and the proof of the rst claim is complete. If we also assume that u satis es Mat ( ) = Mat ( )u n ∩ n OK n ∩ n OK then since z Mat ( ) and zu = x it follows that ∈ n OK ∈ n Mat ( ) = Mat ( )zu = ( Mat ( ))x. n ∩ n OK n ∩ n OK n ∩ n OK This can be the case only if x ( n Matn( K )) so the lemma is proved. ∈ ∩ O ¤ Lemma 4.5.5 For p / S we have M(Z ) U /F = 1. ∈ | p \ p p | Proof: Let p / S and let u U so that ∈ p ∈ p F M(Z ) = F u 1M(Z )u . (4.6) p ∩ p p ∩ p p p By Lemma 4.5.2, there exists F such that p ∈ p F M(Z ) = F M(Z )u . p ∩ p p ∩ p p p Replacing up with up p we may assume that up satis es F M(Z ) = F M(Z )u . (4.7) p ∩ p p ∩ p p We shall show that there exist z M(Z ) and x F such that z u x p ∈ p p ∈ p p p p is the identity element in M(Qp). Let ε be an integral element of F generating F over Q, and let L be an extension of Q containing all the zeros of the minimal polynomial of ε. Let L = L Q , note that neither this nor F are necessarily elds, rather p p p a nite direct product of elds. The embedding F , M(Q) gives us an → embedding Fp , M(Qp). Since p / S we have M(Qp) = Mat4(Qp) so we get an embedding→ ∈ F L , M(Q ) L = Mat (L ). p → p 4 p 4. Division Algebras of Degree Four 103 Let be the ring of integers in L and let = Z . Let denote OL OLp OL p the subspace of diagonal matrices in Mat4(Lp). It follows from our choice of L that ε is diagonalisable in M(L ) by an element A M( ) . Since p ∈ OLp Fp = Qp(ε), the whole of Fp L is simultaneously diagonalisable in M(Lp), 1 that is, we have that A (Fp L)A is a subspace of , and by dimensional 1 considerations we must have that A (Fp L)A = . Then, tensoring (4.6) and (4.7) with and conjugating by A w e get OL M( ) = u 1M( )u (4.8) ∩ OLp ∩ p OLp p and M( ) = M( )u , (4.9) ∩ OLp ∩ OLp p where u p = upA. We are currently working in the matrix algebra M(Lp) = Mat4(Lp). Let p1, ..., pn be the prime ideals of L above p. By virtue of the isomorphism (4.5) we can consider separately each pi-adic component of the entries of the matrices. It then follows from Lemma 4.5.4 and equations (4.8) and (4.9) that there exist z M( Lp ) and x ( M( Lp )) such that z u p = x . Setting ∈ O ∈ ∩ O z = Az M( ) ∈ OLp and x = Ax A 1 ((F L) M( )) ∈ p ∩ OLp we get that zup = x. Consider now the trace map tr L/Q of the eld extension L/Q. The image of L under this map is an ideal in Z. Let Z be the greatest such that O ∈ tr Q( ) p Z. We de ne a linear map L/ OL T : M(Lp) = M(Qp) L M(Qp) Q = M(Qp) → x y x p (tr Q y). 7→ L/ Note that T (M( )) = T (M(Z ) ) M(Z ) Z = M(Z ). OLp p OL p p We denote by the ring of integers of the eld Lp . Let (F Lp ) OL1 1 ∈ 1 ∩ M( L1 ) and let x1 and z1 be the p1 components of x and z. Then z1up = x1. O 1 Now set = x (F Lp ) M( ). From (4.6) we deduce that 1 ∈ 1 ∩ OL1 1 (F Lp ) M( ) = (F Lp ) u M( )u 1 ∩ OL1 1 ∩ p OL1 p 4. Division Algebras of Degree Four 104 1 so there exists M( L1 ) such that = up up, or equivalently up = up. Setting = z ∈ M(O ), we then get 1 ∈ OL1 up = z1 up = z1up = x1 = . 1 Hence we can see that up M( L1 ). We now consider the linear∈ mapO de ned by 0 T : M(Q) Lp1 M(Q) Qp = M(Qp) → x y x p (tr L /Q y), 7→ p1 p where tr is the trace map of the eld extension L / . We note that Lp1 /Qp p1 Qp 0 0 T maps (F Lp1 ) M( Lp ) to Fp M(Zp). The map T induces a linear map ∩ O ∩ M( ) M(Z ) T 00 : OL1 p . p M( ) → pM(Z ) 1 OL1 p Note that M( )/p M( ) is a vector space over the eld /p OL1 1 OL1 OL1 1OL1 and M(Zp)/pM(Zp) is a vector space over the eld Zp/pZp = Fp. By the choice of the integer we can see that the map T 00 is not the zero map, that is, its kernel is a proper subspace of M( L1 )/p1M( L1 ). An element O00 O of M( L1 )/p1M( L1 ) not in the kernel of T corresponds to an element O O 0 of M( L1 ) whose image under T is in M(Zp) . We choose an element O 0 0 1 ∈ (F Lp ) M( ) such that T ( ) (F M(Z )) and T ( u ) M(Z ) . 1 ∩ OL1 ∈ p∩ p p ∈ p We shall write also for the element of (Fp L) M( Lp ) with p1 component equal to and all other components zero. Similarly∩ O we denote 1 by the element of M( Lp ) with p1 component equal to up and all other O 0 components zero. Then up = . From the de nitions of the maps T and T and the isomorphism (4.5) it follows that T ( ) = T 0( 0) (F M(Z )) ∈ p ∩ p and T ( ) = T 0( ) M(Z ) . ∈ p Now set xp = T ( ) (Fp M(Zp)) and zp = T ( ) M(Zp) . We then have that ∈ ∩ ∈ zpup = xp and the lemma is proved. ¤ 4. Division Algebras of Degree Four 105 Lemma 4.5.6 For p S we have M(Z ) U /F = f (F ). ∈ | p \ p p | p Proof: Let p S. We then have that M(Q ) is a division algebra so in ∈ p particular M(Q ) = M(Q ) r 0 . Further, since Q is a local eld, M(Z ) p p { } p p is the unique maximal order of M(Qp) (cf.Proposition 4.1.3(d)) and hence u 1M(Z )u = M(Z ) for all u M(Q ) . Thus the claim is equivalent to p p p p p ∈ p M(Z ) M(Q ) /F = f (F ). | p \ p p | p Let vM be the p-adic valuation on M(Qp) and let e = e(M(Q )/Q ) = v (M(Q ) )/v (Q ) M p p | M p M p | be the rami cation index of Qp in M(Qp). The valuation vM is a surjective group homomorphism of M(Qp) onto Z with kernel M(Zp) . It follows that e = M(Z ) M(Q ) /Q . M | p \ p p | By Proposition 17.7 of [45] we have M(Z ) M(Q ) /Q = e = deg M(Q ) = 4. | p \ p p | M p Let ep(F ) be the rami cation index of p in F . Then by [44], Chapter I, Proposition 8.2, and since p does not split in F , we have ep(F )fp(F ) = [F : Q] = 4. (4.10) Let O be the valuation ring of Fp and let = O be the unique maximal ideal of O, where O is a generator of the principal ideal . Let p = pZ ∈ p be the unique maximal ideal of Z . Then p = Z and p = ep(F ). Thus p ∩ p vM ( ) = vM (p)/ep(F ), which implies that M(Zp) M(Qp) /Qp 4 M(Zp) M(Qp) /Fp = | \ | = , | \ | ep(F ) ep(F ) and hence from (4.10) we get M(Z ) M(Q ) /F = f (F ), | p \ p p | p as required. ¤ 4. Division Algebras of Degree Four 106 Lemma 4.5.7 is ( ) to one. S O ˆ 0 Proof: Recall that OF (S) for some F C(S). Let F = p Fp be the restricted product ovOer ∈all primes p, that is, all∈ but nitely many components Q are integral, and let Uˆ be the set of all u˜ = (up) M(A n) such that 1 ∈ p = Fp up M(Zp)up = p,up for all p. O For I∩ I( ) let U =O 1(IF ). Let u U for some I I( ). We ∈ O I ∈ I ∈ O denote by uˆ the element of M(A n) each of whose components is u. Setting up = u for all primes p, we see that for all p = Z = Z = Op O p Ou p Op,up so uˆ Uˆ. For∈ each I I( ) we de ne the map ∈ O : M(Z) U /F M(Zˆ) Uˆ/Fˆ I \ I → \ by setting I (M(Z) uF ) = M(Zˆ) uˆFˆ . We shall show that I is a bijec- tion. Since from Lemmas 4.5.5 and 4.5.6 we know that M(Zˆ) Uˆ/Fˆ = ( ), the lemma will then follow. | \ | S O First we show surjectivity. Let u˜ = (u ) Uˆ, so that = for all p ∈ Op,up Op primes p. For all primes p, by Lemma 4.5.2 we have I Q = F M(Z )u p p ∩ p p so there exists x F such that I Q = F M(Z )u x . We replace u p ∈ p p p ∩ p p p p with u x . By Lemma 4.4.4 there exist z˜ = (z ) M(Zˆ) and u M(Q) p p p ∈ ∈ such that u˜ = z˜uˆ. It follows that p,u = p for all primes p and hence we get that = , so that u U. FOurthermore,O since Ou O ∈ I Q = F M(Z )u = F M(Z )u p p ∩ p p p ∩ p for all primes p, it follows that I = F M(Z)u, so u U . We have proven ∩ ∈ I that I is surjective. For the proof of injectivity, suppose that u, v U are such that (u) = ∈ I I (v). The assumption that u, v U means that there is an isomorphism I ∈ I of -modules F M(Z)u = F M(Z)v. As we saw above this implies that O ∩ ∩ there exists x F such that F M(Z)u = F M(Z)vx. Replacing v with vx it follows that∈ ∩ ∩ Fˆ M(Zˆ)uˆ = Fˆ M(Zˆ)vˆ. (4.11) ∩ ∩ The assumption that I (u) = I (v) means that there exist z˜ M(Zˆ) and x˜ Fˆ such that uˆ = z˜vˆx˜. Together with (4.11) this implies that∈ ∈ Fˆ M(Zˆ)vˆ = Fˆ M(Zˆ)vˆx˜ ∩ ∩ 4. Division Algebras of Degree Four 107 and hence x˜ vˆ 1M(Zˆ) vˆ. In other words, there exists y˜ M(Zˆ) such that vˆx˜ = y˜vˆ.∈We deduce that ∈ uˆ = z˜vˆx˜ = (z˜y˜)vˆ. Replacing z˜ with z˜y˜ M(Zˆ) , we now have that ∈ z˜ = uˆvˆ 1 M(Zˆ) M(Q) = M(Z) . ∈ ∩ Writing z = z˜ M(Z) we get that u = zv and injectivity is proven. ¤ ∈ Chapter 5 Comparing Geodesics and Orders In this nal chapter we complete the proof of the Main Theorem. We do this by using our results on orders in sub elds of division algebras of degree four to interpret the Prime Geodesic Theorem algebraically. This will then yield information about the class numbers of those orders. All notation is as de ned in previous chapters. In particular we have G = SL4(R) and take = G (Z) as in Chapter 4. 5.1 Regular geodesics Let F be a number eld with r real embeddings and s pairs of complex conjugate embeddings and let t = r + s 1. Let F denote the ring of O integers of F and F the unit group therein. By Dirichlet’s unit theorem ([44], I Theorem 7.4),O there exist units ε , ..., ε such that every ε 1 t ∈ OF ∈ OF can be written uniquely as a product ε = ε 1 ε t , 1 t where is a root of unity and i Z. The set ε1, ..., εt is not uniquely determined. We shall refer to such ∈a set of units as{ a system} of fundamental units. If F is an order of F and ε1, ..., εt a system of fundamental O O { i } units then there exist i N and units ε i = εi such that every ε can be written uniquely as a ∈product ∈ O ε = ε 1 ε t 1 t 108 5. Comparing Geodesics and Orders 109 where is a root of unity and i Z. Once again, the set ε 1, ..., ε t is not uniquely determined. We shall call∈such a set of units a system{ of fundamental} units of . O If 1, ..., t+1 are distinct embeddings of F into C which are pairwise non- conjugate, then the regulator R( ) of an order F is de ned as the absolute value of the determinanOt of an arbitraryO minorO of rank t of the matrix (log (ε ) ) , | i j | ij where 1 i t + 1 and 1 j t. The value of the regulator does not depend on the choice of a system of fundamental units. We shall sometimes refer, somewhat imprecisely, to the elements of a system of fundamental units simply as fundamental units. In the case t = 1, let ε1 be a fundamental unit of an order in F . We shall refer to the elements of the set ε 1 : root of unity inOF as the fundamental units of { 1 } , or in the case that = F the fundamental units of F . O In the case that F isO a totallyO complex quartic eld we have t = 1, so we can choose a fundamental unit ε such that for all ε we have ε = ε 1 , 1 ∈ OF 1 where is a root of unity and 1 Z. Then ε1 is determined up to inversion and multiplication by a root of unit∈ y. In this case the regulator of an order of F with fundamental unit ε satis es O OF 1 R( ) = log (ε ) , O | | 1 | | where is any embedding of F into C. We consider sub elds of the eld F . Apart from Q these are all quadratic. By Dirichlet’s unit theorem, real quadratic elds contain a single fundamental unit (up to inversion and multiplication by 1) and complex quadratic elds contain no fundamental unit. We can see then that F can have at most one real sub eld. Let Cr(S) and Cc(S) be respectively the subsets of C(S) consisting of elds with and without a real quadratic sub eld. Let Or(S) and Oc(S) be respectively the subsets of O(S) consisting of orders in elds in Cr(S) and Cc(S). Then C(S) = Cr(S) Cc(S) and O(S) = Or(S) Oc(S). p,reg ∪ ∪ Let [ ] E ( ). Then by Lemma 4.3.1 the centraliser M(Q) is a ∈ P totally complex quartic eld which embeds into M(Q) and we denote this sub eld of M(Q) by F . We say that an element is weakly neat if the ∈ E p,rw adjoint Ad( ) has no non-trivial roots of unity as eigenvalues. Let P ( ) be the set of primitive, regular, weakly neat elements in EP ( ). 5. Comparing Geodesics and Orders 110 E p,reg c Lemma 5.1.1 Let [ ] P ( ). Then F C (S) if and only if [ ] E p,rw ∈ ∈ ∈ P ( ). Proof: Let be the maximal order in F and let denote the group O O of units. By Dirichlet’s unit theorem we can see that F contains a real quadratic sub eld if and only if R = 1 . Let be a fun- O ∩ 6 { } 1 ∈ O damental unit. The element is a unit F so there exists a root of unity n and n Z such that = 1 . If there also exist a root of unity and ∈ m m Z r 0 such that 1 R then 1 is not weakly neat and so neither is . ∈ { } ∈ Conversely, suppose is not weakly neat. We know that is conjugate in G to an element a b A B. From the assumption that is not weakly ∈ neat it follows that there exists m N such that one of the components of m ∈ m b is diagonal. Since the group AB is abelian we have that is conjugate m m m E m in G to a b . Since P ( ) it follows from Lemma 4.3.1 that b is m ∈ diagonal and hence generates a real quadratic sub eld of F . The lemma follows. ¤ De ne the map : E p,rw( ) Oc(S) P → by : [ ] F M(Z). 7→ ∩ 1 Let , . Then F 1 = F , and since by Lemma 4.4.1 the group ∈ is a subgroup of the multiplicative group M(Z) , we can conclude that 1 F 1 M(Z) = (F M(Z)) = F M(Z). ∩ ∩ ∩ Hence the map is well de ned. Proposition 5.1.2 The map is surjective. Proof: Let Oc(S) be an order in the eld F Cc(S). Since deg F = 4, O ∈ ∈ Proposition 4.1.2(b) implies that F is a maximal sub eld of M(Q). Lemma 4.4.5 tells us that there exists an embedding : F M(Q) such that → = = 1 ( (F ) M(Z)) . O O ∩ Let ε be a fundamental unit in . Let be the image of ε under the O embedding . The unit ε is integral in F , so M(Z). ∈ 5. Comparing Geodesics and Orders 111 First we show that a b for some a A , b B. The eld F G ∈ ∈ has two pairs of conjugate embeddings into C. Let 1, 2 be two distinct, non-conjugate embeddings of F into C. The R-algebra F Q R is isomorphic to the commutative R-algebra C C via the map x (x ( ), x ( )) . 7→ 1 2 We note that Z R = F Q R and we get the series of inclusions O Z R = F Q R M(Q) Q R = Mat (R) Mat (C). O O 4 4 Thus we see that may be considered as an element, which we will denote , of a commutative subalgebra of Mat (C) isomorphic to C C. The matrix 4 has real entries and can be diagonalised in Mat4(C) to the matrix 1( ) ( ) X = 1 , 2( ) ( ) 2 since the eigenvalues of are the roots of its minimal polynomial, ie. the values of under its di erent embeddings into C. Let a, b R+; , ∈ ∈ [ , ] be such that ( ) = aei and ( ) = bei . Since M(Z), we 1 2 ∈ have ( ) = N Q( ) = det = 1, where N Q is the eld norm and the F | F | product is over all embeddings of F into C ([44], I Proposition 2.6(iii)). HenceQa2b2 = 1. But a and b are both real so we must have a2b2 = 1, and so b = a 1 and we have aei ae i X = 1 i . a e 1 e i a Without loss of generality we assume a < b so that a (0, 1). Let R be the 2 2 matrix ∈ 1 i 1 √2 1 i µ ¶ and Y the 4 4 matrix R Mat (C) R ∈ 4 µ ¶ 5. Comparing Geodesics and Orders 112 Then X is conjugate in Mat4(C) via Y to the real matrix aR( ) X0 = A B. 1 R( ) ∈ µ a ¶ So is conjugate in Mat4(C) to an element of A B, hence by [41], XIV 0 0 1 0 Corollary 2.3, the matrix is conjugate in Mat4(R) to X , that is (Z ) Z = 0 0 0 0 X for some Z Mat4(R). Since Z is invertible we have that n = det Z = 0. 1 ∈0 1 0 6 Let Z = n 4 Z . Then det Z = 1 and Z Z = X , so that is conjugate | | in SL4 (R) to an element of A B. Suppose that det Z = 1. Let W be the matrix 0 1 1 0 W = . 1 0 0 1 Then ZW G and (ZW ) 1 ZW = X00, where ∈ aR( ) X00 = A B. 1 R( ) ∈ µ a ¶ Hence is conjugate in G to an element of A B. We also saw above that det = 1 and so and hence EP ( ). It then follows from Lemma 4.3.1 that , and hence∈ also ε, generates∈ either a real quadratic or a totally complex quartic eld over Q. The eld generated by ε over Q is a sub eld of F . However F C c(S) so has no real quadratic ∈ sub elds. Hence Q( ) = Q(ε) is a totally complex quartic eld and Lemma 4.3.1 tells us that is regular. It then follows from Lemma 5.1.1 that is weakly neat. It remains to show that is primitive in . Note that Q(ε) is a quartic eld contained in and hence equal to the eld F . Recall that we write F = M(Q) for the centraliser of in M(Q). Then we have that F = Q( ) = F and = F M(Z) . The fundamental unit ε is primitive in and so O ∩ O = (ε) is primitive in F M(Z) . Suppose is not primitive in . Then ∩ there exists 0 M(Z) such that = 0 for some N. However we ∈ ∈ also have 0 F and so 0 F M(Z) , which is a contradiction. Hence ∈ ∈ ∩ E p,reg is primitive in and we have P ( ). This concludes the proof of the∈surjectivity of . ¤ Lemma 5.1.3 Let be a torsion element of M(Z). Then det ( ) = 1. 5. Comparing Geodesics and Orders 113 Proof: By Proposition 4.1.2(b), the element generates a sub eld Q( ) of M(Q) of degree 1, 2 or 4 over Q. If [Q( ) : Q] = 1 or 2 then by Lemma 4.4.1 we have det ( ) = 1. For a primitive nth root of unity , the degree [Q( ) : Q] is equal to ϕ(n) = # 1 m n : (m, n) = 1 . From [9], Chapter II, 2 we see that { } § r ϕ(n) = pai 1(p 1), i i i=1 Y a1 ar where n = p1 ...pr is the prime decomposition of n. It is then straightfoward to show that [Q( ) : Q] = 4 if and only if n = 5, 8, 10 or 12. In these cases the minimal polynomial of is respectively x4 + x3 + x2 + x + 1, x4 + 1, x4 x3 + x2 x + 1 or x4 x2 + 1. In each of these cases we read o from the constant term of the minimal polynomial that NQ( )|Q( ) = 1. The lemma follows. ¤ For O(S) let O be the number of roots of unity in the order . For [ ] O E∈p,reg( ) let denote the number of roots of unity in the orderO ∈ P F M(Z). Note that F M(Z) is equal to the centraliser M(Z) of in ∩ ∩ M(Z), and by Lemma 5.1.3 the roots of unity in M(Z) are all in and hence in . It follows that is equal to the cardinality of the torsion part of . It is also immediate from the de nitions that, writing = F M(Z), we O ∩ have = O . Lemma 5.1.4 Let O(S) be an order in the eld F C(S) and let O ∈ ∈ f(x) be the minimal polynomial over Q of a fundamental unit in . Then O the number of roots of f(x) in F which are also fundamental units in is independent of the choice of fundamental unit. We call this number O( ). Note that ( ) is equal to 1,2 or 4. O O Proof: Let ε be a fundamental unit in with minimal polynomial f(x) O over Q. Suppose there exists another fundamental unit with = ε which is also a root of f(x). Then there exist embeddings ∈andO of F in6 to C such that (ε) = ( ). Let also be a fundamental unit in with minimal polynomial ∈ O O g(x) over Q. Then = ε 1 for some root of unity . We have ∈ O ( ) = ( ε 1) = ( ) (ε) 1 = ( ) ( ) 1. 5. Comparing Geodesics and Orders 114 Let n be the order of the root of unity . The order is a ring so contains all roots of unity of order n and hence there exists an Onth root of unity such that ( ) = ( ). Let be the fundamental unit 1 . ∈ThenO ( ) = ( ) and so is also a root of g(x). The lemma follows.∈ O ¤ 4h(O) S (O) O Proposition 5.1.5 The map is (O) to one. Proof: The question is, given O(S), how many conjugacy classes [ ] E p,reg O ∈ ∈ P ( ) are there such that F M(Z) = . Suppose is an order in the eld F . We saw in the proof∩of the previousO propositionO that given an embedding : F , M(Q) such that = and a fundamental unit ε → O O E p,reg∈ O the element = (ε) satis es both F M(Z) = and [ ] P ( ). Conversely, it is clear that for [ ] E p,reg( ),∩ the elemenO t is a fundamen∈ tal ∈ P unit in the order F M(Z). ∩ Recall that M(Z) acts on the set ( ) from the left and that, by O Proposition 4.5.1, the cardinality of the set M(Z) ( ) is h( ) ( ). If \ O O S O ε, are fundamental units and , are embeddings of F into M(Q) such ∈ O that = = , then (ε) and ( ) are in the same M(Z) -conjugacy O O O class if and only if there exists an automorphism of F such that ( ) = ε and the embeddings and are in the same class modulo M(Z) . p,reg We will show that for each E ( ) the M(Z) -conjugacy class of ∈ P decomposes into two -conjugacy classes. If , are in the same M(Z) - ∈ conjugacy class we shall write [[ ]] = [[ ]]. For , if is central in M(Q) then clearly the conjugacy class [[ ]] = . Con∈versely, if [[ ]] = then { } { } commutes with every element of M(Z) and, since M(Z) is an order in M(Q), it follows that commutes with every element of M(Q). Hence the conjugacy class [[ ]] consists of a single element if and only if is central in M(Q). Note that, by [49], Theorem 34.9, the image of M(Z) under the map det is equal to Z, so there exist elements of M(Z) whose image under det is 1. p,reg Let E ( ). Then is not central in M(Q) so there exists = ∈ P 6 in [[ ]] such that = 1 for some M(Z) with det ( ) = 1. Suppose further that [ ] = [ ], so that =∈ 1 for some . Then = 1 1 , so = 1 is an element of F with det ( ) ∈= 1. By Lemma 4.4.1 we then have F M(Z) , where = F M(Z) is an ∈ ∩ O ∩ order in the totally complex quartic eld F and = F M(Z) is the O ∩ group of units. But since F is totally complex quartic, Lemma 4.4.1 tells us that det ( ) = 1 and hence, by contradiction, [ ] = [ ]. This shows that each 6 5. Comparing Geodesics and Orders 115 E p,reg M(Z) -conjugacy class of elements in P ( ) decomposes into at least two -conjugacy classes. Suppose that [[ ]] = [[ ]] but [ ] = [ ]. Then there exists M(Z) such that = 1 and det ( ) = 1. 6Let also be such that∈ [[ ]] = [[ ]] ∈ but [ ] = [ ]. Then there exists M(Z) such that = 1 and det ( ) =6 1. Then = 1 1∈, where det ( 1) = 1 so [ ] = [ ]. E p,reg Hence each M(Z) -conjugacy class of elements in P ( ) decomposes into precisely two -conjugacy classes. We conclude that the choice of a fundamental unit ε gives us E p,reg ∈ O 2h( ) S( ) -conjugacy classes [ ] P ( ) such that F M(Z) = . OWe haOve seen that the choice of fundamen∈ tal unit is determined∩ up to O inversion and multiplication by a root of unity so there are 2 O choices for ε. We saw in the previous proposition that for a given embedding of F into M(Q) and choice of a fundamental unit ε there is a conjugacy class E p,reg ∈ O [ ,ε] P ( ) with (ε) = ,ε. However, we saw above that if there exists an automorphism∈ of F such that = (ε), then setting = 1 we have [ ,ε] = [ , ]. Such an automorphism exists if and only if is a root of the minimal polynomial of ε over Q. The claim of the proposition folows. ¤ E p,rw Lemma 5.1.6 Let P ( ) so that is conjugate in G to a b A B where ∈ ∈ a a R( ) a = and b = a 1 R( ) µ ¶ a 1 for some a (0, 1), , R. If ( ) > 1 then either + or is in ∈ ∈ O Z Z. 2 ∪ 3 Proof: The element is a fundamental unit in . We shall write f(x) for its minimal polynomial. The roots of f(x) are Othe eigenvalues of , which i i 1 i 1 i are 1 = ae , 2 = ae , 3 = a e , 4 = a e . Suppose ( ) > 1. Then there exist i, j 1, 2, 3, 4 , i = j and a root O 1 ∈ { } 1 6 of unity such that i = j . If i = j then i j = and either i, j = 1, 2 or i, j = 3, 4 . Without loss of generality take i = 1 and { } { } { }1 {i2 } n j = 2, then = 1 2 = e . It follows that if n N is such that = 1 then 2n = an (0, 1) and so F has a real quadratic∈ sub eld. However, by 1 ∈ 5. Comparing Geodesics and Orders 116 E p,rw Lemma 5.1.1 this contradicts the assumption that P ( ), so we must 1 ∈ have i = j . By considering the possible values of i and j which would satisfy i = 1 j we see that either + or + is equal to q , where q Q is such that = eiq . It remains to show what the possible values for∈q are, that is, which roots of unity may occur in . We saw in the proof of Lemma 5.1.3 that the only roots of unity whichOmay occur in a quartic eld are 1 and roots of order 3, 4, 5, 6, 8, 10 or 12. If is a root of order 5, 8, 10 or 12 then generates a totally complex quartic eld, however, Q( ) has the real quadratic sub eld Q( + 1), which possibility is excluded in our case. So we are left with the possibility that = 1 or is a root of order 3, 4 or 6, which implies the claim of the lemma. ¤ Lemma 5.1.7 Let [ ] E p,reg( ). Then ∈ P 1 1( ) = . Proof: By Theorem 1.4.2, 0 ,A : ( ) = ,A , 1 0 £ : ¤ where 0 is a torsion free subgroup£ of nite¤ index. In the case under consideration we have that is isomorphic as a group to the group of norm one elements in the order = F M(Z). By Lemma 4.4.1 this is equal O ∩ to the group of units . By Dirichlet’s unit theorem and Lemma 5.1.3 it follows that O = εZ ( ), for some generator ε and where ( ) is the nite cyclic group of roots of unity in . It then follows that O 0 kZ = ε , for some k N. Since every torsion element in is sent to the identity ∈ under the projection onto ,A we also have the isomorphisms Z ,A = ε 5. Comparing Geodesics and Orders 117 and 0 kZ ,A = ε . The lemma follows. ¤ E p,reg Let [ ] P ( ). Then is conjugate in G to a matrix a b , where ∈ 1 1 a = diag(a, a, a , a ) for some 0 < a < 1 and R( ) b = R( ) µ ¶ for some , . Recall that the length l of is de ned to be 8 log a and l | | N( ) = e . For O(S) let εO be a fundamental unit of and R( ) = O ∈ 4R(O) O O 2 log εO the regulator of and let r( ) = e . Under the map , the elemen| | t || corresponds to aOfundamentalO unit of the order ([ ]) and it is clear that r ( ([ ])) = N( ). We recall from the proof of Lemma 4.3.1 that ([ ]) is an order in the eld Q(aei , a 1ei ). By Lemma 2.4.2 we have that tr ˜( ) = 4(1 cos 2 )(1 cos 2 ). Let ( ) ( ) = 1 , ( ) Y µ | |¶ where the product is over the embeddings of F into C. A simple calculation then shows that tr ˜( ) = (1 ei )(1 e i )(1 ei )(1 e i ) = ( ). We summarise the results of this section in the following: 4h(O) S (O) O Proposition 5.1.8 The map is surjective and (O) to one. For E p,reg [ ] P ( ) we have that 1( ) = 1/ , that tr ˜( ) = ( ) and that N( ∈) = r ( ([ ])). 5.2 Class numbers of orders in totally com- plex quartic elds We are now in a position to prove our main theorem. 5. Comparing Geodesics and Orders 118 Theorem 5.2.1 (Main Theorem) Let S be a nite, non-empty set of prime numbers with an even number of elements. For x > 0 let (x) = ( )h( ), S S O O O∈Oc(S) RX(O) x Then, as x we have → ∞ e4x (x) . S 8x Proof: By [2], Proposition 17.6 we know that has a weakly neat subgroup 0 which is of nite index. Hence there exists n N such that all roots of unity which are eigenvalues of Ad( ) for some ∈ have order less than or ∈ equal to n . De ne the sets R( ) B0 = B : , / Z Z Z 1 R( ) ∈ ∈ ∪ 2 ∪ ∪ n ½µ ¶ ¾ and R( ) B0 = B0 B : ( + ), ( ) / Z Z . 2 1 ∪ R( ) ∈ ∈ 2 ∪ 3 ½µ ¶ ¾ For x > 0 de ne the functions rw rw 1 (x) = 1( ) and 2 (x) = 1( ). p,rw p,rw [ ]∈E ( ) [ ]∈E ( ); (O )=1 P P NX( ) x NX( ) x Setting B0 = B0 in Theorem 3.0.4 we get as x 1 → ∞ 2x rw(x) . (5.1) 1 log x Setting B0 = B0 in Theorem 3.0.4 we get, using Lemma 5.1.6, as x 2 → ∞ 2x rw(x) . (5.2) 2 log x We also de ne the following functions for x > 0: ( ) (x) = S O h( ) and (x) = ( )h( ). S,1 ( ) O S,2 S O O O∈Oc(S) O O∈Oc(S); (O )=1 RX(O) x RX(O) x 5. Comparing Geodesics and Orders 119 By Proposition 5.1.8 and the fact that N( ) = e4R(O ) we deduce from (5.1) that e4x (x) (5.3) S,1 8x and from (5.2) that e4x (x) . (5.4) S,2 8x For x > 0 de ne ( ) 0 (x) = (x) (x) = S O h( ). S,2 S,1 S,2 ( ) O O∈Oc(S); (O )>1 O RX(O) x Then from (5.3) and (5.4) it follows that e4x 0 (x) = o . S,2 x µ ¶ Since ( ) 4 for all Oc(S) we have O O ∈ (x) (x) (x) + 4 0 (x) S,2 S S,2 S,2 and the theorem follows. ¤ We can also prove the following: Theorem 5.2.2 Let S be a nite, non-empty set of prime numbers with an even number of elements. For x > 0 let ˜ (x) = ( ) ( )h( ), S O S O O O∈Oc(S) RX(O) x where 1 (ε) ( ) = 1 , O 2 (ε) O ε X Y µ | |¶ where the sum is over the 2 di erent fundamental units of and the O O product is over the embeddings of into C. O Then, as x we have → ∞ e4x (x) . S 2x 5. Comparing Geodesics and Orders 120 E p,rnw E p,reg E p,rw Proof: Let P ( ) = P ( )r P ( ) be the set of regular, non weakly E p 0 reg neat elements in P ( ). Setting B = B in Theorem 3.0.4 we get 2x 1( ) p,reg log x [ ]∈E ( ) P NX( ) x and with (5.1) above it follows that x 1( ) = o . p,rnw log x [ ]∈E ( ) P µ ¶ NX( ) x It follows from Lemma 2.4.2 that 0 tr ˜(b ) 16 for all EP ( ), and hence we have ∈ x 1( )tr ˜(b ) = o . p,rnw log x [ ]∈E ( ) P µ ¶ NX( ) x It then follows from Theorem 3.0.3 that 8x 1( )tr ˜(b ) . p,rw log x [ ]∈E ( ) P NX( ) x We can also deduce from (5.1) and (5.2) that x 1( )tr ˜(b ) = o . p,rw log x [ ]∈E ( ); (O )>1 P µ ¶ NX( ) x The theorem then follows from Proposition 5.1.8 using the same arguments as in the proof of the previous theorem. ¤ The methods we have used to deduce the asymptotic result for class num- bers from the Prime Geodesic Theorem were not sharp enough to preserve the error term which was proven there. However, we make the following conjecture: 5. 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Index A, 5, 15, 28, 89 Or(S), 109 A , 15, 29, 89 OF (S), 97 B, 15, 28, 89 P , 29, 88 B0, 57, 71 R( ), 33 1 B , 57, 71 R , (s), 43 nreg B , 71 R , 33 C(S), 97 S, 88 Cc(S), 109 V , 34 r C (S), 109 ZP, (s), 33 Fp, 91, 96 A n, 93 G, 28 (F ), 85 C G , 29 (F ), 85 Dx H (n, K ), 34 C , 3 D H1, 29 n , 2 I( ), 96 EP ( ), 16, 29, 89 O E 0 K, 28 P ( ), 72 E 1 K types, 21 P ( ), 72 E p K- nite, 21 P ( ), 29, 89 E reg KM , 19, 29 P ( ), 62 j E p,reg Ln(s), 73 P ( ), 89 E p,rw M, 4, 15, 29, 88 P ( ), 109 M(R), 87 G , 88 M(Q), 87 G (R), 88 M(Z), 87 p, 91, 97 N, 5, 15, 29, 89 O (f), 30 O N( ), 56 , 95 N(t), 58 OP ,ix, 3 N ( ), 30 , 28, 88 O(S), 97 A, 6 c O (S), 109 , 29 128 Index 129 Ind A, 87 b(X, Y ), 28 G Ind P , 3 f , 10 Inv A, 87 gn, 72 j , 32 gs, 31 , 98 h( ), 96 O ( ), 97 l , 29 GO , 34 m ( ), 34 a, 29 mn, , 73 N , 29 nI , 33 n , 29 rn, 73 1( ), 6, 9 (g,K)-module, 34 I ( ), 33 algebra, 85 gen( ), 6 g, 28 central simple, 85 h, 29 division, 85 Gˆ, 1 simple, 85 ˆ Z, 93 Brauer group, 87 ( ), 113 O Brauer invariant, 87 S( ), 97 li(xO), 56 class number, 96 n, 29 conductor, 92 ( ), vii, 119 O elliptic element, 31 pM , 29 j (x), 76 Euler characteristic n rst higher, 6, 9 n(x), 76 (f), 9 generic, 6 (x), 56 Euler-Poincar e function, 10 1 (x), 57 nite order, function of, 37 K , 21 nite rank, function of, 39 S(x), 118 rst higher Euler characteristic, 6, m, 5 9 (x), 62, 81 full -lattice, 86 OF n(x), 78 fundamental unit, 108 P , 29 ˜(x), 56 generalised Ruelle zeta function, 43 ˜S(x), 119 generalised Selberg zeta function, ˜(x), 62 33 ˜, 44 generic Euler characteristic, 6 Index 130 global character, 34 principal series, 3 standard, 40 in nitesimal character, 21 tempered, 40 integral closure, 86 unitarily equivalent, 1 integral element, 86 unitary, 1 invariant subspace, 1 virtual, 9 Jordan-Zassenhaus Theorem, 96 Ruelle zeta function, generalised, 43 Main Theorem, 118, 119 matrix-coe cient, 49 Schur index, 87 Morita equivalent, 87 Selberg trace formula, 30 Selberg zeta function, generalised, non-decomposed (prime), 92 33 non-regular, 16, 89 singular, 16, 89 Skolem-Noether Theorem, 86 orbital integral, 30 spectral sequence, 48 order, 86 Hochschild-Serre, 50 -order, 86 OF splitting maximal, 86 eld, 92 of a function, 37 of division algebra, 88 strictly maximal sub eld, 86 Phragm en Lindel of Theorem, 60 prime totally complex eld, 89 non-decomposed, 92 Prime Geodesic Theorem, 56 unitary dual, 1 of SL2(R), 4 regular, 16, 89 of SL4(R), 5 regulator, 109 of KM , 19 representation, 1 admissible, 21 weakly neat, 32 complementary series, 3 Wedderburn Structure Theorem, 87 discrete series, 2 Weierstrass Factorisation Theorem, equivalent, 1 39 induced, 3 Weyl group, 6, 21 irreducible, 1 Wiener-Ikehara Theorem, 66 Lie algebra, 1 limit of complementary series, 4 limit of discrete series, 2 Mark Pavey Mathematisches Institut der Universit at Tubingen, Auf der Morgenstelle 10, 72076, Tubingen. Geburtsdatum: 1/11/78 Geburtsort: Cheltenham, UK Lebenslauf und Bildungsgang: Okt.2004-Gegenwart Universit at Tubingen Wissenschaftlicher Angestellter/Doktorand Okt.2002-Sept.2004 Universit at Exeter, UK Doktorand Sept.2001-Sept.2002 RHS Harntec Ltd., Exeter, UK Programmierer Okt.1998-Jun.2001 St Edmund Hall, Universit at Oxford BA(Hons) (Klasse I) Mathematische Wissenschaften 1990-1997 Pates Grammar School, Cheltenham, UK