Quadratic Reciprocity and the Sign of the Gauss Sum Via the Finite Weil Representation,” International Mathematics Research Notices, Vol

Gurevich, S. et al. (2010) “Quadratic Reciprocity and the Sign of the Gauss Sum via the Finite Weil Representation,” International Mathematics Research Notices, Vol. 2010, No. 19, pp. 3729–3745 Advance Access publication February 22, 2010 doi:10.1093/imrn/rnp242

Quadratic Reciprocity and the Sign of the Gauss Sum via the Finite Weil Representation

Shamgar Gurevich1, Ronny Hadani2, and Roger Howe3 1School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA, 2Department of Mathematics, The University of Texas at Austin, Texas 78712, USA, and 3Department of Mathematics, Yale University, New Haven, CT 06520-8283, USA Downloaded from Correspondence to be sent to: [email protected] imrn.oxfordjournals.org

We give new proofs of two basic results in number theory: the law of quadratic reciprocity and the sign of the Gauss sum. We show that these results are encoded in the at Weizmann Institute of Science on July 28, 2011 relation between the discrete Fourier transform and the action of the Weyl element in the Weil representation modulo p, q,andpq.

1 Introduction

Two basic results due to Gauss are the quadratic reciprocity law and the sign of the Gauss sum [8]. The ﬁrst concerns the identity p q p−1 q−1 = (−1) 2 2 , (1.1) q p where p and q are two distinct odd prime numbers and (·/p) (respectively, (·/q))isthe Legendre symbol modulo p (respectively, q), i.e. (x/p) = 1ifx is a square modulo p and −1, otherwise. The latter result asserts that √ π , ≡ ( ), 2 i x2 p p 1 mod 4 G p = e p = √ (1.2) i p, p ≡ 3 (mod 4). x∈Fp

Received December 12, 2009; Revised December 17, 2009; Accepted December 21, 2009 Communicated by Prof. Joseph Bernstein

2 = (− / ) · . In fact, it is easy to show that G p 1 p p Hence, the problem is to determine the exact sign in the evaluation of G p. In this work, we will explain how these results follow from the proportionality relations 01 Fn = Cn · ρn , for n = p, q, and pq, −10

where Fn is the discrete Fourier transform (DFT), ρn is the Weil representation of the group SL2(Z/nZ), both acting on the Hilbert space C(Z/nZ) of complex-valued functions on the ﬁnite ring Z/nZ, and Cn is the proportionality constant. More speciﬁcally, the law of quadratic reciprocity follows from basic properties of the Weil representation and group theoretic considerations, while the calculation of the sign of the Gauss sum is a Downloaded from bit more delicate and it uses a formula for the character of the Weil representation. The

fact that the DFT can be normalized so that it becomes a part of a representation plays imrn.oxfordjournals.org a crucial role in our proof of both statements. The main technical computation that we carry in this paper is of the determinant of the DFT Fn. Interestingly, in this speciﬁc calculation, we use the concrete model at Weizmann Institute of Science on July 28, 2011 of the ring Z/nZ as the set 0, 1,...,n− 1 with the standard addition and multiplication operations, while in the rest of the calculations, we do not use any speciﬁc model of Z/nZ. In his seminal work [10], Andre´ Weil recasts several known proofs of the law of quadratic reciprocity in terms of the Weil representation of some cover of the group

SL2(AQ), where AQ denotes the adele ring of Q. The main contribution of this short note is showing that quadratic reciprocity already follows from the Weil representation over ﬁnite rings and, moreover, establishing a conceptual mechanism, different from that of Weil, which produces the law of quadratic reciprocity.

1.1 Structure of the paper

In Section 2, we recall the Weil representation over the ﬁnite ring Z/nZ. We then describe the relation between the Weil representations associated with the rings Z/n1Z, Z/n2Z, and Z/n1n2Z,forn1 and n2 coprime. Finally, we describe some properties and write the formula of its character in the case n is an odd prime number. In Section 3, we deﬁne the DFT, compute its determinant, and explain its relation to the Weil representation. Quadratic Reciprocity and Sign of Gauss Sum 3731

In Section 4, we prove the quadratic reciprocity law, and in Section 5, we compute the Gauss sum. Finally, in the Appendix, we supply the proofs of the main technical claims that appear in the body of the paper.

2 The Weil Representation

2.1 The Heisenberg group

Let (V,ω) be a symplectic free module of rank 2 over the ﬁnite ring Z/nZ, where n is an odd number. The reader should think of V as Z/nZ × Z/nZ equipped with the standard skew-symmetric form ω (t,w) , t,w = tw − wt. Considering V as an abelian group, it admits a non-trivial central extension H called the Heisenberg group. Concretely, the group H can be presented as the set H = V × Z/nZ with the multiplication given by Downloaded from

(v, ) · (v, ) = (v + v, + + 1 ω(v,v)). imrn.oxfordjournals.org z z z z 2

The center of H is Z = Z(H) = {(0, z) : z ∈ Z/nZ} . The symplectic group Sp(V,ω), at Weizmann Institute of Science on July 28, 2011 which in this case is isomorphic to SL2 (Z/nZ), acts by automorphisms of H through its actionontheV-coordinate.

2.2 The Heisenberg representation

One of the most important attributes of the group H is that it admits a special family of irreducible representations. The precise statement goes as follows. Let ψ : Z → C× be a faithful character of the center (i.e. an imbedding of Z into C×). It is not hard to show

Theorem 2.1 (Stone–von Neuman). There exists a unique (up to isomorphism) irreducible representation (π, H, H) with the center acting by ψ, i.e. π|Z = ψ · IdH.

The representation π which appears in the above theorem will be called the Heisenberg representation associated with the central character ψ. π 2 i z We denote by ψ1(z) = e n the standard additive character, and for every invert- π × 2 i az ible element a ∈ (Z/nZ) , we denote ψa (z) = e n . 3732 S. Gurevich et al.

2.2.1 Standard realization of the Heisenberg representation

The Heisenberg representation (π, H, H) can be realized as follows: the Hilbert space is the space Fn = C(Z/nZ) of complex-valued functions on the ﬁnite ﬁeld, with the standard Hermitian product. The action π is given by

• π(t, 0)[ f] (x) = f (x + t) ; • π(0,w)[ f] (x) = ψ (wx) f (x) ; • π(z)[ f] (x) = ψ (z) f (x) , z ∈ Z.

Here, we are using t to indicate the ﬁrst coordinate of a typical element v ∈ V Z/nZ × Z/nZ and w to indicate the second coordinate. We call this explicit realization the standard realization. Downloaded from

2.3 The Weil representation imrn.oxfordjournals.org A direct consequence of Theorem 2.1 is the existence of a projective representation ρ n :

SL2(Z/nZ) → PGL(H). The construction of ρ n out of the Heisenberg representation π is due to Weil [10], and it goes as follows: considering the Heisenberg representation

g at Weizmann Institute of Science on July 28, 2011 π and an element g ∈ SL2(Z/nZ), one can deﬁne a new representation π acting on the same Hilbert space via π g (h) = π (g (h)). Clearly, both π and π g have the same central g character ψ; hence, by Theorem 2.1, they are isomorphic. Since the space HomH (π, π ) is one-dimensional, choosing for every g ∈ SL2(Z/nZ) a non-zero representative ρ n(g) ∈ g HomH (π, π ) gives the required projective Weil representation. In more concrete terms, the projective representation ρ is characterized by the formula

−1 ρ n (g) π (h) ρ n (g) = π (g (h)) , (2.1)

for every g ∈ SL2(Z/nZ) and h ∈ H. A more delicate statement is that there exists a lifting of ρ n into a linear representation; this is the content of the following theorem

Theorem 2.2. The projective Weil representation ρ n can be linearized into an honest representation

ρn : SL2(Z/nZ) −→ GL(H) that satisﬁes equation (2.1). Quadratic Reciprocity and Sign of Gauss Sum 3733

The existence of a linearization ρn follows from a known fact [1] that any projective representation of SL2(Z/nZ) can be linearized (in case n = p is a prime number, see also [5, 6] for an explicit construction of a canonical linearization).

Clearly, any two linearizations differ by a character χ of SL2(Z/nZ). In addition, we have the following simple lemma:

n Lemma 2.3. Let χ be a character of the group SL2(Z/nZ),thenχ = 1.

For a proof, see Section A.1.

Remark 2.4. In the case when n is not divisible by 3, the group SL2(Z/nZ) is perfect— i.e. does not have non-trivial characters—therefore, the representation ρn is unique. The Downloaded from perfectness of SL2(Z/nZ) can be proved as follows: Let pr : SL2(Z) → SL2(Z/nZ) denote × the canonical projection. Given a character χ : SL2(Z/nZ) → C , its pull-back χ ◦ pr sat- 12 isﬁes (χ ◦ pr) = 1 since the group of characters of SL2(Z) is isomorphic to Z/12Z [3]. imrn.oxfordjournals.org Since pr is surjective, this implies that χ 12 = 1. This combined with the facts that χ n = 1 (Lemma 2.3) and gcd(12, n) = 1impliesthatχ = 1. at Weizmann Institute of Science on July 28, 2011 Notation 2.5. The Weil representation ρn depends on the central character ψ; hence, sometimes we will write ρn [ψ] to emphasize this point. We will denote by ρn the Weil representation associated with the standard character ψ1.

Let n1 and n2 be coprime odd integers. Consider the natural homomorphism

Z/n1n2Z → Z/n1Z × Z/n2Z, which, by the Chinese reminder theorem, is an isomorphism. This isomorphism in- F → F ⊗ F duces an isomorphism of Hilbert spaces n1n2 n1 n2 and an isomorphism of groups SL2(Z/n1n2Z) → SL2(Z/n1Z) × SL2(Z/n2Z). In addition, the character ψ1 ⊗ ψ1 of Z/ Z × Z/ Z ψ Z/ Z n1 n2 transforms to the character n1+n2 of n1n2 . Under these identiﬁca- tions, it is not difﬁcult to show

ρ ⊗ ρ ρ ψ Claim 2.6. The representations n1 n2 and n1n2 n1+n2 , realized on the Hilbert F ⊗ F F (Z/ Z) spaces n1 n2 and n1+n2 , coincide as projective representations of SL2 n1n2 .

+ ∈ (Z/ Z)× ψ Remark 2.7. The element n1 n2 n1n2 , hence, the character n1+n2 is faithful. 3734 S. Gurevich et al.

Remark 2.8. In the case n and n are in addition not divisible by 3, Claim 2.6 1 2 ρ ψ ρ ψ ⊗ ρ ψ and Remark 2.4 imply that the representations n1n2 n1+n2 and n1 [ 1] n2 [ 1] coincide.

2.4 The character of the Weil representation

We denote by chρ : SL2(Fp) → C the character of the Weil representation ρ = ρp[ψ] associated with the non-trivial additive character ψ.

2.4.1 The dependence on the additive character

The character chρ depends on the choice of ψ;however,wehave Downloaded from

Proposition 2.9. The characters chρ[ψ] and chρ[ψ ], associated with two non-trivial additive characters ψ and ψ, agree on regular semisimple elements of SL2(Fp) if p = 3and imrn.oxfordjournals.org differ by a character of SL2(Fp) if p = 3.

For a proof, see Section A.2. at Weizmann Institute of Science on July 28, 2011

2.4.2 Formula

In the case n = p is a prime number, the absolute value of the character chρ was described in [7], but the phases have been made explicit only recently in [5]. The following formula is taken from [5]:

− det (κ (g) + I ) chρ(g) = , (2.2) p

for every g ∈ SL2(Fp) such that g − I is invertible, where (·/p) is the Legendre symbol modulo p and κ (g) = (g + I )/(g − I ) is the Cayley transform. Using the identity κ(g) + I = (2g)/(g − I ), one can write (2.2) in the simpler form

− det (g − I ) chρ(g) = . p

Remark 2.10. Sketch of the proof of (2.2) (see details in [5]). First observation is that the Heisenberg representation π and the Weil representation ρ = ρp combine to give Quadratic Reciprocity and Sign of Gauss Sum 3735 a representation of the semi-direct product τ = ρ π : SL F H → GL(H). Second 2 p observation is that the character of τ,chτ : SL2 Fp H → C, satisﬁes the following multiplicativity property

1 chτ (g · g ) = chτ (g ) ∗ chτ (g ) , (2.3) 1 2 dim H 1 2 where chτ (g) denotes the function on H given by chτ (g, −) and the ∗ operation denotes convolution with respect to the Heisenberg group action. Now, one can easily show that

1 chτ (g)(v, z) = μ · ψ ω (κ (g) v,v) + z , g 4 Downloaded from

for g ∈ SL2 Fp with g − I invertible and for some μg ∈ C. Finally, a direct calcula- imrn.oxfordjournals.org tion reveals that μ must equal (− det(κ(g) + I )/p) for (2.3) to hold. Restricting chτ to g SL2 Fp ⊂ SL2 Fp H, we obtain (2.2). at Weizmann Institute of Science on July 28, 2011 Remark 2.11. Using similar methods to these developed in [5], we can show that

− det (g − I ) chρ (g) = , n n

where ρn is the Weil representation of the group SL2(Z/nZ), the element g is such that det (g − I ) ∈ (Z/nZ)∗,and(·/n) is the Jacobi symbol modulo n (see Section 4.1).

3 The DFT

3.1 The DFT

Given an additive character ψ, there is a DFT operator Fn [ψ] acting on the Hilbert space F = C (Z/nZ) by the formula (Usually, the DFT operator appears in its normalized form n √ n = (1/ n)Fn, which makes it a unitary operator. However, the non-normalized form is better suited to our purposes.)

Fn [ψ] ( f)(y) = ψ (yx) f (x) . x∈Z/nZ 3736 S. Gurevich et al.

2 4 2 It is easy to show that Fn [ψ] ( f) (y) = n· f(−y) and, hence, Fn [ψ] = n · Id.

Notation 3.1. We will denote by Fn the DFT operator associated with the standard character ψ1.

F → F ⊗ F It is not difﬁcult to show that under the isomorphism n1n2 n1 n2 ,wehave

⊗ ψ Lemma 3.2. The operators Fn1 Fn2 and Fn1n2 n1+n2 coincide.

The main technical statement that we will require concerns the explicit evaluation of the determinant of the DFT operator which is associated with the standard character ψ1. Downloaded from

Proposition 3.3 (Main technical statement). For any odd natural number n,wehave imrn.oxfordjournals.org

n(n−1) n det (Fn) = i 2 n2 . (3.1)

For a proof, see Section A.3. at Weizmann Institute of Science on July 28, 2011

3.2 Relation between the DFT and the Weil representation

We will show that for an odd n, the operator Fn is proportional to the operator ρn (w) in the Weil representation, where w ∈ SL2 (Z/nZ) is the Weyl element 01 w = . −10

Lemma 3.4 (Key lemma). The operator ρn (w) does not depend on the choice of linearization ρn. Moreover,

Fn = Cn · ρn (w) , (3.2)

n−1 √ where Cn = i 2 n.

For a proof, see Section A.4. Quadratic Reciprocity and Sign of Gauss Sum 3737

4 The Quadratic Reciprocity Law

We are ready to prove the quadratic reciprocity law. For a natural number n , we deﬁne the Gauss sums [8]

π 2 i ax2 Gn (a) = e n , x∈Z/nZ

× where a ∈ (Z/nZ) , and we denote Gn = Gn (1). Consider two distinct odd prime numbers p and q. Using known identities of Gauss sums, we deduce that

Lemma 4.1. We have Downloaded from · p q = G p Gq . q p G pq imrn.oxfordjournals.org

For a proof, see Section A.5. The Gauss sum is related to the DFT by at Weizmann Institute of Science on July 28, 2011

Gn = Tr(Fn). (4.1)

Hence, we can write ( ) · ( ) · (ρ ( )) · (ρ ( )) p q = Tr Fp Tr Fq = C p Cq · Tr p w Tr q w q p Tr(Fpq ) C pq Tr(ρpq (w)) C p · Cq p−1 · q−1 = = (−1) 2 2 , (4.2) C pq where in the ﬁrst equality, we used Equation (4.1) and Lemma 4.1; in the second and fourth equalities, we used Lemma 3.4. Finally, the third equality follows from the following crucial identity

Proposition 4.2. We have

Tr(ρpq (w)) = Tr(ρp(w)) · Tr(ρq(w)).

For a proof, see Section A.6. 3738 S. Gurevich et al.

This completes our proof of Equation (1.1)—the Quadratic Reciprocity Law.

4.1 Quadratic reciprocity law for the Jacobi symbol

For an odd number n ∈ N,let(·/n) denote the Jacobi symbol of the multiplicative group (Z/nZ)×, which can be characterized [2] by the condition

a G (a) = G (1) , (4.3) n n n for every a ∈ (Z/nZ)×. Downloaded from Remark 4.3. The Jacobi symbol admits the following explicit description: when n = p is an odd prime number, the Jacobi symbol coincides with the Legendre character k1 kl

= × ...× imrn.oxfordjournals.org (Lemma A.3). If n p1 pl is the decomposition of ninto a product of prime numbers, then it is easy to show that for a ∈ (Z/nZ)×,

k1 kl a = a × ...× a . at Weizmann Institute of Science on July 28, 2011 n p1 pl

In particular, this implies that the Jacobi symbol is a character of the multiplicative group (Z/nZ)×, and it takes the values ±1.

The quadratic reciprocity law can be formulated in terms of the Jacobi symbol, for any two coprime odd numbers n1, n2; the general law is

− − n1 n2 n1 1 · n2 1 = (−1) 2 2 (4.4) n2 n1

The proof we just described for the quadratic reciprocity law gives also the more general identity (4.4) without changes: using Equation (4.3), it is not hard to realize that the statement of Lemma 4.1 can be formulated more generally as

n n G · G 1 2 = n1 n2 . n2 n1 Gn1n2 Quadratic Reciprocity and Sign of Gauss Sum 3739

Then applying the same derivation as in (4.2), one obtains

− − n1 n2 Cn · Cn n1 1 · n2 1 = 1 2 = (−1) 2 2 . n2 n1 Cn1n2

4.2 Alternative interpretation

A slightly more transparent interpretation of the above argument proceeds as follows:

Tr Fn Tr Fn C C Tr ρn (w) Tr ρn (w) 1 = 1 2 = n 1 n2 · 1 2 , ψ ψ ρ ψ ( ) Tr Fn1n2 n1+n2 Cn1n2 n1+n2 Tr n1n2 n1+n2 w

where the ﬁrst equality follows from Lemma 3.2 and the second equality appears by Downloaded from substituting imrn.oxfordjournals.org = ρ ( ) , Fn1 Cn1 n1 w

Fn = Cn ρn (w) , 2 2 2 ψ + = ψ + ρ ψ + ( ) . Fn1n2 n1 n2 Cn1n2 n1 n2 n1n2 n1 n2 w at Weizmann Institute of Science on July 28, 2011

Now, by Claim 2.6, we have

Tr ρn (w) Tr ρn (w) 1 2 = 1. ρ ψ ( ) Tr n1n2 n1+n2 w

Hence, the quadratic reciprocity law follows from the equivariance property of the proportionality constant C :

Proposition 4.4. Let n ∈ N be an odd number. We have

a C [ψ ] = C [ψ ] , n a n n 1 for every a ∈ (Z/nZ)×.

Remark 4.5. In our approach, the above identity is a non-trivial corollary of the proof of the quadratic reciprocity law. 3740 S. Gurevich et al.

5 The Sign of the Gauss Sum

The exact evaluation of G p for an odd prime p uses an additional fact about the Weil representation, i.e. the evaluation −2 Tr(ρ (w)) = chρ (w) = , p p p which follows directly from formula (2.2). The explicit evaluation of the Legendre symbol at −2 is a simple and well-known computation (see [9]) which gives (−2/p) = p−1 p2−1 (−1/p)(2/p) = (−1) 2 (−1) 8 . Now using (4.1), we conclude that

G p = Tr(Fp) = C p · Tr(ρp(w)) √ Downloaded from p−1 √ −2 p, p ≡ 1 (mod 4), = i 2 p · = √ p i p, p ≡ 3 (mod 4). imrn.oxfordjournals.org

This completes our proof of equation (1.2)—the Sign of Gauss Sum.

5.1 The sign of the Gauss sum modulo n at Weizmann Institute of Science on July 28, 2011

The exact evaluation of the Gauss sum for a general odd natural number n can also be derived using the above formalism and the more general formula for the character of the Weil representation given in Remark 2.11. We obtain

Gn = Tr(Fn) = Cn · Tr(ρn(w)) √ n−1 √ −2 n, n ≡ 1 (mod 4), = i 2 n· = √ n i n, n ≡ 3 (mod 4).

Acknowledgements

The ﬁrst two authors would like to thank their teacher J. Bernstein for his interest and guidance. They would also like to acknowledge M. Nori for the encouragement to write this paper. They thank C.P. Mok for explaining parts from the known proofs of quadratic reciprocity and T. Schedler for helping with various of the computations. We appreciate the time T.Y. Lam spent with us teaching math and history. Finally, we thank M. Baruch, P. Diaconis, M. Haiman, B. Poonen, and K. Ribet for the opportunities to present this work at the Technion, Israel, and the Mathematical Sciences Research Institute, Research Training Group in Representation Theory, Geometry and Combinatorics, and Number Theory Seminars at Berkeley during February 2008. Quadratic Reciprocity and Sign of Gauss Sum 3741

A Proof of Statements

A.1 Proof of Lemma 2.3

n Let χ be a character of SL2(Z/nZ). The condition χ = 1 follows from the basic fact that the group SL2(Z/nZ) is generated by the unipotent elements [3]:

11 10 u+ = , u− = , 01 11

n n which satisfy u+ = u− = Id.

A.2 Proof of Proposition 2.9 Downloaded from

Let ψ and ψ be two non-trivial additive characters of the ﬁnite ﬁeld Fp. There exists an ∗ ∈ F ψ( ) = ψ( ) ∈ F . ∈ (F ) ( ) = imrn.oxfordjournals.org element a p such that x ax for every x p Let A GL2 p with det A a, and consider the conjugate representation AdAρ[ψ] given by AdAρ[ψ](g) = ρ(dAg).Using the identity (2.1), it is easy to see that AdAρ[ψ] and ρ[ψ] differ by a character of SL2(Fp). (F ) = Moreover, the group of characters of SL2 p is trivial when p 3 (Remark 2.4). at Weizmann Institute of Science on July 28, 2011

It is enough to show that for a regular semisimple element s ∈ SL2(Fp), there ex- ist g ∈ SL2(Fp) such that Adgs = AdAs. This follows from the following general statement about conjugacy classes of regular semisimple elements in SL2(Fp):

Claim A.1. Let s and s ∈ SL2(Fp) be regular semisimple elements. Assume that there exists an element A ∈ GL2(Fp) such that AdAs = s . Then there exists an element g ∈ SL2(Fp) such that Adgs = s .

This concludes the proof of the proposition.

A.2.1 Proof of Claim A.1.

Find A ∈ GL2(Fp) such that As = s A. Let d = det(A). It is enough to ﬁnd a matrix B in the centralizer of s such that det(B) = d−1. The centralizer of s is a semisimple commutative F F × F F algebra K over p, and hence, it is isomorphic either to p p or to p2 .Soweusethe following standard lemma:

: → F∗ Lemma A.2. The norm map N K p is onto. 3742 S. Gurevich et al.

This concludes the proof of the claim.

A.3 Proof of Proposition 3.3

We work with the model of Z/nZ as the set 0, 1, 2,...,n− 1 with addition and multipli- 2πi cation modulo n. If we write ψ in the form ψ (x) = ζ x, x = 0,...,n− 1, and ζ = e n ,then yx thematrixofFn takes the form (ζ : x, y ∈ {0, 1,...,n− 1}); hence, it is a Vandermonde matrix. Applying the standard formula [4] for the determinant of a Vandermonde matrix, we get

y x det(Fn) = ζ − ζ (A.1) ≤ < ≤ − 0 x y n 1 x+y y−x x−y = ζ 2 ζ 2 − ζ 2 Downloaded from 0≤x

= ζ · 2 · 2 ( ( )) , imrn.oxfordjournals.org i 2 sin n j=1

x+y where the equality ζ 2 = ζ 0 follows from the fact that at Weizmann Institute of Science on July 28, 2011 0≤x

1 (x + y) = { (x + y) − (x + y)}=0 − 0 = 0. 2 0≤x

4 = 2 · Now, taking the absolute value on both sides of (A.1), using Fn n Idand the n−1 ( ( π· j ))n− j, positivity of sin n gives us j=1

n−1 n n(n−1) π· j − 2 = 2 ( ( ))n j; n 2 sin n j=1

n(n−1) n hence, det (Fn) = i 2 n2 . This concludes the proof of the proposition.

A.4 Proof of Lemma 3.4

First, we explain why the operator ρn (w) does not depend on the choice of linearization ρn. Any two linearization differ by a character χ of SL2 (Z/nZ); therefore, it is Quadratic Reciprocity and Sign of Gauss Sum 3743 enough to show that χ (w) = 1. By Lemma 2.3, χ n = 1; hence, χ (w) = 1sincew4 = 1and gcd (4, n) = 1.

Next, we explain the relation Fn = Cn · ρn (w). The operator ρn (w) is charac- −1 terized up to a unitary scalar by the identity (see formula (2.1)) ρn (w) π (h) ρn (w) =

π (w (h)) for every h ∈ H. Explicit computation reveals that for every h ∈ H, Fn ◦ π (h) ◦ −1 = π ( ( )) Fn w h , which implies that

Fn = Cn · ρn (w) .

Finally, we evaluate the proportionality coefﬁcient Cn. Computing determinants, ( ) = n · (ρ ( )) χ = ◦ρ one obtains det Fn Cn det n w . Arguing as above, the character det n satis- ﬁes χ (w) = 1. Hence, Downloaded from

( ) = n. det Fn Cn (A.2) imrn.oxfordjournals.org 4 = 2 · ρ ( )4 = 4 = 2 The relations Fn n Id and n w Id imply that Cn n . By Proposition n−1 √ 3.3, Equation (A.2) and using gcd (4, n) = 1, one obtains that Cn = i 2 n. This concludes the proof of the Theorem. at Weizmann Institute of Science on July 28, 2011

A.5 Proof of Lemma 4.1

∼ Consider the isomorphism Z/pZ × Z/qZ → Z/pqZ,givenby(x, y) −→ x · q + y · p. Now write

π π 2 i z2 2 i (x·q+y·p)2 G pq = e pq = e pq ∈Z/ Z ∈Z/ Z ∈Z/ Z z pq x p y q π π 2 i qx2 2 i py2 p q = e p e q = G · G , q p p q x∈Fp y∈Fq where in the last equality, we used

∈ F∗ Lemma A.3. For every a p π π 2 i ax2 a 2 i x2 e p = e p . p x∈Fp x∈Fp

This concludes the proof of the Lemma 4.1. 3744 S. Gurevich et al.

A.5.1 Proof of Lemma A.3

The statement follows from the following basic identity π π 2 i ax2 2 i ax x e p = e p ,(A.3) p x∈Fp x∈Fp

π 2 i ax which can be explained by observing that both sides are equal to e p (1 + (x/p)).

x∈Fp Now using (A.3), we can write π π π −1 2 i ax2 2 i ax x 2 i x a · x e p = e p = e p p p x∈F x∈F x∈F p p p − 1 π π Downloaded from a 2 i x x a 2 i x2 = e p = e p , p p p x∈Fp x∈Fp where, in the second equality, we applied a change of variables x → ax . This concludes imrn.oxfordjournals.org the proof of the lemma.

A.6 Proof of Proposition 4.2 at Weizmann Institute of Science on July 28, 2011

We proceed as in Claim 2.6. For any characters ψ and ψ of Z/pZ and Z/qZ, the representations ρp[ψ]⊗ρq[ψ ] and ρpq [ϕ], where ϕ is a corresponding character of Z/pqZ, differ by a character χ of the group SL2 (Z/pqZ).Sinceχ is of odd order (see Lemma 2.3) and w4 = 1, we have χ (w) = 1. Consequently, we get Tr ρpq [ϕ] (w) = Tr ρp[ψ](w) · Tr ρq[ψ ](w) . (A.4)

Since w is regular semisimple, by Proposition 2.9, the right-hand side of (A.4) does not depend on ψ and ψ. Running over all pairs ψ and ψ, we recover all faithful characters ϕ. Hence, the left-hand side is independent of the speciﬁc additive character ϕ. This concludes the proof of the proposition.

References [1] Beyl, F. R. “The Schur multiplicator of SL(2, Z/mZ) and the congruence subgroup property” Mathematische Zeitschrift 191 (1986): 23–42. [2] Berndt, B. C., R. J. Evans, and K. S. Williams. Gauss and Jacobi Sums. Canadian Mathe- matical Society Series of Monographs and Advanced Texts 21. New York: Wiley-Interscience, 1998. Quadratic Reciprocity and Sign of Gauss Sum 3745

[3] Coxeter, H. S. M., and W. O. J. Moser. Generators and Relations for Discrete Groups. Ergebnisse der Mathematik und ihrer Grenzgebiete 14. Berlin: Springer, 1980. [4] Fulton, W., and J. Harris. Representation Theory. A First Course. Graduate Texts in Mathe- matics 129. Readings in Mathematics. New York: Springer, 1991. [5] Gurevich, S., and R. Hadani. “The geometric Weil representation.” Selecta Mathematica, New Series 13, no. 3 (2007): 465–81. [6] Gurevich, S., and R. Hadani. “Quantization of symplectic vector spaces over ﬁnite ﬁelds.” Journal of Symplectic Geometry 7, no. 4 (2009): 475–502. [7] Howe, R. E. “On the character of Weil’s representation.” Transactions of the American Math- ematical Society 177 (1973): 287–98. [8] Ireland, K., and M. Rosen. A Classical Introduction to Modern Number Theory. Graduate Texts in Mathematics 84. New York: Springer, 1990. [9] Serre,J.P.A Course in Arithmetic. Graduate Texts in Mathematics 7. New York: Springer,

1973. Downloaded from [10] Weil A. “Sur certains groupes d’operateurs unitaires.” Acta Mathematica 111 (1964): 143–211. imrn.oxfordjournals.org at Weizmann Institute of Science on July 28, 2011