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Representations of positive definite Hermitian forms
Prieto-Cox, Juan Pablo, Ph.D. The Ohio State University, 1990
UMI 300N.ZeebRd. Ann Arbor, MI 48106
REPRESENTATIONS OF POSITIVE DEFINITE HERMITIAN FORMS
DISSERTATION
Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy it the Graduate School of The Ohio State University
By
Juan Pablo Prieto-Cox, M.S.
* * * * *
The Ohio State University 1990
Dissertation Committee: Approved by Professor John S. Hsia Professor Paul Ponomarev Advise£r Professor Daniel B. Shapiro epartment of mathematics Dedicad© a la Memoria de mi padre Octavio Prietc Cmwtey-Bcevey (1927-197G) ACKNOWLEDGMENTS
I would like to express my sincere gratitude to Professor John S. Hsia, he not only taught me some good mathematics, but also set forth an example of excellence as a teacher and as a mathematician. I -want to thank him also for the help he provided me in the preparation of this work. In retrospect, I know that I owe him a great debt for bringing me ashore when I was drifting. My trip to Ohio State (as well as that of many others) would not have been possible without the invaluable help of Professor Daniel Shapiro. I would also like to thank Professor Paul Ponomarev for helping me understand some geometry. I want to thank both of them for their capful reading of this thesis. I want to thank Professor Alice Silverberg who introduced me to the fascinating theory of Abelian Varieties. I would also like to thanlc Manuel OTlyan, a long time road companion, who shared his time with me, gave me good ideas and was willing to listen to my elusive arguments and proofs. I would like to thank my mother for her continuous understanding and encouragement Finally and foremost, I want to thank my wife (and companion in a long journey since my undergraduate days) Kenna, and my two amazing daughters, Maria Jesus and Carnila, for accepting the countless hours I could not share with them, for making my life richer and for giving me hope in those, not so rare, moments of anxiety and desperation. Hove you.
iii VITA
April 2, 1959 Born - Santiago, Chile 1983 .LicenciaturaenMatemdrlcas, TJniversidad de Chile, Santiago, Chile 1986 Master of Science, The Ohio Sate University, Columbus, Ohio 1984-Present Teaching Assistant, Dept. of Mathematics, The Ohio State University, Columbus, Ohio Summer 1986, 1988 and 1990 Research Associate, Supported by NSF grants DMS 850 3326, DMS 880 3805 of Professor J.S. Hsia
PUBLICATIONS
(with R. Baeza, D. Leep and M. O'Ryan), Sums of Squares of Linear Forms, Math. Zeit 193, 297-306 (1986)
FIELDS OF STUDY
Major Field: Mathematics Studies in Quadratic Forms; with Professors J.S. Hsia and D. Shapiro Studies in Number Theory with Professors R. Gold, J.S. Hsia, M. Madan, P. Ponomarev, K. Rubin and W. Sinnott,, Studies in Algebraic Geometry with Professors K. Rubin and A. Silverberg
iv TABLE OF CONTENTS
ACKNOWLEDGMENTS iii VITA iv INTRODUCTION 1
CHAPTER PAGE I HERMrriAN SPACES 9 § 1 Basic results and definitions 11 §2 The split case: E = FxF 15 § 3 The induced quadratic form 1 19 §4 Local and Global Hermitian spaces 23 H HERMITIAN LATTICES 28 §5 Generalities 28 §6 Local theory of Hermitian lattices 32 §7 Hermitian lattices over number fields 38 IE REPRESENTATION OF HERMITIAN LATTICES 43 §8 The induced quadratic form II 43 §9 The */-class and the S£/-class 49 § 10 Some local results 54 § 11 Global representations 58
v APPENDIX 73 §A1 The induced quadratic form HI: Going down to (Q 73 § A2 The induced quadratic form IV: The genus and the spinor genus ... .78 §A3 Further research 84 REFERENCES 87
vi INTRODUCTION
Historical background
In studying the representation of a number as a sum of four squares1 Hermite introduces, in 1853, the notion of a (binary) Hermitian form over IQ(^T). He notes that binary Hermitian forms have a remarkable number of common properties with binary quadratic forms and in a short number of pages he proves some important results (fractional equivalence, finiteness of the class number, reduction, etc.). Here is what he had to say about these new forms: Les considerations suivantes, que nous prisentons comme une premiere esquisse d'une thiorie vaste etfeconde sur laquelle nous reviendrons d I'avenir, offriront plusieurs exetnples de cette itroite analogie avec les formes binaires; mais on y verra en mime temps le germe de notions arithmitiques toutes nouvelles, qui miritent peut- etre de fixer 1'attention des Giometres. It is because of its own genesis and the similarities mentioned above that the theory of Hermitian forms has inherited many of the techniques of the quadratic theory. Surprisingly, the two theories have not developed at the same pace, most of the classical problems of representation have not been studied in the Hermitian theory, so in a sense this theory is underdeveloped. We have to admit that some of the classical problems, as
seejHel] 1 2 sums of squares for example, are not as appealing in the Hermitian setting (sums of norms). There is a point worth mentioning. Hermitian forms arose, in part, as a tool to understand a problem in quadratic forms. This has been partially, if not totally, reversed in history. In most cases the theory of Hermitian forms has either been reduced to or followed the theory of quadratic forms. We think that this rich theory can have a useful role in understanding quadratic forms, especially as a testing ground for some open problems, although regrettably, our work is not an example in this direction.
The problems of classification and representation are the two most classical and fundamental problems in the theory of quadratic as well as Hermitian forms. The desire to understand these problems has motivated the larger part of the work on Hennitian forms. The algebraic theory evolved faster than its integral counterpart, in 1936 Landherr proved his classical local-global principle, not only for Hermitian forms over fields but also over quaternion algebras. Soon after that, in 1939, Jacobson introduced the so called trace form of a Hermitian form and proved that we only need to understand quadratic forms, since the trace forms classify Hermitian forms (see theorem 3.1). There is an explicit characterization of all quadratic forms which are trace forms in [Lew] (see also proposition 3.3).
The integral theory is not as well understood, certainly the questions of representation and classification are wide open in the global case, although the local case has been completely solved by Jacobowitz and Johnson (unlike its quadratic counterpart). Except for the work of Raghavan, that we comment upon below, the important results for the theory are somewhat new. In 1962 Jacobowitz gave an answer for classification over local fields using techniques similar to those employed by 3 CMeara for quadratic forms. In 1964, Shimura proved the Strong Approximation for SU(V) when V is indefinite, and as a consequence proved that SU-genus = SU~class. This has an advantage over the corresponding Approximation Theorem for 0+(V) that gives instead spinor genus = class. He also computed the class number of U-genus in the indefinite case. A complete solution to the local representation problem was given by Johnson in 1966. He made extensive use of the invariants provided by a Jordan splitting, very much like the quadratic case. Gerstein studied the decomposition of Hermitian forms in a paper in 1970 and proved (using the result of Shimura previously mentioned) that every global indefinite form is decomposable into components of rank at most 4 (see [Gel]). After the result of Shimura, much of the global work has concentrated on positive definite forms. The computation of the class number was initiated by Iyanaga in [Iy2], he uses the method of neighbor-lattices of Kneser to compute the class number of the standard unimodular lattice /„ over S3(i)t for small values of n. Later, in 1978 Gerstein studied lower bounds for the class number of positive definite forms (see [Ge2]), with special emphasis on unimodular forms. There are other works of Braun (1941) and Otremba (1971) that we have not read (for a comment on them see [Ge2]). Finally in a series of papers in the late 1980's, Hashimoto et al. have studied the class number (of U(V) and SU(V)) using the standard techniques of mass formula (a* la Siegcl), and in particular, have computed the class number of unimodular forms in 2 or 3 variables over an imaginary quadratic number field.
A second question that arose in our study was that of the induced quadratic form (or trace form), which can be defined as follows: If (V, h) is a Hermitian space over a quadratic extension EIF we can define a quadratic space ( $, b) over F by putting, $ = V viewed as an F-space and b = Tr°h. As we previously mentioned, this form was 4 introduced in 1939 by Jacobson. In fact, we believe that this idea can be traced as far back as Hennite, in his second paper on Hermidan forms2 in 1855, he seems to have known some properties of the induced quadratic form when he proves that the characteristic roots of a Hermidan matrix are real. The integral version of this induced form was first studied by Iyanaga in 1968 and then in 1972. He not only works with this form (real part) but also with the alternating form induced by h (imaginary part). His main result is the characterization of modularity under some conditions. We are not aware of any other attempt to investigate the induced quadratic form.
Our work
In this work we have set up to prove a result modelled after "Representations of Positive Definite Quadratic Forms" by Hsia, Kitaoka and Kneser. The main result (corresponding to theorem 3 in JHKK]) is a local global principle with restrictions for positive definite integral Hermitian forms. It can be stated as follows: 3Let Mbea positive definite Hermitian lattice of rank m. There is a constant c = c(M) such that, ifN is any positive Hermitian lattice of rank n, with m£2n + 1.
Then M represents N if and only if Mp represents Np V P; provided min (N) > c(M).
The only previous result in this direction was obtained by Raghavan in 1954 (see [Rag] section 7, theorem 9) using analytic methods (Hermitian theta functions). He was able to prove the main result for F-S}, m£5 and n = 2 with the additional restriction that the two successive minima of N be of the same order of magnitude. On the other 2 see [He2] 3 cf. theorem 11.10 5 hand, our proof is purely arithmetical, except at one point where we have made use of Dirichlet's theorem on primes in arithmetic progressions. There are three main ingredients in the proof: the approximation theorem for SU(V) of Shimura (see [Shi] and theorem 7.6), the strong approximation for S-lattices (see theorem 11.8) and the reduction theory of Humbert (see [Hum] and proposition 11.5 and corollary 11.6).
We have tried to make this work as self contained as possible within the obvious constraint of length (and time), in this framework we have included the most classical results as well as those results directly relatedt o our research, that were used explicitly in this work. The only exceptions being those results which required a great deal of preparation (like the results of Johnson on representation of local Hermitian lattices). The outline of this work is as follows. In Chapter I we build the necessary background on Heimitian spaces. The basic part of the abstract theory of Hermitian forms (i.e. over arbitrary fields of char. * 2) is presented in section 1. We have relied on the yellow book of Scharlau, [Sch], which contains everything we present in this section (and a lot more). Unfortunately it does not treat the split case, the subject of section 2, for this we have used Shimura's paper [Shi] plus a few straight forward extensions of the non-split case. In section 3 we present a most fruitful idea introduced by Jacobson (see [Jac]), relating a Hermitian form with a naturally associated quadratic form, the so called trace form. We also include a result of Lewis characterizing such trace forms. Those two results give the whole picture of this "association" and thus there is not much left to say. Finally the focus of section 4 is the presentation of the more number theoretic setting, that is, when the fields involved are local or global. Particularly we discuss a peculiar (for the uninitiated) form of localization introduced by Shimura. In the Hermitian theory there are two algebraic structures involved, a field, say F, and separable algebra of rank two over F, say E (more generally, E could be a quaternion algebra as well). Assume 6 for the sake of the argument that E is afield. The usual way to localize (or complete) E
is to take a spot in E, say p% and construct E p in the standard way (say Cauchy
sequences). Now, when the prims p \ E = p is split in E, E & - Fp and therefore the Hermitian structure is lost It is for this reason that we view all structures as being based on the field F; under this perspective, E becomes an algebra of rank 2 and as such, localization (as a process based on F) simply means extension of scalars, i.e., for
a prime p of F we define Ep= E ®FFp (all the other structures, as £-vector spaces, ideals, etc. are treated in the same way). The local classification and representation problems are reduced, using Jacobson's method, to the corresponding results on quadratic forms. A complete set of invariants is {det, dim) (see proposition 4.1). Finally we present the classical global-local principle of Landherr (theorem 4.3), which gives a complete answer to the global classification and representation problems.
Chapter II has the similar purpose of presenting the common ground of the integral theory. We have adopted the more modern geometrical terminology of lattices (as we have done in the algebraic theory by working with spaces) instead of forms or matrices. Although we have not completely disregarded the useful approach of using matrices. A more desirable situation, for a self contained work, would have been to include what correspond to the second part of chapter 8 of O'Meara (§ 82), but certainly that was beyond the length constraint of this work. At any rate, in section 5 we have included what we estimated were the more relevant results, and even then we have refrained from proving them as those proofs are completely analogous to the proofs found in CMeara's book. We also include here the definition of 17-class and SU— class. Section 6 treats the standard local theory, with special emphasis on the construction of Jordan splittings (including the split case). We could not include the results of Jacobowitz on classification as well as those of Johnson on representation, 7 except for the special, but important cases of modular and maximal lattices. The core of section 7, about the global theory, is the Strong Approximation Theorem for SU(V) of Shimura and its consequences. Other results that we have included are analogs of the quadratic theory (like the existence of global lattices with prescribed localizations).
Chapter m is not as homogeneous as the previous two chapters. We start by trying to understand the induced form. For a moment our hope was that the induced form together with the results in [HKK] could lead us to a solution of the main result. As it turned out, that is not the case, although for forms of rank one it is possible to use the induced form. The main result in section 8 is the computation of the volume (or determinant) of the induced form (see proposition 8.4). We briefly study the modularity of the induced form, and except for the unramified or split case, the modularity is lost in the going down process. In section 9 we study the relationship between the U-class and the SU-class. Unlike the quadratic case, the number of SU-classes in a U-class (as well as the number of SU-genera in the U-genus of a form) is in general infinite (see proposition 9.4 and proposition 9.8). As it is usual in the global theory, we need a number of local results that, aside from the standard results which appear in section 6, we have collected in section 10. Section 11 is the core of this work. It resembles very much the work in [HKK]. As we have already mentioned, the main result is theorem 11.10. We have also proved an approximation result for positive definite lattices and we have given a conceptually easier, although longer, proof of a weaker version of it due to Kneser, which was announced in [HKK].
In the Appendix we collect some of our investigations on the induced form. First, in section Al, we prove a general formula for the determinant of the induced Z- lattice (see formula (Al.l)), generalizing a similar formula for quadratic forms of Milnor 8 (see lemma.2.2 in [Mil]). Secondly, in section A2, we study the behavior of the U- genus and SU-genus under the trace map, and determine that the U-genus is mapped to the genus and the SU-genus goes to the spinor genus. Pinally, in section A3, we propose some further avenues for research, which for the most part are natural continuations of our work. CHAPTER I HERMITIAN SPACES
In this chapter we will concentrate on those results that form the basis for the theory of Hermitian spaces. In the first section we will introduce the necessary definitions and classical results. The second section contains those results that are particular for the case FxF. Then we will discuss Jacobson's theorem and its consequences for the problems of classification and representation of Hermitian spaces, as well as a few more recent results. Finally in section 4 we present the classical results for local and global fields, which are the fields we will be working with when we discuss the integral theory in Chapters II and IE. Throughout this work F wilt denote afield of characteristic not 2.
Notations F: a field of characteristic not 2 (usually a number field or a p-adic field) E: a quadratic extension of F or a sum of two copies of F R: the ring of integers of F S: the ring of integers of £ Z, SQ, 25, € : the set of integer, rational, real and complex numbers respectively
Zp, tQp: the p-adic integers andp-adic numbers respectively V: a finitely generated free E-module Ox: the set of invertible elements of a ring O 9 10 6 =V3,when£ = i?fV3), di F2
Let E be either a quadratic extension of F or the direct sum of two copies of F. In both cases E has a non-trivial involution whose fixed field is F (that we usually call conjugation and denote by ~ ), in the first case the generator of the Galois group and in the second defined by (a, b) := (b, a) (here F is embedded into E diagonally, i.e. a i—*• {a, a)). Associated with this involution we define the norm and trace of an element a ofEby: M(a) = aa; Tr(a)=a + a.
Let V be a free E-module of finite rank. A map h:VxV *• E is a Hermitian form if (i) h is linear in the first component and (ii) h(x, y) = h(y, x) .
The pair (V, h) is called a Hermitian space. Two clear consequences are: (a) h(ax, ay) = N(a)h(x, y); (b) h(x, x)e F for any x in V. Sometimes we will denote the Hermitian space by V alone or by h if there is no fear of confusion. We will also talk about a Hermitian form with the understanding that there is an underlying vector space.
We will study Hermitian spaces (respec. Hermitian lattices) over E (respec. over S). Due to the peculiarities of the case E = F xF and the inability, on the part of the author, to find a suitable unifying view and also due to some expository concerns we have decided to separate the basic results (as well as some other sections) in two different parts. 11 § 1 Basic results and definitions
In this section we will concentrate on the case E = F(^[d) (even though most of the results hold true in general) and leave the case E = F xF for the next section..
There are two standard ways to studying Hermitian forms. The one we just described is the geometrical one, the other is the study of Hermitian matrices, that is ioatrices H in
Mn(E) satisfying H* = H, where for a matrix H - (a^) , H* := (a^) . To establish a correspondence between the two let us start by taking a Hermitian space (V, h) and choosing a basis of V over E. The matrix of h in this basis, say //, is clearly a Hermitian matrix. For the converse take a Hermitian matrix H in Mn(E) and associate the Hermitian form h(x, y) = y*Hx, where x, y are column vectors on a vector space V of dimension n over E. This "correspondence" is certainly not one-to-one, on the one hand it depends on the basis chosen to construct the matrix and on the other it depends on the vector space used to support the Hermitian form. For our purposes we say that two Hermitian spaces (V, h) and (W, TJ) are isometric (or equivalent) if there is an E-linear isomorphism Throughout this work we will adopt the geometric view point but we will use matrices whenever it seems more convenient 12 Let H be a matrix associated with a Hermitian form (V, h). If K is any other matrix
associated with h, H = T*KT, for a suitable matrix T in GLn(E). Hence det H = N(det T)- det K, i.e. det H and det K differ by a factor of N(EX) and so det H is well defined modulo N(EX). Accordingly, we define the determinant (some authors use discriminant, but we reserve this for the field discriminant) of h as the class of det H in FX/N(EX) u {0} and denote it by dh or dV (when there is no fear of confusion we use a representative to write the determinant of V).
Definition 1.1. We say that (V, h) is regular (or non-singular or non-degenerate) if dV^O. Otherwise, h is called singular (or degenerate).
Orthogonality
Two subspaces U, W of V are said to be orthogonal if h(x, y) = 0 for x e U and y e W. We say that V is the orthogonal sum of U and W if V = U © W and U, W are
orthogonal. In this case we write V = U ± W. Also if (V]t hj) and (V2, h2) are two Hermitian spaces, their external orthogonal sum, say (V, h), is defined by
V = Vj © V2 and ftCx7 © *2, y7 © y2) = hjfXj, y2) + h2(x2, y2). A space V= U J. Wis regular iff both U and W are regular
Any Hermitian space V has an orthogonal basis, that is, a basis x]f ..., xn, where
h(xit Xj) = 0 if 19*y. In this case we write V - [aj\ X ... _L [an] = [aj,..., an], where a,-
= A(x(-, xf7. The orthogonal complement of a subset W of V is defined as [x e V: Afo y) = 0 for all. yeW) and denoted by W^. In particular, VL is called the radical of V. F is regular whenever rod V* {0}. Every space V can be written as V - V X rad V where V is regular. It is a well known fact that two spaces V=V'± rod V and W = W X rod W are 13 isometric iff V" = W and rank (rod V) = rank (rod W). From now on all spaces will be assumed to be regular (though subspaces arising from our discussion could be singular). If W is any regular subspace of V we can write V = W _L W1- and W11 = W.
Isotropy
A vector 0 & x in V is called isotropic if h(x, x) = 0; otherwise, it is called anisotropic. A space V is called isotropic if it contains an isotropic vector, otherwise it is called anisotropic. A subspace W of V is called totally isotropic if h(x,x) = 0 for all vectors x inW. A hyperbolic plane is a 2 dimensional regular isotropic space, usually denoted by H. The following statements are equivalent for a plane (2-dimensional space) V: (i) Vis hyperbolic (ii)V has a matrix of the form (JQ)
(iu) dV = -l
Any isotropic space V contains a hyperbolic plane, i.e. V = M _L U. A hyperbolic space is an orthogonal sum of hyperbolic planes. The following theorem gives what is called the Witt decomposition of a space (cf. [Sch] corollary 7.9.2).
Theorem 1.2. Let Vbea regular Hermitian space. V can be decomposed uniquely (up to isometry) asV=V0±Vj, where V0 is anisotropic and Vjisa hyperbolic space.
Another classical result due to Witt is (see theorem 7.9.2 in [Sch]): 14 Theorem 1.3. (Witt cancellation) Let U, W, W be Hermitian spaces. If U ±W = U±W'thenW~W.
It follows that the number of hyperbolic planes in theorem 1.2 (= 112 dim Vj) is an invariant of V. This number is called the Witt index ofV.
The unitary group
A group that plays a fundamental role in studying Hermitian forms is the so called unitary group of space (V, h). It is the set of all £-linear maps o~: V *• V which satisfy h(a(x), o(y)) = h(x, y) for all x, y in V and it is denoted by U(V). The elements of U(V) are called isometries or unitary transformations. If we use a basis for V over E and let H represent the matrix of h and T the matrix of o*e U(V), then they satisfy the matrix equation T*HT = H.A determinant computation shows \hsXN(det a) -N(det T) = 1. The following is a classical result:
Proposition 1.4. Let (V, h) be a Hermitian space over EIF. Then the map U(V) ** > {aeE: N(a) = 1 ;.-= E1 is onto (the kernel is coiled the special unitary group and is denoted by SU(V)).
Proof. Fix a with N(a) = 1. Write V = Eex 1W and define the map a: V •> V by a(ej) = aej and o(w) = w for all w in W. Then it is easy to check that <7e U(V) anddeta=a. El 15 Unitary reflections
Let x be an anisotropic vector in V and let W - (Ex) . If a :V >V is a unitary transformation with a\w = Id then a(x) = ax with aSc = 7. Such a transformation is called a unitary reflection (Dieudonne" calls them quasi-symmetry) and will be denoted by sx a (sx _j is the standard symmetry or reflection, and sxJ = id). To find an explicit formula for sx a we need to look at the component of a vector v in Ex, that is v = ax + w, where we W. As h(v, x) - ah(x, x) we get a = . ;v' x\. Then for the unitary reflection sx a we have:
sx,a = acoc +w = aax +v-ax
Clearly sxasxa = id anddetsXitx = a. A well-known result (cf. [Sch] Theorem 7.9.5 or [Die] chapter n, § 3) is
Theorem 1.5. Let EIF be a quadratic extension then the unitary group U(V, h) is generated by the unitary reflections. In fact any unitary transformation is the product of at most n + 1 such reflections.
§ 2 The split case: E = F x F
In this section we will present those results that are particular to the case E-F xF, that from now on we will refer to as the split case. As before we let (V, h) represent a Hermitian space over E. Since V is a free E-moduIe of finite rank we can use a basis of V to associate a matrix to h so that the notions of determinant and regularity are the same 16 as in § 1, although in this case N(EX) = Fx so that dV= lorO for any Hermitian space V. Also we can define isometry, representation, isotropy, etc. as in the previous section. Let us assume that (V, h) is a regular Hermitian space of rank n. As N(EX) = Fx, any regular space V has detV= 1. In contrast with the field case, every space is isotropic because of the existence of non-zero elements of norm zero. Let e = (1,0) and e = (0,1) so that e + e = 1. Since h: V * F xF we can write h(x, y) = f(x, y)e + g(x, y)e As h is Hermitian it follows that g(x, y) ~f(y, x) andf(x, y) is F-bilinear. To compute the determinant of h let us take a basis Vj, .... vfl of V and put h(vitv) = hy and f(vit Vj) - fy. Then h^ = f^-e + fye, and since the arithmetic operations are defined t componentwise, det(hjj) = det(ftj)e + det(fjj) e = det(f^ (recall that F is embedded diagonally into F xF). The regularity of V implies that defff^) *0, so let (a;-) = fly7 n and put xt = ^(a^-e + S(je)Vj for i = 1,..., n (here Sy is Rronecker's delta). Clearly
(dij-e + Sije) is an invertible matrix, therefore xlt..., xn is a basis for V and the matrix of h with respect to this basis is /„ e GLn(E). Thus we have proved
Proposition 2.1. Every regular Hermitian space over F x F has an orthonormal basis.
As a consequence we obtain
Corollary 2.2. (i) Two regular Hermitian spaces over F xF are isometric if and only if they have the same rank. 17 (ii) V represents W if and only if rank V £ rank W.
Note that based on the above corollary, theorem 1.2, theorem 1.3 and the characterization of hyperbolic planes hold trivially in the split case. The next result is easily proved adapting the proof known for Hermitian spaces over fields (e.g. see [Sch] theorem 7.1.4)
Proposition 2.3. Let W be a regular subspace ofV, then V = W1 W1.
The unitary group of (V, h) is also generated by unitary reflections. Since a proof of this fact is not in the literature we will give a proof based on the standard proof for the field case (see, e.g. [Sch] theorem 7.9.5).
Theorem 2.4. Let E = F xF (assume F has more than 5 elements) and let (V, h) be a regular Hermitian space over E. Let W cV be a regular subspace. For any unitary isometry a: W •• V there is a product of unitary reflections that extends G. In particular any unitary isometry is a product of unitary reflections.
Proof. We use induction on dim W. The case W = 0 is trivial. Let* e W with h(x, x)
# 0 and write W= ExXWj. We can then apply the induction hypothesis to cr| w. (Wj = (Ex) is regular). Hence there is a product of unitary reflections, say 2>, 1 extending a\ w , i.e. E\ w = o"l w.. Lety := iT cr(x), clearly y e Wj . We claim that there is a product of at most two reflections (actually two if allow id as a reflection, sxl - id), say A, such that A \ w. = idW] and A(x) = y. If this is true, EA(x) = 2(y) = a(x) and ZA(w) = E(w) = o(w), for we.Wlt that is XA extends a. 18
Write y = fix + yQ with y0 orthogonal to x. The above claim follows from the following: (fk) There is an element ye E1:- {elements of norm 1} such that (i) x - yy is anisotropic (ii) h(x, x - yy) is invertible. Suppose first that (<) is true and define a:- 1 ffi—^'"~.^. A computation gives: Mx -yy.x- yy) = (2-Tr(yP))h(x, x) (A) h(x,x-yy) = (l-yp)h(x,x) (B)
(using the equation h(XjX) = h(y,y) = pfih(x,x) + h(y0,y0)) So, a - 1 - -ZJEzJE- = _ < ~ W) and hence Affa) = 1. This allows us to define a-rP) (l-rP) sx _ yya, computing this reflection at x
(by definition of a). Consequently syr.j°sx_yyJx) = y and as y andx-yy are 1 elements of Wj- ,syr.i°sx_yya(w) = w forw e Wj.
To complete the proof take A :=*s rj °sx_ „, #. It only remains to prove our claim (#), by (A) and (B) we have (i) <=> Tr(yp) *2 (ii) <=> 1 - yp is invertible 1 1 x As ye E we can write y= (c, c" ), ce F . Put fi = (bv b2) with bJt b2 e F. Then 1 (i) <=> 2-cb1-c~ b2*0 1 1 (ii) <=> 1 -yfi=(l - cbj, I - c~ b2) is invertible, i.e. 1 -cbj&Q^l- c~ b2 2 Now, 2 - cbj - c~*b2 = 0 <=> c ^ - 2c + b2 = 0. But this equation has at most two solutions in F, say ctj, a2 . On the other hand, 1 - cbj =0 <=> c = bj (assuming bj J ?*0,ifbj - 0 then 1 -cbj = 1) and J - c~ b2 -0 <=> c~b2. Therefore if we take any 19 x 1 c in F with c * a1% a2, bf , b2 (when bj = 0, erase it frcm the list), the element y = (c, (T1) satisfies (i) and (ii) in (#). E
Fortunately in the split case the structure of the unitary group is much easier to describe.
Let Vj = Ve and V2 = Vei Then h(xe, xe) = h(xe, xe) = 0, and h(xe, xe) = h(x, x)e e
F x(0). Let a e U(V, k) and denote by av eVjxV2 we can find § 3 The induced quadratic form
Two fundamental problems in the theory of Hermitian forms are the following: Classification: Can we determine whether or not two Hermitian spaces are isometric? This has been answered in a number of cases, most notably when F is a local or a global field. Representation: We say that a Hermitian space (V, h) represents another Hermitian space (Vj, hj) if there is an E-linear embedding cr: Vj > +V with h(ox, oy) = hj(x, y). In this case we put Vj —>— V. Now we can state the question of representation as: given two Hermitian spaces V, Vj, can we determine whether Vrepresents Vj ? Again this is fully understood in the local and global cases. 20
None of these questions has a completely satisfactory answer for general fields. Nevertheless the trace from E to F provides ns with a natural way to ohtain a quadratic form (whose theory has been extensively studied) fiooi a Herroitian form. This idea was introduced by Jacobson in 1939 (cf. [Jac], page 267).
To each Hermitian form (V, h) Jacobson associated a quadratic form ( V, bh), called the trace form, the real part or the form induced byk, in the following way (i) ^ = V setwise but viewed as a vector space over F.
(ii) bfl = Tr°h When there is no fear of confusion we will write simply b for b^ 3a a few instances
we will use the related form bh ~ | Tr° h. Specifically when dealing with Hermitian
spaces, in this section and in the next If we denote fcy qh the quadratic form associated with h in this manner (i.e. qh(x) = bh(x, x)), then qh(x) = h(x, x) and so qh and h represent the same elements. Note also that bh(axr ay) = N(a)bh(x, y).
N.B. It appears that Hermite1, as early as 1855, may have used the same underlying form.
Let us study first the field case, that is, when E = F(^d). If H is a matrix for h we can write H = B + -{dA where B is symmetric and A alternating. Then if we denote by ft a matrix forty, (= ^ Tr°h) a computation shows that
£y _(B -dA\ M -\dA-dB/
1 C. Hermite, Remarque sur un thior&me de Caucky, (Envies \. 1 r Gaulhiei-ViJIars (1905). 479—4-81. 21 On the other hand, the Hemnitian form h can be recovered from its induced form b as follows: If b = =7>°fc then cleaily h(x, y) = b(x, y) +• a(x, y)5, where a(x, y) is an alternating 2 form. Now b(Sx, y) = !fr(Sh(xt y)) = jTr(6b(x, y) 4 a(x, y)&) = a(x, yJS , that is, jj>{Sxry) = a(x,y)5. Therefore
h(x,y) = b(x,y)¥ -sb(5x,y) Obviously when b = 7r° h, h(x, y) = ~ [bfx, y) •+ ^b(cx, y)].
An important step towards the classification problem is the following
Theorem 3.1 (Jacobson). Let EiF be a quadratic extension and let (V, h), (W, g) be Herrnitian spaces over E. Then
(i) (V, ft) is regular iff ( $, qh) is regular.
(ii)(V,h) s (W>£) iff (9, qh) = f ft, qg).
(Hi) (V, h) is isotropic ijf (P, qh) is isotropic
With this result in hand, the problem of classification of Hermitian spaces is reduced to the corresponding prohlern for quadratic spaces.
Remark 3-2- In the split case, as we saw in section 2, we can write h(x,y) =f(x,y)e *J(y> x)eT withj(x, y) F-bilinear. As in the previous case we can recover hjrom its induced form b = Jnh as follows: b(xe,y) = Tr{ek(x,y}) = Tr(f(x, y)e) = f(x,y). Similarly b(x, ye) =f(y, x). Therefore h(x, y) = b(xe, y)e +- b(x, ye)e
Tj h has a matrix of the form (kjj = {f^-e -+ f^e) then bh has a matrix of the form G/ ^\ In. the split case we have shown that every space has an orthonormal basis and 22 therefore bh always has a matrix of the form L £ j That is, bh is a hyperbolic space.
Consequently we also have
(V, h) = (W, g) iff (9, qh) s (ft, qg).
A natural question that arises from this construction is: which quadratic spaces are obtained in this way? This question was answered by Lewis1 in 1979 (cf. [Lew] page 266), in this work he characterized, in the case E = F(-Jd), all quadratic spaces that arc induced by a Hermitian space. He based his proof on the following exact sequence of Witt rings 0 • W(E, ~) » W(F) > W(E) • W(F)
h l - qh l • qh®E I * s*(qh®E) where s = xTr.
As we will see, it turns out that the obvious necessary conditions are also sufficent.
Assume V has an orthogonal £-basis v^, .„, vn, write h = [afi J L [an]. Then $ has an orthogonal basis Vj,..., vn, bvj,..., bvn over F and qh = [aj] -L L [an] ±
Hfai]±...±Hfa„].
Since d is a square in £ it follows that qh is hyperbolic over £, i.e $®E is hyperbolic. Also it is clear that d V s (-df modF2.
Proposition 3.3.(Lewis) Let E = F(^[d) be a quadratic extension. The quadratic space (W, q) over F is the induced space of a Hermitian space over E iff
(a) dimFW=2n (b) WQE is hyperbolic
1 added in proof: I was told that this result is a consequence of much more general results of Frflhlich- Mc Evett dating back to 1968. 23 (c) dW^i-dfmodF2.
Note that in the split case only the hyperbolic forms are obtained in this way.
§ 4 Local and Global Hermitian spaces
We start this section with the classification of Hermitian spaces over a p-adic fields. Here we use essentially Jacobson's reduction (theorem 3.1) and the corresponding classification of quadratic spaces. Let F be a p-adic field and let E = F(4d) be a quadratic extension.
It is well known that every 4-dimensional quadratic space over F is universal (i.e. represents all the elements of F), it follows then, from section 3, that every Hermitian plane is universal and therefore every Hermitian space of dimension 3 or more is isotropic. In fact we have
Proposition 4.1. Let V, W be Hermitian spaces over E. Then, (a) If dim V>2,V is isotropic. If dim V = 2,V is isotropic or anisotropic according as -dV == 1 mod N(E*) or not. If dim V = I, V is anisotropic.
(ty w —, y ifand oniy if dim W < dim V or dim W = dim V and dW = dV mod N(EX).
A short proof of this involves the computation of the complete set of invariants of the underlying quadratic space. They are as follows (cf. [Sch] page 350):
dimF V~2n, dV = d", where n = dimEV and if we use O'Meara's definition of the Hasse invariant (see [O'm] §63B), 24
* _[(d, det V) if n = 0, 3 mod 4 S( )=\(-l, -d)(d, det V) if n = 1,2 mod 4
Now we turn our attention to the global case. Let E = F(*Jd ) be a quadratic extension of number fields. Denote by 5 and J? the rings of integers of E and F respectively. The 7 different oiEIF, &E/F, is defined by J9~ £/F ={a<= E: Tr(a S) cJ?) and the discriminant (ideal) disc(EIF) := N(&EfF). When there is no fear of confusion we will write JD for JS>E/F. The primes of F ramify,remai n inert or split completely in E, In the first two cases the local degree n* := [Ep: FA = 2 ( where p is the unique prime in E above p) and we say that p is a non-split prime, in the third case Up := [E&: Fp] = 1 (where p is any of the two primes in E above P) and we say that p is a split prime. Then n« = / or 2 according as d is a square in Fp or not. It is well known that there are infinitely many split as well as non-split primes in F (see [O'm] § 65).
For a prime p of F (finite or infinite) define the localization Ep by:
Ep = E®FFp
For a vector space Vover E we define V« = V <8>FFp. Then Vp is just the extension of scalars EpV. Let us assume first that np = 2, then it is well known that Ep = E& = Fp (V5 ) where p is the unique prime of E dividing p. Likewise if we define 5« to be the integral closure of 5 at p then it is also its topological closure. The involution extends to a unique F«-involution on Ep that coincides with the involution of F«fV5 ) over Fp, and the Hermitian form extends to a unique Hermitian form, that we denote by h, over Vp. 25
Assume now that np= 1, that is p splits in E, then Ep ^Ep x Ep =Fp xFp where
are me p j, p2 two primes in £ dividing J> (cf. [C-F] page 57). In order to better understand the nature of this identification it is necessary to look at the isomorphisms above more carefully. By identifying E with F[x]/ (JT — d) it is easy to construct an explicit isomorphism (canonical) Ep = E®F Fp —=—*• FpXFp, this map is given by
a®p \—*-(ap, afi) where ae E- F(Jd) c Fp(^) = Fp and p e Fp. If we take the product topology on the RHS the map becomes a topological isomorphism. Clearly E is dense in E «., moreover E is dense in Ep. Let S be the ring of integers of E and let Sp be its integral closure in Ep. Then we can see that Sp = Sp x Sp =RpX Rp. By strong approximation S is dense in Sp, for the RHS has the product topology. Again the involution extends uniquely to an F„-involution of Ep given by (e€f) = e<8f. So the effect of the involution in FpXFp is (a, b) = (b, a).
In the split case we can define the different in the same manner as we did for quadratic extensions, i.e. &EplFp = {ae Epi Tr(aSp) £R p}but in this case &Ep/Fp *s toi-vizA
(= Sp), therefore the discriminant disc(Ep/Fp) =N(SSE^FJ - Rp.
Finally we present the classical resulto f Landheir (local-global principle) and introduce the notion of definite and indefinite spaces.
Definition 42. The extension EIF is called a CM-extension ifF is totally real and E is totally imaginary quadratic extension ofF.
It is well known that the involution of EIF, that for obvious reasons we now denote by a, commutes with all embeddings of E c ». £t and moreover for any embedding 26 x, i{{aa)) =T(a), where "~ denotes complex conjugation. That is, if we fix an algebraic closure E c Cthe involution is just complex conjugation.
Let (V, h) be a regular Hermitiaii space over E. Denote by XJt ...kt the distinct infinite
primes (spots). As before, denote by E«. the localization E ®F Fi.. Reorder the spots
so that for 1 £i VM-
Theorem 4.3.l (Landherr) Let V, W be two regular Hermitan spaces over E of dimension m and n respectively. Then, V represents W if and only if
Case 1, m>n.O ZrJV^) - r_(Wx.) £m -nforl £i £s.(<=* Wx. —>— Vx.)
Case 2,m = n. r_(Vx.) = r_(Wk.)for 1 £i£s, anddV = dWmodN(Ep)for every prime ideal P (i.e. finite prime) ofF (o Wp=Vp).
The theorem can also be stated as a local-global Hasse principle, i.e. W—>— y <^
Wp^>—Vp for all p.
We say that a Hermitian space Vover an extension EIF is positive definite if:
(i) EXIFX = CfJR at each infinite prime X of F (i.e. the extension EIF is a CM- extension)
(ii) Vx is positive definite (i.e. in any diagonalization of Vx the diagonal entries are all positive)
1 cf. 5.8 in [Shi] 27 Similarly we can define negative definite. We say that V is definite if it is either positive definite or negative definite. Otherwise we say that V is indefinite. Note that an equivalent definition is: V is definite <=> V^ is anisotropic for each infinite prime X.
Let P represent as before the quadratic space underlying (V, h), and assume that E/F is a CM-extension then we have
Lemma 4.4. Let Vbea Hermitian space over E. Then (V, h) is positive definite iff ( V, b) is a positive definite quadratic space over F.
Proof. Let E ~ F(^fd ), d a totally negative number. Let [aJf ..., an] be a diagonalization of h over £^=C(Aa real prime). Then [1, -d] [av ..., ocn] is a diagonalization of b over F^=1R. Since -dis positive b is positive definite. Moreover sgn h = -j sgn b. E3 CHAPTER n HERMITIAN LATTICES
In Chapter I we have developed our background knowledge of Hermitian spaces or what is called the algebraic theory of forms. Now we will present the arithmetic theory of Hennitian forms, the one concerned with the integral properties of those forms. As we said in the introduction, the content of sections 5 and 6 are the bare bones of the arithmetic theory. We basically follow some works of Gerstein, Jacobowitz, Johnson and Shimura.
§ 5 Generalities
In this section we will introduce most of the definitions that will be used throughout Chapters II and III, those definitions that are particular to the local or global case will be introduced in sections 6 and 7.
Let F be a number field or a p-adic field. In the first case let E - F(-fd), in the second case we let E be either F(*Jd) OTFXF. AS before let 8 = 4d. Denote by S (resp. R) the ring of integers of E (resp. F). l Let &Etf = (fle£: Tr(aS) c R). The different of E/F is defined as the integral S- ideal £>£# (when there is no fear of confusion we will write 49 for 49£/f.). It follows then that when E is unarmified or split, 49 = 5. For an 5-ideal Ct we define the trace of
28 29 We will present two equivalent definitions of an S-lattice. Although we will usually view S-lattices in the context of the first definition, it is sometimes more natural to take the second approach, this is the one used by Shimura in [Shi], especially when we look at the underlying quadratic lattice.
Definition 5.1. (1) Let V be an E-space. An S-lattice LonV is a finitely generated S-submodule of V with EL = V. Or equivalently, (2) Let Wbea vector space over F. We say that K is an R-lattice on W ifK is a finitely generated R-submodule ofWandFK - W. IfV is an E-space. We call an R-lattice on 9 (= V, viewed as an F-space) an S-lattice onVifSK^K. We say that an S-lattice LisfaV if EL £ V, If(V, h) is a Hermitian E-space, a Hermitian S-lattice on V is a pair (L, h) consisting of an S-lattice L on V, together with the Hermitian form h. We usually denote this by just L, with the assumption that the Hermitian form is understood.
Any S-lattice L can be written in the form L - Definition 5.2. Let (V, h) and (W, t\) be two Hermitian spaces over E. Let L, K be S-lattices on W and V respectively. We say that L represents K, and write K —>— L, if there is a representation Whenever we study classification or representation of one lattice by another we can assume a priori that (at least in the local or global cases, because of the Hasse principle) they live in a common space (i.e. EK £ V). In this setting then, K is represented by L iff there exists a e U(V) with a(K) c L, and K & L iff there exists a e U(V) with a(K) = L. We also say that K is properly represented by h, and write K —*su — L, if there is Next we define a few of the invariants (under integral equivalence) of Hermitian lattices. The scale of L, /6(L), is defined as /6(L) = {h(x, y):x,y e L) (an 5-ideal) and the norm of L, %(L), as the 5-ideal generated by {h(x, x) : x e L}. By definition /6(L) is ambiguous, that is, /6(L) - /6(L) . The following inclusions hold true: Tr(ML)) '1 (5.1) The second inclusion being trivial we will check the first and last. Let x,y e Z, and a e S. Tr(ah(x,y)) = h(ax + y, ax + y) ~ h(ax, ax) - h(y, y) e %(L) . Therefore h(x, y) %(L)'1 £ J9"1 and thus we prove our claim.
Definition 5.3. An S-lattice L is said to be normal if %(L) = /6(L). Otherwise is called subnormal.
It follows from (5.1) that whenever EIF is unramified or split all S-Iattices are normal.
If we write L = CtjXj + ... + <*„*„, the volume of L, yL, is the fractional ideal
QjCtj—SJCJ +... + Sxn then vL = det(h(xvXi)) is called the determinant ofLandis denoted by dL (note that dL is well defined modulo norms of units). It is not difficult to see that vL c (/6 Lf. We also define the dual of a lattice L as if - {x e V: h(x, L) c 5} = {x e V: h(L, x) c 5}. It is easy to see that M — -1 — -J 1. L = We will state the following results without proof, for they follow at once from the proofs for quadratic forms (see [O'm] § 82G) and can be found explicitly in the mimeographed notes of A. Johnson (Notre Dame University, c. 1966).
Proposition 5.4. An S-lattice L is Ct-modular if and only if Olf = L.
Corollary 5.5. IfLis O-modular then L = {xe EL: h(x, L) c 0}
Definition 5.6. Let M be a sublattice ofL. We say that M splits L ifL = M J.J. In this case M (as well as J) is called a component ofL.
Proposition 5.7. Let Lbe a Hermitian S-lattice and M be an Q-modular sublattice. Then M splits L if and only ifh(M, L) £ 0.
Corollary 5.8. IfM is an Ct-modular sublattice ofL with /6(L) = 0 then M splits L. 32
A lattice L is called Q-maximal if %(L) c <*, and if L c K with %(K) c 0; implies L = /ST. The fruitful idea of maximal lattices is well developed in [Shi]. Here we present a general result about the existence of maximal lattices (see proposition 2.14 in [Shi]), and in the next section we give an Eichler-type theorem for the non-split as well as the split case.
Proposition 5.9. (Shimura) Let L be an S~lattice on V. Then there exists a maximal S-lattice Mon V such that %(M) = %(L) andM^L.
§ 6 Local Theory of Hermitian lattices
Throughout this section F will denote a p-adic field and E a quadratic extension of F or the sum of two copies of F (the split case). As before S, R denote the ring of integers of £ and F respectively. We let K0 stand for a generator of the maximal ideal p of/?, and when £ is a quadratic extension of/7, # denotes the generator of the maximal ideal of S.
We start with a result of a general nature, and then study the non-split and the split cases separately. This next result states what is expected in this case, if two local lattices are sufficiendy close then they are indeed isometric. We have taken the proof from the book of Cassels an adapted it to fit our Hermitian setting (considering the split case as well) (cf. [Cas] chapter 8, lemma 5.1). 33
Proposition 6.1.(adapted from Cassels) LetXj,..., xn ,ylt..., yR be vectors in a
Hermitian S-lattice L. Ifxi is sufficiently close to yrfor all i, then
Sfxj,..., xn] s S[yJt..., ynJ.
Proof. Let A, B be the Hermitian matrices (h(xit Xj)) and (h(y{, yj)) respectively. Let us consider three cases. (a) p inert. Write (dA) = p5 and (2) » p*- (X=Otfp %2). Suppose A =B mod ps+2X+K Putfi=5 + 2X + 1 and define C = j A'^B - A) = 2° dA (B~A)' nenceC is integral. Moreover C = 0 modpfi~s~\ thus / + C s I modp, so/+C is invertible in S. Put Aj = (I+C)*A(I + C) thenAj = A. Now a direct computation of Aj-B shows that Aj-B = 0 mod p*1*1.
Taking now Cj - ^Af^B-Aj) and A2 = (I + C1fA1 (I + Cj) we haveA2=A
ti + 2 and A2=Bmodp . In this way we finally obtain B =A. (b) p ramified. Choose 7rin E with (it)2 = PS and write (dA) = (it)5, (2) = (n)x. Assume A =B mod (nf and repeat the argument
(c) p split. S = R xR with the product topology. Let it = (nQ, n0)S = pjp2 where p1% p2 are the two only prime ideals of S, namely pj = (KQ, 1)S and p2 = (1, n0)S, corresponding to the two primes in E dividing p. If two elements a, b of S are close s then a-b e (n0 , iu0')S for s, t sufficiently large. Let m be the smallest of the two then a-b e (itf1. Thus a, bare sufficiently close iff a - b e (ic)n, for n large. We know that dA e R, so (dA)S » (nf and (2) » (itf-. Assume A =B mod (iff as before and repeat the argument. E) 34 Non-split case
Assume now that £ is a quadratic extension off. LetLbeaHeimitianS-latdce. If Lis normal then there is a vector xe L with h(x, x)S = ML), that is M:- Sx is an /6(L)~ modular sublattice of L and hence by corollary 5.8 it splits L. If L is subnormal, then %(L) c /6(L). Put /t{L) = {nf and %(L) = (xf, with/> s. Let JC, y e L so that Afjt, yjS = ,4(Z,;. We claim that M := Sx + Sy is an /6(h)- modular sublattice of L (and thus it splits L). Clearly, /6(M) = ML). Now, v(L) = (h(x, x)h(y, y) - h(x, y) h(x, y) )S
= (i?fa - z?su)S = 7l?SS i.e. v(L) = /6(Mr, and this proves our claim. Then we have
Proposition 6.2. Let L be a Hermitian S-lattice. Then L-Ljl ...±Lr where all LL are modular of rank 1 or 2. Moreover, when EIF is unramified we can take all Li to be one-dimensional.
Definition 6.3. (Jordan splitting) By grouping together the lattices with the same scale we can write
L = Lj±...J.Lt where Lt are modular and MLj) z> /6(L2) z> ••• z> /6(Lt). Such a decomposition is called a Jordan splitting.
The previous proposition shows that any Hermitian S-lattice has a Jordan splitting.
Two Jordan splittings L = Lj J. L Lt and M = Mj± L MT are said to be of the same type if / =T, dimLi = dimKit ML$ = MM() andL(- and Mi are both normal or both subnormal for every i. 35
Proposition 6.4. l Two Jordan splittings of isometric lattices are always of the same type.
Jordan splittings play a crucial role in the problems of classification and representation. It is from these splittings that Jacobowitz and Johnson obtained their invariants for the complete solution of those two problems in the local case.
The next result is the Hermitian equivalent of proposition 91:1 in [O'm] (see proposition 4.5 in [Shi]).
Proposition 6.5. (Shimura) Suppose that V is an anisotropic space (and hence dim V = 1 or 2). LetL be an C-maximal lattice on V. Then L = {xe V: h(x, x) e 0}
The next result is Eichler's theorem for Hermitian forms (see proposition 4.13 in [Shi]).
Proposition 6.6. (Shimura) Let L and M be maximal S-lattices on V. Then, L=M if and only if %(L) = %(M).
1 see [Ja] or [O'm] § 91C 36 Split case
Now we will look at the split case. LetE = FxF and letL be a Hermitian S-lattice. The next result shows how much more simple is the structure of S-lattices in the split case (cf. [Shi] proposition 3.2).
Proposition 6.7. (Shimura) Assume that E - F xF. Let L be an S-lattice on V.
There exists a basis {z$ ofV over E and fractional ideals a2R 2 a^R ^... ^ a^ such that: n (i) L = ^j(Re + Ra&Zi (ii) h(zit z-) = &, i=i J J
(Hi) %(L) - /6(L)= a2S (iv) L maximal <=> ajR = ... = ajt
Proof. Write V, = eV, V2 = eV, so V = Vj Q V2.. Similarly Lj = eh, L2-eL, so
Li = LIJ ©In.
Define £,/ = [ye V2: h(L,y) zeR } = {y <= V2: h(y, L) zeR } = {yeV2:h(L,y) £eS). Using the standard technique, it is easy to check that Lj = (L/ f -[xeVj: h(x, L/) £ e R } = {x e V2: h(x, L/ ) ce S }. We can now apply the theorem of elementary divisors (see [O'm] 81:11) to L/ andL2 to find a basis [y,-} of V2 and ideals (uniquely determined by Lj and L2) ajR 2 a2R ^ • • • .2 a^i. such that
i=l i=l
Let {xA be a basis for Vj so that h(xv yA = ed^- (dual basis with respect to the bilinear n forminducedbyAonVjXV^)- ThenL7 = ^Rx; (see [O'm] § 82F). 37 n Put zf s= jcf + yf ({z(.} is an orthonormal basis for V), then L - ^(Re + Raie)zi. If i=l n we write L = £/R xR)w;, where w/ - (7, a^Z; then the matrix of h with respect to j=i (al ° ~\ this basis is K° anj
It is clear that %(L) = ajS. Construct the lattice V = ^JR xR)(ltajUi»^en clearly
LszU and TiCL'J = ajS. Hence if L is maximal L - L', i.e. a7# = ... = anR. The converse is obvious since by the first part any lattice with norm ajS can be written in n the form 2//J xR)(l,bi)zi where a2R = 677? ^... Z2 bnR hence contained in L = U. 1=7
Now we can write L = (aj) ± ... X with a;/? ,2... ^ a„R and call such a splitting a canonical decomposition forL (cf. [Ge2] § 1).
Corollary 6.8. Let L- (aj)l... J. (an)andM = &j)-L... 1 (bn) be canonical decompositions for L and M repectively. Then L =M if and only if m~ n and aft = bfl.
Gerstein has proved a similar result for representation, which we state as our next lemma (cf. [Ge2] lemma 1.1)
Lemma 6.9. (Gerstein) Let L= (aj)l L (an)andM = foj)J- L (bm) be canonical decompositions forL and M repectively. Then M—»— L if and only if m £n and aft 2 bfi for 1 £i £m. 38
A lattice L is f aj-modular if aR = ajR = a2R = ... = a^, i.e. if it is fa>-maximal. Moreover, from proposition 6.8 we have the following Eichler-type result
Cortollary 6.10. There is only one (a)-modular (maximal) lattice of a given rank (up to isometry).
As we did for the non-split case, we can define the Jordan splitting
L = Lj.L LLt with L(- modular, and /6(L$ z> /6(Li+1)
Also we define the Jordan type as we did for the non-split case. From the previous discussion we have
Corollary 6.11. Two lattices are isometric if and only if they have the same Jordan type.
§ 7 Hermitian lattices over number fields
Perhaps one of the most important results to date in the integral theory of Hermitian forms is the Strong Approximation theorem of Shimura (1964). For an extensive generalization to algebraic linear groups see [Kne]. The aim of this section is, however, to present Shimura's result, some of its consequences, and other results which, being direct analogs of the quadratic form theory, do not seem to appear explicitly in the literature. 39 Throughout this section F will denote a number field and E a quadratic extension of F. Let (V, h) be a regularHermitia n space over E and L be a Hermitian S-lattice on V.
As we have seen in section 4, for a prime spot p of F we have
Ep = E QpFp, Vp = Ve>FFp = EpV
We also defined Sp as the integral closure of R «. It is not difficult to see that
Sp=RpS (=S<8>RRp)
Similarly we define Ctp=Rp(t (= Sp-ideal). When rcp= 2 (non-split), the above are the usual localizations of lattices and ideals, in particular Sp = Sp, where p is the only prime ideal above p. When np = l (split) e denote by p1 and p2 *h tw** primes in E above />, then ej 62 Sp = Rpx Rp and <*p = p x p where £,. is the order of Remark 7.1. Z^f L, M be two S-lattices on V, then Lp = Mpfor all except a finite number of p.
The next two results are classical and can be found in [Shi] (cf lemma 5.2 and lemma 5.3 in [Shi]).
Proposition 7.2. Let V be a vector space over E and let L be an S-lattice on V. Let
M(P) be an Sp-lattice on Vpfor each prime ideal p ofF. Then there exists an S-lattice
MonV such that Mp = M^for every p if and only if M,p) = Lpfor all except a finite number of p. If such a lattice exists then Af = /"*) (M(P\ /"> V). 40
Lemma 7.3. Let Vp be a Hermitian space over Ep and letLp be an Sp-lattice on Vp.
Let a, 7] be elements ofAutE(Vp). If (a- r\)Lp c poLp then oLp = TJ Lp.
Definition 7.4. We define the U-class of a lattice L, as the set U-cls L= {Kon V: K = oL, a e U(V)}. Likewise we define the SU-class ofL (sometimes called the proper class) as the set SU-cls L- {K onV:K= aL,ae SU(V)}. We also define the U- genus ofL and the SU-genus ofL by U-gen L = {Kon V: Kp = Op Lp, ap e U(Vp)} and SU-gen L = {Kon V: Kp = ap Lp,