Edited by N. Mukunda, Indian Academy of Sciences, 2009 The Sato–Tate conjecture and generalizations∗ MRAMMURTY1 and V KUMAR MURTY2 1Department of Mathematics, Queen’s University, Kingston, Ontario, K7L 3N6, Canada. e-mail: [email protected] 2Department of Mathematics, University of Toronto, Toronto, Ontario, M5S 2E4, Canada. e-mail: [email protected] 1. Introduction the analogue of the Riemann hypothesis for the function field zeta function turns out to be Consider the elliptic curve E defined by the equivalent to (1). In his thesis, Artin verified his equation conjecture for many small primes p but could not prove it. In February, 1933, Hasse [14] proved y2 = x3 + ax + b, a, b ∈ Z. the conjecture using techniques from algebraic geometry. One could say that understanding this function field analogue of the Riemann hypoth- 3 2 Let Δ = −16(4a +27b ). For each prime p with esis was an important step in the annals of (p, Δ) = 1, we can consider the congruence mathematics. The reader is referred to the his- torical document [11] for further discussions on y2 ≡ x3 + ax + b(mod p), this. Artin’s thesis was seminal in many ways. First, it opened up the study of algebraic geome- and count the number Np of solutions (x, y). This quantity was first studied by Emil Artin [1] in his try over finite fields and connected it to the 1924 doctoral thesis in which he conjectured that study of exponential sums that occur in classi- cal analytic number theory. Second, it inspired √ Weil [36] to formulate in 1949 general conjec- |N − p|≤ p, p 2 (1) tures that led Grothendieck [13] to chart out a visionary program in algebraic geometry ulti- for all such primes. In many ways, his study was mately leading to a resolution of the Weil conjec- motivated by the classical Riemann hypothesis. To tures in the fundamental work on Deligne [8] in understand the nature of zeta functions in general, 1974. Artin defined the analogue of the Dedekind zeta Around 1960, Mikio Sato and John Tate [32] function in the setting of a function field over a (independently) asked about the distribution of the finite field. In the case of a quadratic extension of numbers Fp(x) defined by √ Np − p/ p y2 = x3 + ax + b, ∗Research of both authors partially supported by in the interval [−2, 2], as p tends to infinity. For Natural Sciences and Engineering Research Council example, is it reasonable to expect that these (NSERC) grants. numbers are uniformly distributed in the interval? Keywords. Elliptic curves; Sato–Tate; symmetric power L-functions; Chebotarev density theorem. 639 640 M RAM MURTY AND V KUMAR MURTY In other words, is it true that for any interval On March 18, 2006, Taylor [33] (see also [5]) [a, b] ⊆ [−2, 2], we have just published a proof of this conjecture, when √ E has at least one prime of multiplicative reduc- #{p ≤ x :(Np − p)/ p ∈ [a, b]} tion. He was building on his earlier work with lim = b − a? x→∞ #{p ≤ x} Clozel, Harris and Shepherd-Barron (see Carayol’s S´eminaire Bourbaki article [4]). This question is the genesis of the Sato–Tate In this paper, we give an informal exposition conjecture. Numerical evidence seemed to suggest of this recent development. We also indicate some otherwise. More precisely, Sato and Tate were led modest generalizations that are obtained by slight to predict that for a ‘generic’ elliptic curve E the modifications in the proof. Our first result is a following is true. If we write hybrid Chebotarev–Sato–Tate theorem. √ Theorem 1. Let E be an elliptic curve defined (Np − p)/ p =2cosθp, 0 ≤ θp ≤ π, over a totally real number field K with at least M/K and [α, β] ⊆ [0,π], then, their conjecture says one prime of multiplicative reduction. If is a solvable Galois extension of finite degree with G =Gal(M/K),andC is a conjugacy class of G, {p ≤ x θ ∈ α, β } β p # : p [ ] 2 2 then the density of prime ideals for which the lim = sin θdθ σ ∈ C θ ∈ α, β x→∞ #{p ≤ x} π α Artin symbol p and the angle p [ ] with 0 ≤ α ≤ β ≤ π is β − α 1 = − (sin 2β − sin 2α). π 2π 2|C| β sin2 θdθ. π|G| α What ‘generic’ means is that the elliptic curve should be without complex multiplication (see [20] for details). If the elliptic curve has complex multi- In particular, we have the following corollary: plication, then the (essentially) uniform distribu- E tion law for the angles was worked out by Deuring Corollary 2. Let be an elliptic curve defined [10] building on earlier work of Hecke [15,16]. over the rational number field with at least one prime of multiplicative reduction. Let q be a natu- One can formulate a more general conjecture. a a, q Let E be an elliptic curve defined over a number ral number and an integer with ( )=1.For K v K E 0 ≤ α ≤ β ≤ π, the density of primes p for which field . For each place of where has good θ E ∈ α, β p ≡ a q reduction, we may consider the group of points of p( ) [ ] and (mod ) is E mod v. Its cardinality (including the identity element) can be written as 2 β sin2 θdθ. πϕ(q) α Nv +1− av, where Nv denotes the norm of v and av is an inte- It is evident that by similar arguments, one 1/2 ger satisfying Hasse’s inequality |av|≤2(Nv) . can handle the joint distribution of angles of any As before, one can therefore write finite set of elliptic curves provided that there is at most one elliptic curve in the set without CM 1/2 (and having at least one prime of multiplicative av =2N(v) cos θv, reduction). One can also study the joint distribution of where θv(E):=θv satisfies 0 ≤ θv ≤ π. The Sato– angles of a finite collection of pairwise non- Tate conjecture now is a statement about how isogenous elliptic curves. This looks like a difficult θ ,π the angles v are distributed in the interval [0 ] question and Harris has recently announced some as v varies. When E has complex multiplication progress in this direction. (CM), the distribution law is known and again Though our treatment is informal, the back- follows from the classical work of Deuring on Hecke ground needed for a total understanding is quite L-series (see [26] for details). In the non-CM case, formidable spanning representation theory, arith- one expects that the angles are uniformly distri- metic algebraic geometry, analytic and algebraic buted with respect to the measure number theory. Still, we hope that the presenta- tion given here will enable the non-expert to see 2 2 how the proof is put together and appreciate the μST := (sin θ)dθ. π interplay of ideas. THE SATO–TATE CONJECTURE AND GENERALIZATIONS 641 Recently, ‘friendly’ expos´es of the Sato–Tate 2. Symmetric power L-series of conjecture have appeared in various places. The elliptic curves papers by Mazur [22,23] are a good place to begin for the totally uninitiated reader. Here, our goal Let K be an algebraic number field. Let E be an is more mathematical. We hope to give (with- elliptic curve defined over K.LetS be the (finite) out proofs) the main mathematical ingredients set of places where E has bad reduction. For each that enter into such equidistribution questions so finite place v/∈ S of K, we know that the number that the reader may have a conceptual understand- of points on E mod v is given by ing of the results. Because of the celebrated Taniyama conjecture N(v)+1− av, (now proved by Wiles [37] and others [3]), one can view the av’s as Fourier coefficients of certain a modular forms of weight 2, at least in the case that where v is an integer satisfying Hasse’s inequality |a |≤ N v 1/2 E is defined over the rational number field. In the v 2 ( ) .Foreachprime , the action ¯ general number field case, it is still open whether of Gal(K/K)onthe-adic Tate module gives rise the av’s can be viewed as coming from automorphic to an -adic representation representations as predicted by Langlands. Thus, ¯ one can view the Sato–Tate conjecture as a special ρ := ρ :Gal(K/K) → GL2(Q), case of a more general statement concerning distri- bution of eigenvalues of Hecke operators. (See [6] which is integral, that is, the characteristic poly- for more details.) nomial of ρ(Fv )whereFv denotes the Frobenius One can formulate a function field analogue of automorphism at v/∈ S has integer coefficients, the Sato–Tate conjecture and this has been proved K independent of . In fact, this characteristic in various contexts. Let be a rational function X2 − a X N v F E polynomial is v + ( ). Let us write field in one variable over a finite field and let 1/2 1/2 αvN(v) ,βvN(v) for the two roots of the denote an elliptic curve over K with nonconstant quadratic polynomial j-invariant. Let Y denote the projective line over F and consider the N´eron model E−→Y .Thisis 2 a smooth group scheme whose general fibre is E X − avX + N(v). and outside of a finite set S of points y ∈ Y ,the E y fibre y at is an elliptic curve (the ‘reduction’ The (partial) m-th symmetric power L-function is E y of modulo the residue field corresponding to ).