ELLIPTIC CURVES, RANDOM MATRICES AND ORBITAL INTEGRALS 1. INTRODUCTION

Transcription

1 ELLIPTIC CURVES, RANDOM MATRICES AND ORBITAL INTEGRALS JEFFREY D. ACHTER AND JULIA GORDON ABSTRACT. An isogeny cass of eiptic curves over a finite fied is determined by a quadratic Wei poynomia. Gekeer has given a product formua, in terms of congruence considerations invoving that poynomia, for the size of such an isogeny cass (over a finite prime fied). In this paper, we give a new, transparent proof of this formua; it turns out that this product actuay computes an adeic orbita integra which visiby counts the desired cardinaity. This answers a question posed by N. Katz in [13, Remark 8.7] and extends Gekeer s work to ordinary eiptic curves over arbitrary finite fieds. 1. INTRODUCTION The isogeny cass of an eiptic curve over a finite fied F p of p eements is determined by its trace of Frobenius; cacuating the size of such an isogeny cass is a cassica probem. Fix a number a with a 2 p, and et I(a, p) be the set of a eiptic curves over F p with trace of Frobenius a. Further suppose that p a, so that the isogeny cass is ordinary. Gekeer proposes ([9]; see aso [13]) a random matrix mode to compute the size of I(a, p). For each rationa prime = p, et # {γ GL 2 (Z/ (1.1) ν (a, p) = n ) : tr(γ) a mod n, det(γ) p mod n } im n # SL 2 (Z/ n )/ n. For = p, et # {γ M 2 (Z/p (1.2) ν p (a, p) = n ) : tr(γ) a mod p n, det(γ) p mod p n } im n # SL 2 (Z/p n )/p n. On average, the number of eements of GL 2 (Z/ n ) with a given characteristic poynomia is # GL 2 (Z/ n )/(#(Z/ n ) n ). Thus, ν (a, p) measures the departure of the frequency of the event that a random matrix γ satisfies f γ (T) = T 2 at + p from the average (over a possibe characteristic poynomias). It turns out that [9, Thm. 5.5] (1.3) #I(a, p) = 2 1 pν (a, p) ν (a, p), where ν (a, p) = 2 π 1 a2 4p, #I(a, p) is a count weighted by automorphisms (2.1), and we note that the term H (a, p) of [9] actuay computes 2 #I(a, p) (see [9, (2.1) and (2.13)] and [13, Theorem 8.5, p. 451]). This equation is amost miracuous. An equidistribution assumption about Frobenius eements, which is so strong that it can t possiby be true, eads one to the correct concusion. JDA s research was partiay supported by grants from the Simons Foundation (24164) and the NSA (H and H ). JG s research was supported by NSERC. 1

2 2 JEFFREY D. ACHTER AND JULIA GORDON In contrast to the heuristic, the proof of (1.3) is somewhat pedestrian. Let a,p = a 2 4p, et K a,p = Q( a,p ), and et χ a,p be the associated quadratic character. Cassicay, the size of the isogeny cass I(a, F p ) is given by the Kronecker cass number H( a,p ). Direct cacuation [9] shows that, at east for unramified primes, ν (a, p) = 1 1 χ a,p() is the term at in the Euer product expansion of L(1, χ a,p ). More generay, a term-by-term comparison shows that the right-hand side of (1.3) computes H( a,p ). Even though (1.3) is striking and unconditiona, one might sti want a pure thought derivation of it. (We are not aone in this desire; Katz cas attention to this question in [13, Rem. 8.7].) Our goa in the present paper is to provide a conceptua expanation of (1.3). We wi show that Gekeer s random matrix mode (i.e., the right-hand side of (1.3)) directy cacuates #I(a, p), without appea to cass numbers. A further payoff of our method is that we extend to Gekeer s resuts to the case of ordinary eiptic curves over an arbitrary finite fied F q. Our method reies on the description, due to Langands (for moduar curves) and Kottwitz (in genera), of the points on a Shimura variety over a finite fied. A consequence of their study is that one can cacuate the cardinaity of an ordinary isogeny cass of eiptic curves over F q using orbita integras on the finite adeic points of GL 2 (Proposition 2.1). Our main observation is that one can, without expicit cacuation, reate each oca factor ν (a, q) to an orbita integra (1.4) 1 GL2 (Z )(x 1 γ x) dx, G γ (Q )\ GL 2 (Q ) where γ is an eement of GL 2 (Q ) of trace a and determinant q, G γ is its centraizer in GL 2 (Q ), and 1 GL2 (Z ) is the characteristic function of the maxima compact subgroup GL 2 (Z ). Here the choice of the invariant measure dx on the orbit is crucia. On one hand, the measure that is naturay reated to Gekeer s numbers is the so-caed geometric measure (cf. [8]), which we review in On the other hand, this measure is inconvenient for computing the goba voume term that appears in the formua of Langands and Kottwitz. The main technica difficuty is the comparison, which shoud be we-known but is hard to find in the iterature, between the geometric measure and the so-caed canonica measure. We start ( 2) by estabishing notation and reviewing the Langands-Kottwitz formua. We define the reevant, natura measures in 3, and study the comparison factor between them in 4. Finay, in 5, we compete the goba cacuation. It is perhaps not surprising that one can use a simiar method to give an anaogous product formua for the size of an isogeny cass of simpe ordinary principay poarized abeian varieties over a finite fied. (The fact that the group controing the modui probem is GSp 2g rather than GL 2 means that, for exampe, conjugacy and stabe conjugacy no onger coincide; the expicit invocation of the fundamenta emma is more invoved; the comparison of measures (Proposition 4.5) is more difficut; the goba voume cacuation is ess immediate; etc.) We take up this chaenge in a companion work. It turns out that [8, 3] has much of the information one needs for the crucia comparison of measures. This is expained in the appendix ( A) by S. Ai Atuğ. As we were finishing this paper, the authors of [6] shared their preprint with us, which takes Gekeer s random matrix mode as its starting point; we invite the interested reader to consut that work.

3 ELLIPTIC CURVES, RANDOM MATRICES AND ORBITAL INTEGRALS 3 Notation. Throughout, F q is a finite fied of characteristic p and cardinaity q = p e. Let Q q be the unique unramified extension of Q p of degree e, and et Z q Q q be its ring of integers. We use σ to denote both the canonica generator of Ga(F q /F p ) and its ift to Ga(Q q /Q p ). Typicay, G wi denote the agebraic group GL 2. Whie many of our resuts admit immediate generaization to other reductive groups, as a rue we resist this temptation uness the statement and its proof require no additiona notation. Shorty, we wi fix a reguar semisimpe eement γ G(Q) = GL 2 (Q); its centraizer wi variousy be denoted G γ and T. Conjugacy in an abstract group is denoted by. Acknowedgment. We have benefited from discussions with Bi Casseman, Cifton Cunningham, David Roe, Wiiam Sawin, and Sug-Woo Shin, as we as from suggestions from the referee. We are particuary gratefu to Luis Garcia for sharing his insights. It is a great peasure to thank these peope. 2. PRELIMINARIES Here we coect notation concerning isogeny casses (2.1) as we as basic information on Gekeer s ratios (2.3) and the Langands-Kottwitz formua (2.4) Isogeny casses of eiptic curves. If E/F q is an eiptic curve, then its characteristic poynomia of Frobenius has the form f E/Fq (T) = T 2 a E/Fq T + q, where a E/Fq 2 q. Moreover, E1 and E 2 are F q -isogenous if and ony if a E1 /F q = a E2 /F q. In particuar, for a given integer a with a 2 q, the set { } I(a, q) = E/F q : a E/Fq = a is a singe isogeny cass of eiptic curves over F q. Its weighted cardinaity is 1 (2.1) #I(a, q) := # Aut(E). E I(a,q) A member of this isogeny cass is ordinary if and ony if p a; henceforth, we assume this is the case. Fix an eement γ G(Q) with characteristic poynomia f (T) = f a,q (T) := T 2 at + q. Newton poygon considerations show that exacty one root of f a,q (T) is a p-adic unit, and in particuar f a,q (T) has distinct roots. Therefore, γ is reguar semisimpe. Moreover, any other eement of G(Q) with the same characteristic poynomia is conjugate to γ. (Here and esewhere, we use the fact that in a genera inear group, two eements are conjugate if and ony if they are staby conjugate.) Let K = K a,q = Q[T]/ f (T); it is a quadratic imaginary fied. If E I(a, q), then its endomorphism agebra is End(E) Q = K. The centraizer G γ of γ in G is the restriction of scaars torus G γ = RK/Q G m. If α is an invariant of an isogeny cass, we wi variousy denote it as α(a, q), α( f ), or α(γ ), depending on the desired emphasis.

4 4 JEFFREY D. ACHTER AND JULIA GORDON 2.2. The Steinberg quotient. We review the genera definition of the Steinberg quotient. Let G be a spit, reductive group of rank r, with simpy connected derived group G der and Lie agebra g; further assumpe that G/G der = G m. (In the case of interest for this paper, G = GL 2, r = 2, and G der = SL 2.) Let T be a spit maxima torus in G, T der = T G der (note that T der is not the derived group of T), and et W be the Wey group of G reative to T. Let A der = T der /W be the Steinberg quotient for the semisimpe group G der. It is isomorphic to the affine space of dimension r 1. Let A = A der G m be the anaogue of the Steinberg quotient for the reductive group G, cf [8]. We think of A as the space of characteristic poynomias. There is a canonica map (2.2) G c A Since G/G der = G m, we have A = A r 1 G m A r Gekeer numbers. We resume our discussion of eiptic curves, and et G = GL 2. As in 2.1, fix data (a, q) defining an ordinary isogeny cass over F q. Reca that, to each finite prime, Gekeer has assigned a oca probabiity ν (a, q) (1.1)-(1.2). We give a geometric interpretation of this ratio, as foows. Since G is a group scheme over Z, for any finite prime, we have a we-defined group G(Z ), which is a (hyper-specia) maxima compact subgroup of G(Q ), as we as the truncated groups G(Z / n ) for every integer n. Reca that, given the fixed data (a, q), we have chosen an eement γ G(Q). Since the conjugacy cass of a semisimpe eement of a genera inear group is determined by its characteristic poynomia, γ is we-defined up to conjugacy. Let be any finite prime (we aow the possibiity = p); using the incusion Q Q we identify γ with an eement of G(Q ). In fact, if = p, then γ is a reguar semisimpe eement of G(Z ). For a fixed (notationay suppressed) positive integer n, the average vaue of #c 1 (a), as a ranges over A(Z / n ), is #G(Z / n )/#A(Z / n ). Consequenty, we set (2.3) ν,n (a, q) = ν,n (γ ) = # {γ G(Z / n ) : γ (γ mod n )} #G(Z / n )/#A(Z / n, ) and rewrite (1.3) (and extend it to the case of F q ) as (2.4) ν (a, q) = im n ν,n (a, q). Again, we have expoited the fact that two semisimpe eements of GL 2 are conjugate if and ony if their characteristic poynomias are the same. Note that the denominator of (2.4) coincides with that of Gekeer s definition [9, (3.7)]. Indeed, (2.5) #G(Z/ n )/#A(Z/ n ) = ( 1)(2 1) 4n 4 ( 1) n 1 n = ( 2 1) 2n 2. For = p, γ ies in GL 2 (Q p ) Mat 2 (Z p ). We make the apparenty ad hoc definition { # γ Mat2 (Z p /p n ) : γ (γ mod p n ) } (2.6) ν p (a, q) = im n #G(Z p /p n )/#A(Z p /p n, )

5 ELLIPTIC CURVES, RANDOM MATRICES AND ORBITAL INTEGRALS 5 where we have briefy used to denote simiarity of matrices under the action of GL 2 (Z p /p n ). In the case where q = p, this recovers Gekeer s definition (1.2). Finay, we foow [9, (3.3)] and, inspired by the Sato-Tate measure, define an archimedean term (2.7) ν (a, q) = 2 π 1 a2 4q The Langands and Kottwitz approach. For Shimura varieties of PEL type, Kottwitz proved [15] Langands s conjectura expression of the zeta function of that Shimura variety in terms of automorphic L-functions on the associated group. A key, abeit eementary, too in this proof is the fact that the isogeny cass of a (structured) abeian variety can be expressed in terms of an orbita integra. The specia case where the Shimura variety in question is a moduar curve, so that the abeian varieties are simpy eiptic curves, has enjoyed severa detaied presentations in the iterature (e.g., [5], [2] and, to a esser extent, [1]), and so we content ourseves here with the reevant statement. As in 2.1, fix data (a, q) which determines an isogeny cass of ordinary eiptic curves over F q, and et γ G(Q) be a suitabe choice. If E I(a, q), then for each q there is an isomorphism H 1 (E F q, Q ) = Q 2 which takes the Frobenius endomorphism of E to γ. There is an additive operator F on Hcris 1 (E, Q q). It is σ-inear, in the sense that if a Q q and x Hcris 1 (E, Q q), then F(ax) = a σ F(x). To F corresponds some δ G(Q q ), we-defined up to σ-conjugacy. (Reca that δ and δ are σ-conjugate if there exists some h G(Q q ) such that h 1 δh σ = δ.) The two eements are reated by N Qq /Q p (δ ) γ. Let G γ be the centraizer of γ in G. Let G δ σ be the twisted centraizer of δ in G Qq ; it is an agebraic group over Q p. Finay, et A p f denote the prime-to-p finite adees, and et Ẑp f A p f be the subring of everywhereintegra eements. With these notationa preparations, we have Proposition 2.1. The weighted cardinaity of an ordinary isogeny cass of eiptic curves is given by #I(a, q) = vo(g γ (Q)\G γ (A f )) 1 G γ (A p G(Ẑ p f )\G(Ap f ) f )(g 1 γ g) dg (2.8) 1 ( G δ σ(q p )\G(Q q ) G(Zq ) 1 )G(Z q ) (h 1 δ h σ ) dh. p Here, each group G(Q ) has been given the Haar measure which assigns voume one to G(Z ) (this is the so-caed canonica measure, see 3.1.2). The choice of nonzero Haar measure on the centraizer G γ (Q ) is irreevant, as ong as the same choice is made for the goba voume computation. Simiary, in the second, twisted orbita integra, G(Q q ) is given the Haar measure which assigns voume one to G(Z q ). Since we sha need to abe to say something about the voume term ater, we need to fix the measures on G γ (Q ) for every. We choose the canonica measures µ can on both G and G γ at every pace. These measures are defined beow in The idea behind Proposition 2.1 is straight-forward. (We defer to [5] for detais.) Fix E I(a, q) and H 1 (E F q, Q ) = Q 2 as above. This singes out an integra structure H 1 (E F q, Z ) Q 2. If E is any other member of I(a, q), then the prime-to-p part of an F q -rationa isogeny induces E E gives a new integra structure H 1 (E F, Z q ) on Q 2. Simiary, p-power isogenies give rise to new integra structures on the crystaine cohomoogy Hcris 1 (E, Q q). In this way, I(a, q) is identified with K \Y p Y p, where Y p ranges among γ -stabe attices in Y 1 (E F q, A p ), and Y p ranges among

6 6 JEFFREY D. ACHTER AND JULIA GORDON attices in Hcris 1 (E, Q q) stabe under δ and pδ 1. It is now straight-forward to use an orbita integra to cacuate the automorphism-weighted, or groupoid, cardinaity of the quotient set K \Y p Y p (e.g., [12, 6]). We remark that most expositions of Proposition 2.1 refer to a geometric context in which 1 G(Ẑ p f ) is repaced with the characteristic function of an open compact subgroup which is sufficienty sma that objects have trivia automorphism groups, so that the corresponding Shimura variety is a smooth and quasiprojective fine modui space. However, this assumption is not necessary for the counting argument underying (2.8); see, for instance, [5, 3(b)]. 3. COMPARISON OF GEKELER NUMBERS WITH ORBITAL INTEGRALS The cacuation is based on the interpay between severa G-invariant measures on the adjoint orbits in G. We start by carefuy reviewing the definitions and the normaizations of a Haar measures invoved Measures on groups and orbits. Let π n : Z Z / n be the truncation map. For any Z - scheme X, we denote by πn X the corresponding map induced by π n. π X n : X (Z ) X (Z / n ) Once and for a, fix the Haar measure on A 1 (Q ) such that the voume of Z is 1. We wi denote this measure by dx. For our cacuations, the key observation is that, with this normaization, the fibres of the standard projection π Ad n : A d (Z ) A d (Z/ n Z) have voume nd. There are two fundamenta approaches to normaizing a Haar measure on the set of Q -points of an arbitrary agebraic group G: one can either fix a maxima compact subgroup and assign voume 1 to it; or one can fix a voume form ω G on G with coefficients in Z, and thus get the measure ω G on each G(Q ). For the Q -points of a genera variety, one aso has the Serre-Oesteré measure; it is this measure which naturay arises in studying Gekeer-type ratios. In the case of GL 2, this measure comes from the voume form which Gross cas canonica. We now review these constructions and the reations between them Serre-Oesteré measure. Let X be a smooth scheme over Z. Then there is the so-caed Serre- Oesteré measure on X, which we wi denote by µ SO X. It is defined in [21, 3.3], see aso [23] for an attractive equivaent definition. For a smooth scheme that has a non-vanishing gauge form this definition coincides with the definition of A. Wei [24], and by [24, Theorem 2.2.5] (extended by Batyrev [3, Theorem 2.7]), this measure has the property that vo µ SO(X (Z X )) = #X (F ) d, where d is the dimension of the generic fiber of X. In particuar, µ SO is the Haar measure on the affine A 1 ine such that vo µ SO(A 1 (Z A )) = 1 = 1, i.e., µ SO A 1 (Q ) coincides with dx. Simiary, on any d-dimensiona affine space A d, the Serre-Oesteré measure gives A d (Z ) voume 1. The agebraic group GL 2 is a smooth group scheme defined over Z; in particuar, for every, GL 2 Z Z is a smooth scheme over Z, so µ SO gives GL 2 (Z ) voume vo µ SO GL2 (GL 2 (Z )) = # GL 2(F ) d = ( 1)(2 1) 4.

7 ELLIPTIC CURVES, RANDOM MATRICES AND ORBITAL INTEGRALS The canonica measures. Let G be a reductive group over Q ; then Gross [11, Sec. 4] defines a canonica integra mode G/Z. If G is unramified and connected, then G(Z ) is a hyperspecia maxima compact subgroup of G(Q ). If T is a (possiby ramified) torus, then T is the identity component T of the weak Néron mode T of T (discussed in more detai in 4.1). The measure most commony used in orbita integras, µ can, is the Haar measure which assigns voume 1 to G(Z ). In fact, Gross uses G to define a canonica voume form ω G, which does not vanish on the specia fiber G κ of G. If G is unramified over Q, then ω G recovers the Serre-Oesteré measure, insofar as ω G = #Gκ (F ) dim G [11, Prop. 4.7]. G(Z ) The geometric measure. We wi use a certain quotient measure µ geom on the orbits, which is caed the geometric measure in [8]. This measure is defined using the Steinberg map c (2.2); we return to the setting of 2.2. For a genera reductive group G and γ G(Q ) reguar semisimpe, the fibre over c(γ) is the stabe orbit of γ, which is a finite union of rationa orbits. In our setting with G = GL 2, the fibre c 1 (c(γ)) is a singe rationa orbit, which substantiay simpifies the situation. From here onwards, we work ony with G = GL 2. Consider the measure given by the form ω G on G, and the measure on A = A 1 G m which is the product of the measures associated with the form dt on A 1 and ds/s on G m, where we denote the coordinates on A by (t, s). We wi denote this measure by dω A. The form ω G is a generator of the top exterior power of the cotangent bunde of G. For each orbit c 1 (t, s) (note that such an orbit is a variety) there is a unique generator ω geom of the top exterior c(γ) power of the cotangent bunde on the orbit c 1 (c(γ)) such that Then for any φ Cc (G(Q )), φ(g) dω G = G(Q ) ω G = ω geom c(γ) ω A. A(Q ) c 1 (c(γ)) φ(g) dω geom c(γ) dω A (t, s). This measure aso appears in [8], and it is discussed in detai in 4 beow Orbita integras. There are two kinds of orbita integras that wi be reevant for us; they differ ony in the normaization of measures on the orbits. Let γ be a reguar semisimpe eement of G(Q ), and et φ be a ocay constant compacty supported function on G(Q ). Let T be the centraizer G γ of γ. Since γ is reguar (i.e., the roots of the characteristic poynomia of γ are distinct) and semisimpe, T is a maxima torus in G. First, we consider the orbita integra with respect to the geometric measure: Definition 3.1. Define Oγ geom (φ) by O geom γ (φ) := T(Q )\G(Q ) φ(g 1 γg)dµ geom γ, where µ geom γ is the measure on the orbit of γ associated with the corresponding differentia form ω geom c 1 (c(γ)) as in above.

8 8 JEFFREY D. ACHTER AND JULIA GORDON Second, there is the canonica orbita integra over the orbit of γ, defined as foows. The orbit of γ can be identified with the quotient T(Q )\G(Q ). Both T(Q ) and G(Q ) are endowed with canonica measures, as above in Then there is a unique quotient measure on T(Q )\G(Q ), which wi be denoted µ can γ. The canonica orbita integra wi be the integra with respect to this measure on the orbit (aso considered as a distribution on the space of ocay constant compacty supported functions on G(Q )): Definition 3.2. Define Oγ can (φ) by O can γ (φ) := By definition, the distributions O geom γ T(Q )\G(Q ) and O can γ φ(g 1 γg)dµ can γ. differ by a mutipe that is a function of γ. This ratio (which we fee shoud probaby be we-known but was hard to find in the iterature, see aso [8] and the appendix A) is computed in 4 beow. We wi first reate Gekeer s ratios to orbita integras with respect to the geometric measure, in a natura way, and from there wi get the reationship with the canonica orbita integras, which are more convenient to use for the purposes of computing the goba voume term appearing the formua of Langands and Kottwitz Gekeer numbers and voumes, for not equa to p. From now on, G = GL 2, γ = γ a,q, and is a fixed prime distinct from p. Our first goa is to reate the Gekeer number ν (a, q) (2.4) to an orbita integra Oγ geom (φ ) of a suitabe test function φ with respect to dω geom c(γ). (Reca that γ is the eement of G(Q ) determined by E, and in this case since = p, it ies in G(Z ).) In order to do this we define natura subsets of G(Q ) whose voumes are responsibe for this reationship. Reca (2.3) the definition of ν,n (γ ). For each positive integer n, consider the subset V n of GL 2 (Z ) defined as (3.1) (3.2) and set V n = V n (γ ) := {γ GL 2 (Z ) f γ (T) f (T) mod n } { } = γ GL 2 (Z ) πn A (c(γ)) = πn A (c(γ )). (3.3) V(γ ) := n 1 V n (γ ). We define an auxiiary ratio: (3.4) v n (γ ) := vo µ SO GL2 (V n (γ )) 2n. Now we woud ike to reate the imit of these ratios v n (γ ) both to the imit of Gekeer ratios ν,n (γ ) and to an orbita integra. Let φ = 1 GL2 (Z ) be the characteristic function of the maxima compact subgroup GL 2 (Z ) in GL 2 (Q ). Proposition 3.3. We have im n v n(γ ) = O geom γ (φ ).

9 ELLIPTIC CURVES, RANDOM MATRICES AND ORBITAL INTEGRALS 9 Proof. Because equaity of characteristic poynomias is equivaent to conjugacy in GL 2 (Q ), V(γ ) is the intersection of GL 2 (Z ) with the orbit O(γ ) of γ in G = GL 2 (Q ). Then the orbita integra Oγ geom (φ ) is nothing but the voume of the set V(γ ), as a subset of O(γ ), with respect to the measure dµ geom γ. Let a = c(γ ) = (a, q) A 1 G m (Q ), and et U n (a ) be its n n -neighborhood. Its Serre- Oestere voume is vo µ SO A (U n (γ )) = 2n. Moreover, V n (γ ) = c 1 (U n (γ )) GL 2 (Z ). Consequenty, (3.5) vo (c 1 (U im v n(γ ) = im µso n (γ )) GL 2 (Z GL )) 2 n n vo µ SO(U n (γ )) A by definition of the geometric measure. vo dωg (c 1 (U n (γ )) GL 2 (Z )) = im = vo geom n vo dωa (U n (γ )) µ (V(γ γ )), Next, et us reate the ratios v n to the Gekeer ratios. Proposition 3.4. The ratios v n (γ ) (and thus, aso ν,n (γ )) stabiize, in the sense that when n is arge enough, v n (γ ) = im n v n (γ ), and we have im v n(γ ) = # SL 2(F ) n 3 im ν,n (γ ) n = ν (a, q). Remark 3.5. We note that we do not need the caim that Gekeer s ratios ν,n stabiize for arge n in order to reate them to the orbita integras. However, we have incuded this caim in order to point out that this behaviour (aso proved by Gekeer by direct computation) is a specia case of a very genera phenomenon (which can be thought of as a muti-variabe version of Hense s emma) that has appeared in the work of Igusa, Serre, and ater Veys, Denef, and others, and was at the foundation of the theory of motivic integration (cf. [22] for reated resuts), but does not appear to be widey known. We provide more specific references in the proof of the proposition. Proof. Let π n = π GL 2 n : GL 2 (Z ) GL 2 (Z/ n ). To ease notation sighty, et V n = V n (γ ). Let S n GL 2 (Z/ n Z) be the set that appears in the numerator of (2.3): S n := {γ GL 2 (Z/ n ) f γ (T) f (T) mod n }. First, observe that for a n 1, we have V n = πn 1 (S n ). Indeed, taking characteristic poynomias commutes with reduction mod n, since the coefficients of the characteristic poynomia are themseves poynomia in the matrix entries of γ, and reduction mod n is a ring homomorphism. We caim that, for arge enough n (with he restriction depending on the discriminant of f ), the foowing hod: (i) π n Vn : V n S n is surjective; (ii) we have the equaity (3.6) vo µ SO GL2 (V n ) = 4n #S n. (iii) the number 2n vo µ SO GL2 (V n ) does not depend on n.

10 1 JEFFREY D. ACHTER AND JULIA GORDON We ony need the second and third caims to estabish the Proposition; we have singed out the first caim since it is key to the proof of the caims (ii) and (iii). First, et us finish the proof of the Proposition assuming (ii) hods. Handing the denominator of Gekeer s ratio as in (2.5) above, we get: (3.7) v n (γ ) = 4n #S n 2n as required. = #S n 2n = #S n # SL 2 (F ) # SL 2 (F ) 3(n 1) n 3 = # SL 2(F ) 3 ν,n (γ ), Thus, it remains to address the three caims. The set V(γ ) is the subset of A 2 (Z ) cut out by the agebraic equations tr(γ) = tr(γ ) and det(γ) = det(γ ). Since γ is a reguar semisimpe eement, these equations define a 2-dimensiona -adic anaytic submanifod of A 4 (namey, the orbit of γ ). For such submanifods, a three caims were proved by J.-P. Serre in [21](see Theorem 9 in 3.3 and remarks foowing it; see aso [23, Proposition.1], and the discussion before Coroary in the survey [7]). We note that (i) is key, and the other two caims foow easiy. Indeed, since GL 2 is smooth over the residue fied F, a fibres of π n have voume equa to 4n. The set V n is a disjoint union of fibres of π n, and by (i), the number of these fibres is #π n (V n ) = #S n. Thus, the voume of V n is exacty 4n times the number of points in the image of the set in the numerator under this projection. Caim (iii) foows in a simiar fashion by considering π n+1 (V n ) = S n+1 as a fibration over S n. Combining Propositions 3.3 and 3.4, we immediatey obtain: Coroary 3.6. The Gekeer numbers reate to orbita integras via ν (a, q) = 3 # SL 2 (F ) Ogeom γ (φ ) = p revisited. We now consider ν p (a, q) in a simiar ight. Since det(γ ) = q, γ ies in Mat 2 (Z p ) GL 2 (Q p ) but not in GL 2 (Z p ), and we must consequenty modify the argument of 3.2. ( ) p m For integers m and n, et λ m,n = p n, and et C m,n = GL 2 (Z p )λ m,n GL 2 (Z p ). The Cartan decomposition for GL 2 asserts that GL 2 (Q p ) is the disjoint union GL 2 (Q p ) = C m,n, so that Mat 2 (Z p ) GL 2 (Q p ) = m n n m We now express ν p (a, q) as an orbita integra. Reca that q = p e. Since we consider an ordinary ( isogeny cass, the eement γ GL 2 (Q p ) actuay can be chosen to have the form γ = u1 p e ), where u u 1, u 2 Z p are units, and thus in particuar, γ C e,. 2 Lemma 3.7. Let φ q be the characteristic function of C e, = GL 2 (Z p ) ν p (a, q) = C m,n. p 3 # SL 2 (F p ) Ogeom γ (φ q ). ( ) q GL 1 2 (Z p ). Then

11 ELLIPTIC CURVES, RANDOM MATRICES AND ORBITAL INTEGRALS 11 Proof. The proof is simiar to the case = p, with one key modification. There, we use the reduction mod n map π n defined on G(Z ). Here, we need to extend the map π n to a set that contains γ. Let πn M : Mat 2 (Z p ) Mat 2 (Z p /p n ) be the projection map, and et c : GL 2 (Q p ) A(Q p ) be the characteristic poynomia map. As in 3.2 above, we define the sets U n := {a = (a, a 1 ) A(Z p ) a i a i (γ ) mod p n, i =, 1} S n := {γ Mat 2 (Z p /p n ) : γ π M n (γ )} V n := (π M n ) 1 (S n ) Mat 2 (Z p ) GL 2 (Q p ). As before, informay, we think of U n as a neighbourhood of the point given by the coefficients of the characteristic poynomia of γ in the Steinberg-Hitchin base, and we think of V n as the intersection of the corresponding neighbourhood of the orbit of γ in GL 2 (Q p ) with Mat 2 (Z p ). In the case = p we had GL 2 (Z ) in the pace of Mat 2 (Z p ) in this description, and so it was cear that the evauation of the voume of V n woud ead to the orbita integra of φ, the characteristic function of GL 2 (Z ). Here, we need to make the connection between the set V n and our function φ q. We caim that if n > e, then V n C e,. Indeed, suppose γ V n. Then, since the characteristic poynomia of γ is congruent to that of γ, the trace of γ is a p-adic unit. Then γ cannot ie in any doube coset C m,n with both m, n positive, because if it did, its trace woud have been divisibe by p min(m,n). Then γ has to ie in a doube coset of the form C e+m, m for some m ; but if m >, then such a doube coset has empty intersection with Mat 2 (Z p ), so m = and the caim is proved. As in the proof of Proposition 3.4 (iii), the voume of the set V n equas p 4n #S n. The rest of the proof repeats the proofs of Proposition 3.4 and Coroary 3.6. We again set V(γ ) = n 1 V n C e,. Since π M n is surjective, V(γ ) = O(γ ) C e,. By (3.5), Oγ geom (φ q ) = im n vo µso V n (γ ) GL 2 vo µ SO(U n ) A = im n #S n (γ )p 4n p 2n, and the statement foows by (3.7), which does not require any modification. Reca that, in terms of the data (a, q), we have aso computed a representative δ for a σ-conjugacy cass in GL 2 (Q q ). It is characterized by the fact that, possiby after adjusting γ in its conjugacy cass, we have N Qq /Q p (δ ) = γ. (Here we expoit the fact that, in a genera inear group, conjugacy and stabe conjugacy coincide.) The twisted centraizer G δ σ of δ is an inner form of the centraizer G γ [14, Lemma 5.8]; since γ is reguar semisimpe, G γ is a torus, and thus G δ σ is isomorphic to G γ. Using this, any choice of Haar measure on G δ σ(q p ) induces one on G γ (Q p ). If φ is a function on G(Q q ), denote its twisted (canonica) orbita integra aong the orbit of δ by TOδ can (φ) = φ(h 1 δ h σ ) dµ can. G δ σ(q p )\G(Q q ) Lemma 3.8. Let φ p,q be the characteristic function of GL 2 (Z q )λ,1 GL 2 (Z q ). Then TOδ can (φ p,q ) = Oγ can (φ q ).

12 12 JEFFREY D. ACHTER AND JULIA GORDON Proof. The asserted matching of twisted orbita integras on GL 2 (Q q ) with orbita integras on GL 2 (Q p ) is one of the eariest known instances of the fundamenta emma ([17]; see aso [19, Sec. 4], [1, (E.4.9)] or even [1, Sec. 2.1]). Indeed, the base change homomorphism of the Hecke agebras matches the characteristic function of GL 2 (Z q )λ 1, GL 2 (Z q ) with φ q + φ, where φ is a inear combination of the characteristic functions of C a,b with a + b = e and a, b >. As shown in the proof of the previous emma, the orbit of γ does not intersect the doube cosets C a,b with a, b >, and thus the ony non-zero term on the right-hand side is Oγ can (φ q ). 4. CANONICAL MEASURE VS. GEOMETRIC MEASURE Finay, we need to reate the orbita integra with respect to the geometric measure as above to the canonica orbita integras. A very simiar cacuation is discussed in [8] (and as the authors point out, surprisingy, it seemed impossibe to find in earier iterature). Since our normaization of oca measures seems to differ by an interesting constant from that of [8] at ramified finite primes, we carry out this cacuation in our specia case Canonica measure and L-functions. Here we briefy review the facts that go back to the work of Wei, Langands, Ono, Gross, and many others, that show the reationship between convergence factors that can be used for Tamagawa measures and various Artin L-functions. Our goa is to introduce the Artin L-factors that naturay appear in the computation of the canonica measures. To any reductive group G over Q, Gross attaches a motive M = M G [11]; foowing his notation, we consider M (1) the Tate twist of the dua of M. For any motive M we et L (s, M) be the associated oca Artin L-function. We wi write L (M) for the vaue of L (s, M) at s =. The vaue L (M (1)) is aways a positive rationa number, reated to the canonica measure reviewed in In particuar, if G is quasi-spit over Q, then (4.1) µ can G ([11, 4.7 and 5.1]). = L (M (1)) ω G We sha aso need a simiar reation between voumes and Artin L-functions in the case when G = T is an agebraic torus which is not necessariy anisotropic. Here we foow [4]. Suppose that T spits over a finite Gaois extension L of Q ; et κ L be the residue fied of L, and et I be the inertia subgroup of the Gaois group Ga(L/Q ). Let X (T) be the group of rationa characters of T. Let T be the Néron mode of T over Z, with the connected component of the identity denoted by T. This is the canonica mode for T referred to in Let Fr L be the Frobenius eement of Ga(κ L /F ). The Gaois group of the maxima unramified sub-extension of L, which is isomorphic to Ga(κ L /F ), acts naturay on the I-invariants X (T) I, giving rise to a representation which we wi denote by ξ T (and which is denoted by h in [4]), ξ T : Ga(κ L /F ) Aut(X (T) I ) GL di (Z), where d I = rank(x (T) I ). Then the associated oca Artin L-factor is defined as ( L (s, ξ T ) := det 1 di ξ ) T(Fr L ) 1 s. Proposition 4.1. ([4, Proposition 2.14]) L (1, ξ T ) 1 = #T (F ) dim(t) = T (Z ) ω T.

13 ELLIPTIC CURVES, RANDOM MATRICES AND ORBITAL INTEGRALS 13 We observe that by definition [11, 4.3], since G = T is an agebraic torus, the canonica parahoric T is T ; the canonica voume form ω T is the same as the voume form denoted by ω p in [4]. We aso note that the motive of the torus T is the Artin motive M = X (T) Q. If T is anisotropic over Q, by the formua (6.6) (cf. aso (6.11)) in [11], we have L (M (1)) = L (1, ξ T ). As in the first paragraph of 3.1.3, et G be a reductive group over Q with simpy connected derived group G der and connected center Z, and assume that G/G der = G m. Lemma 4.2. Let T G be a maxima torus; et T der = T G der. Then (4.2) L (M G (1)) L (1, ξ T ) = L (M G der (1)) L (1, ξ T der). Proof. The motive M H of a reductive group H, and thus L (M H (1)), depends on H ony up to isogeny [11, Lemma 2.1]. Since G is isogenous to Z G der, L (M G (1)) = L (M Z(1))L (M G der (1)). Because G der Z is finite [8, (3.1)], so is T der Z. Therefore, the natura map T der T/Z is an isogeny onto its image. For dimension reasons it is an actua isogeny, and induces an isomorphism X (T der ) Q = X (T/Z) Q of Ga(Q )-modues. Therefore, L(s, ξ T der) = L(s, ξ T/Z ), and thus Identity (4.2) is now immediate. L(s, ξ T ) = L(s, ξ T/Z )L(s, ξ Z ) = L(s, ξ T der)l(s, ξ Z ) Wey discriminants and measures. Our next immediate goa is to find an expicit constant d(γ) such that µ can γ = d(γ)µ geom γ. We note that a simiar cacuation is carried out in [8]. However, the notation there is sighty different, and the key proof in [8] ony appears for the fied of compex numbers; hence, we decided to incude this cacuation here. Let G be a spit reductive group over Q ; choose a spit maxima torus and associated root system R and set of positive roots R +. Definition 4.3. Let γ G(Q ), et T be the centraizer of γ, and t the Lie agebra of T. discriminant of γ is D(γ) = (1 α(γ)) = det(i Ad(γ 1 )) g/t. α R Then the Wey integration formua, revisited. As pointed out in [8] (the paragraph above equation (3.28)), since both µ can γ and µ geom γ are invariant under the center, it suffices to consider the case G = G der. So for the moment, et us assume that the group G is semisimpe and simpy connected; et φ Cc (Q ). On one hand, the Wey integration formua (we write a group-theoretic version of the formuation for the Lie agebra in [16, 7.7]) asserts that 1 (4.3) φ(g) dω G = D(γ) φ(g 1 γg) dωt\g dω T, G(Q ) T W T T(Q ) T(Q )\G(Q ) by our definition of the measure dω T\G. (Here, the sum ranges over a set of representatives for G(Q )-conjugacy casses of maxima Q -rationa tori in G, and W T is the finite group W T = N G (T)(Q )/T(Q ).)

14 14 JEFFREY D. ACHTER AND JULIA GORDON On the other hand we have, by definition of the geometric measure, φ(g) dω G = φ(g) dωγ geom (g) dω A. G(Q ) A(Q ) c 1 (a) To compare the two measures, we need to match the integration over A(Q ) with the sum of integras over tori. Up to a set of measure zero, A(Q ) is a disjoint union of images of T(Q ), as T ranges over the same set as in (4.3); and for each such T, the restriction of c to T is W T -to-one. It remains to compute the Jacobian of this map for a given T. Over the agebraic cosure of Q this cacuation is done, for exampe, in [16, 14]; over Q, this ony appies to the spit torus T sp. The answer over the agebraic cosure is c T α> (α(x) 1), where c T F is a constant (which depends on the torus T). We compute c T in the specia case where T comes from a restriction of scaars in GL 2. Lemma 4.4. Let T be a torus in GL 2 (Q ), and et c T be the constant defined above. Then c T = 1 if T is spit or spits over an unramified extension, and c T = 1/2 if T spits over a ramified quadratic extension. In particuar, if γ GL 2 (Q) and T = R K/Q G m is the centraizer of γ as in 2.1, then c T = K 1/2. Proof. We prove the emma by direct cacuation for GL 2. First, et us compute c T for the spit torus. Here we can just compute the Jacobian of the map T der T der /W by hand. Since we are working with invariant differentia forms, we can just do the Jacobian cacuation on the Lie agebra; it suffices to compute the Jacobian of the map from t to t/w. Choose coordinates on the spit torus in SL 2 = GL der 2, so that eements of t are diagona matrices with entries (t, t); then the canonica measure on t is nothing but dt. Now, the coordinate on t/w is y = t 2 ; the form ω A 1 is dx. The Jacobian of the change of variabes from t/w to A 1 is 2t. Thus, for the spit torus c = 1: note that 2t is the product of positive roots (on the Lie agebra). Thus, c T = 1. Now, consider a genera maxima torus T in GL 2. Let T sp be a spit maxima torus; we have shown that c T sp = 1. The torus T is conjugate to T sp over a quadratic fied extension L. Let us briefy denote this conjugation map by ψ. Then the map c T can thought of as the conjugation ψ : T T sp (defined over L) foowed by the map c T sp. Then ω T c T = c T sp ψ (ω T sp), where ψ (ω T sp) is the puback of the canonica voume form on T sp under ψ and the ratio is a constant in L. We thus have (4.4) c T = ω T ψ (ω T sp), L where L is the unique extension of the absoute vaue on Q to L. ω T ψ (ω T sp ) At this point this is just a question about two tori, no onger requiring Steinberg section, and so we pass back to working with the group GL 2 rather than SL 2. Now T is obtained by restriction of scaars from G m, and so we can compute ψ (ω T sp) by hand. By definition, T = R L/Q G m ; T sp = G m G m. The form ω T sp is ω T sp = du u dv v, where we denote the coordinates on G m G m by (u, v). Let L = Q ( ɛ), where ɛ is a non-square in Q (assume for the moment that = 2). Then every eement of T is conjugate in GL 2 (Q ) to

15 [ x ɛy y x ELLIPTIC CURVES, RANDOM MATRICES AND ORBITAL INTEGRALS 15 ], and using (x, y) as the coordinates on T, the map ψ can b written as ψ(x, y) = (x + ɛy, x ɛy). Then one can simpy compute Thus we get (for = 2), ψ ( du u dv v ) = 2 dx dy ɛ x 2 ɛy 2 = 2 ɛω T. c T = 2 ɛ L = { 1 L is unramified L is ramified, which competes the proof of the emma in the case = 2. There is, however, a better argument, which aso covers the case = 2. Namey, to find the ratio ω T of (4.4), we just need to find the ratio of the voume of T (Z ) with respect to the ψ (ω T sp ) L measure dω T to its voume with respect to dψ (ω T sp). This is, in fact, the same cacuation as the one carried out in [24, p.22 (before Theorem 2.3.2)], and the answer is that the convergence factors for the pu-back of the form ω T sp to the restriction of scaars is ( K ) dim(gm), in this case. Finay, summarizing the above discussion, we obtain Proposition 4.5. Let γ GL 2 (Q) be a reguar eement. Let T be the centraizer of γ, and et K be as in 2.1. Abusing notation, we aso denote by γ the image of γ in GL 2 (Q ) for every finite prime. Then for every finite prime, as measures on the orbit of γ. µ geom γ = L (1, ξ T ) L (M G (1)) K 1/2 D(γ) 1/2 µ can γ 5. THE GLOBAL CALCULATION In this section, we put a the above oca comparisons together, and thus show that Gekeer s formua reduces to a specia case of the formua of Langands and Kottwitz. In the process we wi need a formua for the goba voume term that arises in that formua. We are now in a position to give a new proof of Gekeer s theorem, and of its generaization to arbitrary finite fieds. Theorem 5.1. Let q be a prime power, and et a be an integer with a 2 q and gcd(a, p) = 1. The number of eiptic curves over F q with trace of Frobenius a is q (5.1) #I(a, q) = 2 ν (a, q) ν (a, q). Here, ν (a, q) (for = p), ν p (a, q), and ν (a, q) are defined, respectivey, in (2.4), (2.6), and (2.7), and the weighted count #I(a, q) is defined in (2.1). Proof. Reca the notation surrounding γ and δ estabished in 2.1. Given Proposition 2.1, it suffices to show that the right-hand side of (5.1) cacuates the right-hand side of (2.8). Let G = GL 2. First, et φ p = =p 1 G(Z ) be the characteristic function of G(Ẑ p f ) in G(Ap f ). The first integra appearing in (2.8) is equa to O γ (φ p ) = φ p dω G = O can (1 G(Z )). G(A p ) =p

16 16 JEFFREY D. ACHTER AND JULIA GORDON Combining Coroary 3.6, reation (4.2) and Proposition 4.5, we get, for = p, ν (a, q) = 3 #G der (F ) Ogeom γ (1 G(Z )) = 3 #G der (F ) L (1, ξ T der) L (M G der (1)) K 1/2 D(γ ) 1/2 O can γ (1 G(Z )) = L (1, ξ T der) D(γ ) 1/2 K 1/2 Oγ can (1 G(Z )). ( ) 1 Second, et φ q be the characteristic function of G(Z p ) G(Z q p ) in G(Q p ), and et φ p,q be the ( ) 1 characteristic function of G(Z q ) G(Z p q ) in G(Q q ). Using Lemmas 3.7 and 3.8, we find that ν p (a, q) = = = p 3 #G der (F p ) Ogeom γ (φ q ) p 3 L p (1, ξ T der) #G der (F p ) L p (M (1)) K 1/2 p D(γ ) 1/2 p G der p 3 L p (1, ξ T der) #G der (F p ) L p (M (1)) K 1/2 G der Taking a product over a finite primes, we obtain: (5.2) ν (a, q) = L(1, ξ T der) < p D(γ ) 1/2 p O can γ (φ q ) TO can δ (φ p,q ). K D(γ ) TOcan δ σ (φ p,q)o can γ (φ p ). Reca that f (T), the characteristic poynomia of γ, is f (T) = T 2 at + q. The (poynomia) discriminant of f (T) and the (Wey) discriminant of γ are reated by D(γ ) det(γ ) = disc( f ) = 4q a 2. Consequenty, qν (a, q) = 1 D(γ ). π Since L(1, ξ T der) = L(1, ξ T/Z ) (Lemma 4.2), to deduce (5.1) from (5.2) it suffices to show that (5.3) K 2π L(1, ξ T/Z) = vo(t(q)\t(a f )). On one hand, L(s, ξ T/Z ) coincides with L(s, K/Q), the Dirichet L-function attached to the quadratic character of K. Therefore, the anaytic cass number formua impies that the eft-hand side of (5.3) is h K /w K, the ratio of the cass number of K to the number of roots of unity in K. On the other hand, the right-hand side of (5.3) is aso we-known to be h K /w K (e.g., [25, Prop. VII.6.12]); we defer to the appendix (Lemma A.4) for detais.

17 ELLIPTIC CURVES, RANDOM MATRICES AND ORBITAL INTEGRALS 17 APPENDIX A. ORBITAL INTEGRALS AND MEASURE CONVERSIONS S. Ai Atuğ In this appendix, we expain how to deduce the comparison factor of Proposition 4.5 from [8] and certain computations in [18] as we as cacuate the voume factor that goes into the proof of Theorem 5.1. We aso remark that the same measure comparison aso appears in [2] (athough impicity) in the passage from equation (2) to (3). Comparison of measures. Let G = GL 2. Let ω G be the same voume form as in For a torus T G, et ω T be as in Proposition 4.1. Reca that T is the connected component of the Néron mode of T. Lemma A.1. Let be a finite prime, et γ G(Q ) be reguar semisimpe, and et T = G γ be its centraizer. Let µ G and µ T be nonzero Haar measures on G(Q ) and T(Q ), respectivey. Then µ geom vo( ω G γ, = D(γ) ) vo(µ T, ) vo( ω T ) vo(µ G, ) µ T\G,, where D(γ) = tr(γ) 2 4 det(γ) and µ = µ can GL 2, /µ T. Proof. By equation (3.3) of [8], we have µ geom γ, = D(γ) ω T\G, where we note that the eft hand side of (3.3) of oc. cit. is what we denoted by µ geom γ. Since the Haar measure is unique up to a constant we have ω G = c (G)dµ G, and ω T = c (T)dµ T,. The constants can be cacuated easiy by comparing the voumes of the integra points: c (G) = vo ω G (G(Z )) vo µg, (G(Z )) and c (T) = vo ω T (T (Z )) vo µt, (T (Z )). Therefore, the quotient measures µ T\G, and ω T\G are reated by The emma foows. As an immediate coroary to Lemma A.1 we get Coroary A.2. Let µ can G, et the rest of the notation be as in Lemma A.1. Then ω T\G = c (G) c (T) µ T\G,. and µcan T, be normaized to give measure 1 to G(Z ) and T (Z ) respectivey, and µ geom γ, = D(γ) vo ωg (G(Z )) vo ωt (T (Z )) µ T\G,. We now quote a resut of [18]. Let ζ (s) = 1/(1 s ). Lemma A.3. We have vo( ω G ) = ζ (1) 1 ζ 1 (2) ζ (1) 2 vo( ω T ) = K ζ (2) 1 ζ (1) 1 K/Q is spit at K/Q is unramified at, K/Q is ramified at where K/Q is the quadratic extension which spits T and K is the discriminant of K.

18 18 JEFFREY D. ACHTER AND JULIA GORDON Proof. The resut for odd primes is given on pages 41 and 42 of [18]. The case for = 2 foows the same ines. The ony point to keep in mind is the extra factor of 2 that appears in the cacuation of the differentia form on page 42 of [18]; we eave the detais to the reader. Coroary A.2 and Lemma A.3 then gives the conversion factor between the two measures. Cacuation of vo(k \A,fin K ). Let (a, p) be such that a 2 4p <. Let dµ can T, be the Haar measure normaized to give measure 1 to T(Z ) and set dµ can T, f in := = dµ can T,. Lemma A.4. We have µ can T, f in (T(Q)\T(A f in )) = h K ω K, where K/Q is the quadratic extension which spits T, ω K is the number of roots of unity in K, and h K is its cass number. Proof. By identifying T = G γ with G m over the quadratic extension K we have µ can T,fin (T(Q)\T(Afin )) = µ can K,fin (K \A,fin K ), where the measure on the right is such that µ can K,fin (O v ) = 1 for each pace v. Let Ô K = v O v. Reca that 1 (K Ô K )\Ô K K \A K C(K) 1, which impies that µ(k \A K ) = h Kµ(K Ô K )\Ô K ) = h K ω K. MASSACHUSETTS INSTITUTE OF TECHNOLOGY, CAMBRIDGE, MA E-mai address: atug@mit.edu

19 ELLIPTIC CURVES, RANDOM MATRICES AND ORBITAL INTEGRALS 19 REFERENCES [1] Jeffrey D. Achter and Cifton L. R. Cunningham, Isogeny casses of Hibert-Bumentha abeian varieties over finite fieds, J. Number Theory 92 (22), no. 2, MR (22k:1189) [2] S. A. Atuğ, Beyond Endoscopy via the Trace Formua-I: Poisson Summation and Isoation of Specia Representations, Compositio Mathematica 151 (215), no. 1, [3] Victor V. Batyrev, Birationa Caabi-Yau n-fods have equa Betti numbers, New trends in agebraic geometry (Warwick, 1996), 1999, pp MR (2i:1459) [4] Rony A. Bitan, The discriminant of an agebraic torus, J. Number Theory 131 (211), no. 9, MR (212g:11118) [5] Laurent Coze, Nombre de points des variétés de Shimura sur un corps fini (d après R. Kottwitz), Astérisque (1993), no. 216, Exp. No. 766, 4, Séminaire Bourbaki, Vo. 1992/93. MR (95c:1175) [6] Chanta David, Dimitris Koukouopouos, and Ethan Smith, Sums of Euer products and statistics of eiptic curves, 215. arxiv: [7] Jan Denef, Arithmetic and geometric appications of quantifier eimination for vaued fieds, Mode theory, agebra, and geometry, 2, pp [8] Edward Frenke, Robert Langands, and Báo Châu Ngô, Formue des traces et fonctoriaité: e début d un programme, Ann. Sci. Math. Québec 34 (21), no. 2, MR (212c:1124) [9] Ernst-Urich Gekeer, Frobenius distributions of eiptic curves over finite prime fieds, Int. Math. Res. Not. (23), no. 37, [1] Jayce Getz and Mark Goresky, Hibert moduar forms with coefficients in intersection homoogy and quadratic base change, Progress in Mathematics, vo. 298, Birkhäuser/Springer Base AG, Base, 212. MR [11] Benedict H. Gross, On the motive of a reductive group, Invent. Math. 13 (1997), [12] Thomas C. Haes, The fundamenta emma and the Hitchin fibration [after Ngô Bao Châu], Astérisque (212), no. 348, Exp. No. 135, ix, Séminaire Bourbaki: Vo. 21/211. Exposés MR [13] Nichoas M. Katz, Lang-Trotter revisited, Bu. Amer. Math. Soc. (N.S.) 46 (29), no. 3, MR MR [14] Robert E. Kottwitz, Rationa conjugacy casses in reductive groups, Duke Math. J. 49 (1982), no. 4, MR 6833 (84k:22) [15], Points on some Shimura varieties over finite fieds, J. Amer. Math. Soc. 5 (1992), no. 2, MR (93a:1153) [16], Harmonic anaysis on reductive p-adic groups and Lie agebras, Harmonic anaysis, the trace formua, and Shimura varieties, 25, pp MR (26m:2216) [17] Robert P. Langands, Base change for GL(2), Annas of Mathematics Studies, 198. [18], Singuarités et transfert, Ann. Math. Qué. 37 (213), no. 2, MR [19] Gérard Laumon, Cohomoogy of Drinfed moduar varieties, part I, Cambridge University Press, [2] Peter Schoze, The Langands-Kottwitz approach for the moduar curve, Int. Math. Res. Not. IMRN (211), no. 15, MR (212k:119) [21] Jean-Pierre Serre, Queques appications du théorème de densité de Chebotarev, Inst. Hautes Études Sci. Pub. Math. (1981), no. 54, [22] Wiem Veys, Arc spaces, motivic integration and stringy invariants, Singuarity theory and its appications, 26, pp [23], Reduction moduo p n of p-adic subanaytic sets, Math. Proc. Cambridge Phios. Soc. 112 (1992), no. 3, MR (93i:11142) [24] André Wei, Adees and agebraic groups, Progress in mathematics, vo. 23, Birkhäuser. [25], Basic number theory, Cassics in Mathematics, Springer-Verag, Berin, Reprint of the second (1973) edition. MR COLORADO STATE UNIVERSITY, FORT COLLINS, CO E-mai address: achter@math.coostate.edu UNIVERSITY OF BRITISH COLUMBIA, VANCOUVER, BC E-mai address: gor@math.ubc.ca