The Determinant and the Discriminant
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CHAPTER 2 The determinant and the discriminant In this chapter we discuss two indefinite quadratic forms: the determi- nant quadratic form det(a, b, c, d)=ad bc, − and the discriminant disc(a, b, c)=b2 4ac. − We will be interested in the integral representations of a given integer n by either of these, that is the set of solutions of the equations 4 ad bc = n, (a, b, c, d) Z − 2 and 2 3 b ac = n, (a, b, c) Z . − 2 For q either of these forms, we denote by Rq(n) the set of all such represen- tations. Consider the three basic questions of the previous chapter: (1) When is Rq(n) non-empty ? (2) If non-empty, how large Rq(n)is? (3) How is the set Rq(n) distributed as n varies ? In a suitable sense, a good portion of the answers to these question will be similar to the four and three square quadratic forms; but there will be major di↵erences coming from the fact that – det and disc are indefinite quadratic forms (have signature (2, 2) and (2, 1) over the reals), – det and disc admit isotropic vectors: there exist x Q4 (resp. Q3) such that det(x)=0(resp.disc(x) = 0). 2 1. Existence and number of representations by the determinant As the name suggest, determining Rdet(n) is equivalent to determining the integral 2 2 matrices of determinant n: ⇥ (n) ab Rdet(n) M (Z)= g = M2(Z), det(g)=n . ' 2 { cd 2 } ✓ ◆ n 0 Observe that the diagonal matrix a = has determinant n, and any 01 ✓ ◆ other matrix in the orbit SL2(Z).a is integral and has the same determinant. Thus Lemma. For any n Z, Rdet(n) is non empty and in fact infinite. 2 31 32 2. THE DETERMINANT AND THE DISCRIMINANT We have exploited the faithful action of the infinite group SL2(Z) on (n) M2 (Z) to establish its infiniteness; therefore to “count” the number of representations it is natural to consider the number of orbits under this action. (n) Proposition 1.1. For n =0,thequotientSL2(Z) M (Z) is finite and 6 \ 2 ↵+1 (n) p 1 (1.1) SL2(Z) M (Z) = d = − . | \ 2 | p 1 d n p↵ n X| Yk − Therefore (n) 1+o(1) (1.2) SL2(Z) M (Z) = n . | \ 2 | Proof. It is easy to verify that a set of representatives is given by ab M2(Z),ad= n, 0 b d 1 . { 0 d 2 − } ✓ ◆ ⇤ Written in this form the ressemblence between formulas (1.1) and (2.1) is pretty striking, the two number agreeing as long as 4 - n.Thismaybe “explained” by the fact that the Q-algebras B and M2 are “forms” of each other, and precisely, for any prime p = 2, one has 6 B(Zp):=B(Z) Zp M2(Zp). ⌦Z ' 1.1. The algebra of matrices as a quaternion algebra. As we see, the algebra of 2 2 matrices play the same role as the Hamilton quaternions ⇥ for sums of four squares. In fact M2(Q)isaquaternion algebra (in the sense of Chap. 10) and is the simplest possible one, the split (unramified) quaternion algebra. For instance, M2(Q) may be written into the form M2(Q)=QId + QI + QJ + QK with 10 01 01 I = ,J= ,K= 0 1 10 10 ✓ − ◆ ✓ ◆ ✓ − ◆ satisfying I2 = J 2 =Id,IJ= JI = K. − The canonical anti-involution on M2(Q) is given by t ab d b 1 ab g = g = − = w− w cd 7! ca cd ✓ ◆ ✓ − ◆ ✓ ◆ 0 1 with w = , 10− ✓ ◆ and corresponding reduced trace and reduced norm are just the usual trace and determinant (up to identifying Q with the algebra of scalar matrice Z = Q.Id): m + m =(a + d)Id = tr(m)Id,mm =det(m)Id; 1. EXISTENCE AND NUMBER OF REPRESENTATIONS BY THE DETERMINANT 33 and, again the“trace”and the“determinant”of m M2(Q) acting on M2(Q) by left multiplication is twice and the square of the2 usual trace and deter- minant. The group of units M2⇥(Q) is the linear group GL2(Q), and the subgroup (1) of units of norm one M2 (Q) is the special linear group SL2(Q). Considering (M2(Q), det) as a quadratic space, one has an isomorphism of Q-algebraic groups GL GL /∆Z⇥ SO 2 ⇥ 2 ' M2 (∆Z⇥ the subgroup of scalar matrices diagonally embedded in GL2 GL2) induced by ⇥ GL2 GL2 SOM2 ⇢ : ⇥ 7! 1 (g, g ) ⇢ : m gmg − 0 7! g,g0 7! 0 1.1.1. Trace zero matrices. As for Hamilton quaternions, the stabilizer of the subspace of scalar matrices in GL GL /∆Z is 2 ⇥ 2 ⇥ ∆GL2/∆Z⇥ =PGL2, and the orthogonal subspace to the scalars is the space of trace-zero matrices 0 M (Q)= m M2(Q), tr(m)=0 : 2 { 2 } 0 in other terms the action of GL2 on M2 by conjugation induces the isomor- phism PGL2 SOM 0 ⇢ : 7! 2 . g m gmg 1 7! 7! − 1.1.2. The order of integral matrices. The order corresponding to the integral Hamilton quaternions B(Z) is the ring of 2 2 integral matrices ⇥ Id + I + J + K M2(Z)=OM2 = Z[I,J,K, ]. 2 Its groups of units, and of units of norm one are, respectively, (1) ⇥ =GL ( ), ( )=SL ( ). OM2 2 Z OM2 Z 2 Z The analog of Theorem ?? and its corollary is Proposition 1.2. One has – The order OM2 is a maximal order and any maximal order of M2(Q) is conjugate to M2(Z). – It is principal: any left (resp. right) OM2 -ideal I M2(Q) is of the ⇢ form OM2 .g (resp. g.OM2 )forsomeg GL2(Q) uniquely defined 2 up to left (resp. right) multiplication by an element of GL2(Z). Proof. We merely sketch the proof: the main point is the introduction of the lattices in Q2 (ie. the finitely generated Z-modules of Q2 of maximal 2 rank, for instance the square lattice Z ) and the fact that GL2(Q) act tran- sitively on the space of lattices. One show that any order O M2(Q)is contained in ⇢ OL := EndZ(L) 34 2. THE DETERMINANT AND THE DISCRIMINANT 2 where L Q is a lattice (check that OL is an order). For instance, O OL for L the⇢ lattice ⇢ 2 2 L := x Z ,xO Z . 2 { 2 ⇢ } Writing L = Z .g, g GL2(Q), one obtain that 2 1 gOg− O = O 2 . ⇢ M2 Z Similarly, if I M2(Q)isaleftOM2 -ideal, ⇢ 2 L = Z .I 2 2 is a lattice and HomZ(Z ,L)=I.WritingL = Z .g one has I = OM2 .g. We refer to [Vig80, Chap. 2, Thm. 2.3 ] for greater details (there the above statements are proven for non-archimedean local field, but the proof carry over since Z is principal.) ⇤ 2. The distribution of integral matrices of large determinant Having counted the “number” of representation on an integer by the determinant (and found that there are “more and more” asthe integer grows) we adress the third question: How are these many representations distributed as n ? (n!1) ( n) Firstly we may assume that n is non-negative since M2 (Z)=m.M2− (Z) where m is any integral matrix of determinant 1. Next we may proceed 1/2 (n) − as before, and, dividing by n ,projectM2 (Z) on the set of matrices of determinant 1 1/2 (n) n− M (Z) SL2(R). 2 ⇢ Now SL2(R) is a locally compact (unimodular) group and endowed with some Haar measure (well defined up to multiplication by a positive scalar) µSL2 . One has the following equidistribution theorem 1/2 (n) Theorem 2.1. As n + , n− M (Z) becomes equidistributed into ! 1 2 SL2(R) w.r.t. µSL2 in the following sense: for '1,'2 Cc(SL2(R)) such that µ (' ) =0,then 2 SL2 2 6 1/2 (n) '1( det g − g) g M (Z) µSL ('1) 2 2 | | 2 ,n . 1/2 (n) '2( det g g) ! µSL ('2) !1 Pg M (Z) − 2 2 2 | | More precisely,P there is a positive constant λ>0 depending only on the choice of the measure µSL2 such that for any ' Cc(SL2(R)), 2 1/2 (2.1) '( det g − g)= | | (n) g M (Z) 2 X2 (n) (n) λµSL ( )(') SL2(Z) M (Z) + o( SL2(Z) M (Z) ). 2 R | \ 2 | | \ 2 | Remark. This definition of equidistribution takes care of the fact that µSL2 is a not a finite measure. 2. THE DISTRIBUTION OF INTEGRAL MATRICES OF LARGE DETERMINANT 35 2.0.3. Sketch of the proof of Theorem 2.1. Clearly the first part of the theorem follows from the second one. Let G =SL2(R) and Γ= SL2(Z); this is a discrete subgroup therefore acting properly on G and the (right-invariant) quotient of the Haar measure µG by the counting measure on Γ, µΓ G is finite; in a way this is a measure \ (n) analog of the fact that the quotient Γ M (Z) is finite. Up to multiplying \ 2 µG by a scalar, we will therefore assume that µΓ G is a probability measure. \ For g GL2(R)weset 2 1/2 g˜ = det g − g. | | Let ' be a smooth compactly supported function on G, one has '(˜gn)= 'Γ(˜gn) (n) (n) gn M (Z) gn Γ M (Z) 2X2 2 \X2 where ' (g) is the function on Γ G defined by Γ \ (2.2) 'Γ(g)= '(γg), γ Γ X2 (n) (the notationg ˜n for gn Γ M (Z)is(well)definedintheevidentway).