The Sato–Tate Conjecture and Generalizations∗

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 classical 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 Séminaire 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’ exposés 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 distribution 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 Néron 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 ). defined as Thus, as an elliptic curve over a finite field, its zeta function is of the form m m j m−j −s −1 LS(s, Sym ρ):= (1 − αvβv N(v) ) . − α T − α T v/∈S j=0 (1 y )(1 y ). (1 − T )(1 − qdeg yT ) Clearly, the product converges absolutely for (s) > 1. In [29], Serre showed that if for all (deg y)/2 iθ(y) m Here αy = q e ,where0≤ θ(y) ≤ π.Let m, LS(s, Sym (ρ)) extends to (s) ≥ 1 and does Fn denote the unique extension of F of degree n. not vanish there, then the Sato–Tate conjec- Then, the Sato–Tate conjecture in this context says turefollows.In[26],KMurtyshowedthat that as n −→ ∞ ,wehave the non-vanishing assumption is unnecessary. Thus, in this way, the Sato–Tate conjecture was reduced to a problem of analytic continuation of #{y ∈ Y (Fn):α ≤ θy ≤ β} m LS(s, Sym (ρ)) to the region (s) ≥ 1, for all values of m. β 2 In 1970, Langlands [21] outlined a method ∼ 2 θdθ |Y F |. sin ( n) of attacking the problem of analytic continua- α π tion. He suggested the existence of an automorphic representation πm attached to GLm+1(AK ), This was proved by Yoshida in [34] and in a dif- where AK denotes the adele ring of K,such m ferent way by K Murty in [26]. Very general the- that LS(s, πm)=LS(s, Sym (ρ)) where LS(s, πm) orems of Sato–Tate type (in which the base Y is is the automorphic L-function attached to πm with replaced by an arbitrary variety and E by fami- the Euler factors corresponding to the places v ∈ S lies of -adic sheaves) are proved in Deligne [9], removed. In fact, it is conjectured that one can section 3.5. define the local factors for v ∈ S in such a way that 642 M RAM MURTY AND V KUMAR MURTY the completed L-function, L(s, Symm(ρ)) (which Serre [30] formulated a general conjecture about we shall abbreviate as Lm(s)) satisfies a func- representations tional equation relating s to 1 − s. (See for example, [7] for details.) When K = Q and m =1,this ρ :Gal(Q¯ /Q) → GL (F), is the Taniyama conjecture. Langlands’ functoria- 2 lity conjecture predicts that the symmetric powers F of π1 are automorphic. This has been proved for with a finite field, which are odd, and absolutely some small values of m (see [6] for a survey irreducible. More precisely, he predicted that such of the present state of knowledge). If the Lang- representations arise from classical modular forms. lands conjecture about the existence of πm is true, Serre [30] showed that his conjecture implies the then by the theory of automorphic representa- Taniyama conjecture. On the other hand, Frey tions, one immediately has analytic continuation [12] had a remarkable insight which was completed m of LS(s, Sym (ρ)) to the entire complex plane and and made more precise by Ribet [27] who showed by the result of K Murty [26], the non-vanishing on that the Taniyama conjecture implies Fermat’s last the line (s) = 1 follows and the Sato–Tate conjec- theorem. Thus, Serre’s conjecture offered a new ture follows. The non-vanishing of the L-function approach to Fermat. Wiles [37] proved an impor- on the line (s) = 1 can also be deduced from tant case of Serre’s conjecture that was enough to a celebrated result of Jacquet and Shalika [17] deduce Taniyama. This work led to further deve- who showed that for any automorphic represen- lopments. Most recently, Chandrashekhar Khare tation π,wehaveL(s, π) =0,for (s)=1.So, [18] proved the level 1 case of the Serre conjec- what is known about the analytic continuation of ture. In very recent work, Khare and Wintenberger m LS(s, Sym (ρ))? [19] have settled the odd conductor case. This last For m = 1, this is the (partial) Hasse–Weil development not only gives a new proof of Fermat’s L-series attached to the elliptic curve E.Itispossi- last theorem, but it also implies the strong Artin ble to define the Euler factors at places in S as conjecture [2] for 2-dimensional Galois representa- well so that the completed L-function conjecturally tions of odd conductor. admits an analytic continuation to the entire com- In his recent paper, Taylor [33] made substantial plex plane and satisfies a suitable functional equa- progress towards this conjecture. If K is a totally tion. In the case K = Q, the Taniyama conjecture, real field and m is odd, he showed that there is a finite, totally real Galois extension L/K such proved by Wiles (in the semistable case) [37] and m by Breuil, Conrad, Diamond and Taylor (in the that (Sym ρ) restricted to L is automorphic over general case) [3] asserts that there is a classical L. One can choose an L that works simultane- cusp form f of weight 2 and level N (the con- ously for any finite set of odd numbers. One can ductor of E) such that the Hecke L-series L(s, f) also choose it to be unramified at any finite set attached to f agrees with L1(s − 1/2). If πf is the of places. Once this theorem is in hand, Taylor automorphic representation associated to f, then, uses standard results from the theory of automor- in the context of the Langlands program, we have phic L-functions to deduce the Sato–Tate conjec- L(s, πf )=L1(s). The series L1(s) is essentially the ture. We will give an outline of this deduction L-function attached to ρ, coming from the action below. of Gal(K/K¯ ) on the Tate module. Before we do this, let us review some essen- More generally, Lm(s) is essentially the tial theorems from the theory of automorphic L-function attached to the representation L-functions. We refer the reader to [24] for details, Symm(ρ), which comes from the action of definitions and additional references to the litera- Gal(K/K¯ )onthem-fold symmetric product ture. More precisely, we highlight pages 119 and of the Tate module. As alluded to above, one 215 of [2] for the exact definitions of the notions of expects the existence of an automorphic repre- base change and automorphic induction. Here are sentation πm attached to GLm+1(AK ) satisfying the key theorems we will need. L(s, πm)=Lm(s). This expectation is far from First is the theorem of base change and auto- being realized, though important advances have morphic induction, due to Arthur and Clozel [2]. been made in this direction. This says the following. Suppose that L/K is a What Taylor proves is not the automorphy of cyclic extension and π, Π are cuspidal represen- Lm(s) but rather its ‘potential automorphy.’ This tations of GLn(AK )andGLn(AL), respectively. fact,combinedwithotherresultsintheana- Then, the base change of π, denoted B(π)andthe lytic theory of automorphic L-functions, leads to automorphic induction I(Π) of Π exist. the Sato–Tate conjecture. This result of ‘potential The second fact we need is a celebrated theorem automorphy’ builds on a massive collection of ear- of Jacquet and Shalika [17] which states that for lier work that can be traced back to the celebrated any unitary cuspidal automorphic representation conjecture of Serre. π,ofGLn(AK ), we have L(1 + it, π) =0. THE SATO–TATE CONJECTURE AND GENERALIZATIONS 643

The third fact needed is a non-vanishing theorem with ai integers and ψi 1-dimensional characters of due to Shahidi [31]. This states that the Rankin– nilpotent subgroups Hi of G. Then, Selberg L-function L(s, π1 × π2) does not vanish on s π π the line ( ) = 1, whenever 1 and 2 are unitary G L s, mρ ⊗ L s, mρ ⊗ ψ ai . cuspidal automorphic representations. ( (Sym ) 1)= ( (Sym ) IndHi i) A fourth fact needed is the Artin reciprocity law i which states that every abelian Artin L-function of any Galois extension of K is a Hecke L-function By Frobenius reciprocity, and corresponds to a cuspidal automorphic repre- m G G m sentation of GL1(AK ). ρ ⊗ ψ ρ | H ⊗ ψ . (Sym ) IndHi i =IndHi ((Sym ) L i i)

m 3. An outline of Taylor’s theorem Hi By step 1, (Sym ρ)|LHi is automorphic over L . By Artin reciprocity, ψi is a Hecke character χi of m Here is a brief outline of Taylor’s proof of the Sato– Hi L .Thus,(Sym ρ)|LHi ⊗ ψi is automorphic over K H Tate conjecture. His main theorem is: let be a L i .ByinvarianceofL-series under induction, we E/K totally real field and an elliptic curve with deduce multiplicative reduction at some prime. For any odd number m, there is a finite, totally real Galois m m m ai extension L/K such that Sym ρ becomes auto- L(s, Sym ρ)= L(s, (Sym ρ)|LHi ⊗ χi) , m morphic over L. In other words, (Sym ρ)|L is auto- i morphic over L. (One can also choose an L that will work simultaneously for any finite set of odd and the product on the right hand side, being positive numbers.) a product of automorphic L-functions by step 1, From this result, he deduces the Sato–Tate con- represents a meromorphic function of s.Inthisway, jecture in three steps. Here is an outline. one derives the meromorphic continuation of the L E Step 1: For any intermediate field K ⊂ F ⊂ L, odd symmetric power -functions attached to . with L/F solvable, Symmρ is automorphic over F . m Step 3: If we apply the Jacquet–Shalika theo- In other words, (Sym ρ)|F is automorphic over F rem which assures us that there are no poles on for every F with L/F solvable. (s) = 1 for a cuspidal automorphic L-function L s, π L s, π This is proved in his earlier paper: M Harris, ( ), as well as the non-vanishing of ( ) on (s) = 1 for any automorphic representation N Shepherd-Baron and R Taylor, Ihara’s lemma π and potential automorphy, (preprint available at , we obtain from the above product the ana- ∼ lytic continuation and non-vanishing on (s)=1 www.math.harvard.edu/ rtaylor). L s, mρ m m Essentially, it applies the Arthur–Clozel theory of ( Sym )for odd. To treat even, one of base change, the key idea being that the base uses induction and the identity change lift of Symmρ to L, which exists by Taylor’s main theorem, is Galois invariant and so must be Symm−1ρ ⊕ Symm+1ρ =Symmρ ⊗ Sym1ρ, the base change lift of an automorphic representa- π F L/F tion F for every intermediate ,with solv- which is essentially the trigonometric identity able. This is because one can find a chain F F ⊂ F ⊂ F ⊂···⊂F L sin mθ sin(m +2)θ sin(m +1)θ sin 2θ = 0 1 2 m = + = sin θ sin θ sin θ sin θ of extensions so that Fi+1/Fi is cyclic for 0 ≤ i ≤ m − 1 and apply the Arthur–Clozel the- (or the Clebsch–Gordon branching rule for SL2). orem for automorphic induction successively, Thus, in stages, to each of the cyclic extensions Fm/Fm−1,...,F1/F0. We refer the reader to m−1 m+1 [2] for precise details concerning automorphic L(s, Sym ρ)L(s, Sym ρ) induction. = L(s, (Symmρ) ⊗ Sym1ρ). Step 2: Let G =Gal(L/K). Now apply Brauer induction to write Now apply step 1, with the two odd numbers 1,m to get the base change to L of both Symmρ and a G ψ , 1 1= i IndHi i Sym ρ automorphic. By the same Brauer induc- i tion trick of step 2 applied to the right hand side, 644 M RAM MURTY AND V KUMAR MURTY one deduces that the right hand side has a mero- the present state of knowledge. However, Harris has morphic continuation for all complex s. To get ana- recently announced some progress in this direction. lytic continuation to (s) = 1, one needs to apply If however, one of the curves has CM and corres- Shahidi’s results on the non-vanishing of Rankin– ponds to a Hecke character ψ, then one needs to Selberg L-functions which appear on the right hand study the L-series side. Poles on the line (s) = 1 can also be ruled m1 m2 out by the same theory. This completes the proof. L(s, ψ ⊗ Sym (ρ2)), In many respects, this is the elliptic analogue of Brauer’s theorem of the meromorphy of Artin and establish analytic continuation and non- L-series. As can be seen, the non-vanishing of the s ≥ L s vanishing in the region ( ) 1. This can be done -series on the line ( ) = 1 is essential in the since only Hecke characters intervene and these proof. This was ensured by an application of can be base-changed to any field by a well-known theorems of Jacquet, Shalika and Shahidi. The theorem of Weil [36]. It is also clear one can take Jacquet–Shalika theorem and the Shahidi theorem any number of CM elliptic curves and derive a rely on the theory of Eisenstein series (à la Lang- similar theorem for the same reasons. lands). There are other ways of establishing non- Here is the proof of theorem 1. vanishing of the L-series concerned without using the theory of Eisenstein series. Indeed, if one is By standard Tauberian theory, as discussed in willing to admit Rankin–Selberg theory and exis- [29], we need to show that for any irreducible tence and analyticity of these L-functions, then representation τ of G =Gal(M/K), the L-function classical non-vanishing techniques (of the type used by Hadamard and de la Vallée Poussin) actually L(s, τ ⊗ Symmρ) work. This method is outlined in a recent paper of Sarnak [28]. is analytic and non-vanishing in the region (s) ≥ 1. Using Brauer induction, we write 4. A Chebotarev–Sato–Tate theorem and G τ = ciInd φi, generalizations Hi i There are some natural generalizations of the Sato– where the ci are integers, and φi is an abelian Tate conjecture that one can consider. We indicate H H briefly how this can be done and what can actually character of i,with i certain nilpotent subgroups of G. In Taylor’s theorem, we can be proved. Firstly, we can take two non-isogenous L L M elliptic curves and consider the joint distribution choose so that and are disjoint. Thus, Gal(LM/L)=G and of the angles. For instance, if E1 and E2 are non- isogenous elliptic curves, both without CM, defined G Q θ E θ E L s, mρ ⊗ τ L s, mρ ⊗ φ ci . over ,and p( 1)and p( 2)aretheangles ( Sym )= ( Sym IndHi i) respectively, it is reasonable to expect that the dis- i tribution of the pair of angles (θp(E1),θp(E2)) is given by the product distribution Since we are viewing G as the Galois group of LM/L, we can re-write this, by Frobenius reci- 4 procity, as sin2 θ sin2 θ dθ dθ . π2 1 2 1 2 G m ci L s, ρ| Hi ⊗ ψ , ( IndHi (Sym (LM) i) ) i To prove such an assertion, we can use the formalism of Serre [29]. Using the formalism of [29], it where ψi is the Hecke character corresponding to is not difficult to show that this involves the study m φi via Artin reciprocity. As (Sym ρ)|L is automor- of certain L-series. Indeed, if both curves are non- m phic by Taylor’s theorem, (Sym ρ)|LM is automor- CM and have associated Galois representations ρ1 ρ phic by the theory of base change applied to the and 2 respectively, then one needs to show that solvable extension LM/L.Asinstep1ofTaylor’s the L-series m theorem, we deduce that (Sym ρ)|(LM)Hi is automorphic over (LM)Hi by an application of the m1 m2 L(s, Sym (ρ1) ⊗ Sym (ρ2)) Arthur–Clozel theory. Since ψi is a Hecke character, we deduce that extends to (s) ≥ 1 and does not vanish there. m ρ | H ⊗ ψ This looks like a difficult question to answer with (Sym ) (LM) i i THE SATO–TATE CONJECTURE AND GENERALIZATIONS 645 is automorphic over (LM)Hi . Consequently, by the References invariance of L-series under induction, we deduce that [1] Artin E 1924 Quadratische Körper im Gebiete der höheren Kongruenzen, I, II; Math. Zeit. 19 153–246. [2] Arthur J and Clozel L 1989 Simple algebras, base G m H change and the advanced theory of the trace formula; L(s, IndH ((Sym ρ)|(LM) i ⊗ ψi)) i Annals of Math. Studies, 120, Princeton University Press. is automorphic. Thus, it is analytic and non- [3] Breuil C, Conrad B, Diamond F and Taylor R 2001 On vanishing for (s) ≥ 1. This proves that the modularity of elliptic curves over Q : wild 3-adic exercises; J. Am. Math. Soc. 14(4) 843–939. m [4] Carayol H 2006–07 La conjecture de Sato–Tate L(s, (Sym ρ) ⊗ χ) (d’après Clozel, Harris, Shepherd-Barron, Taylor), Sem. Bourbaki 59 Exp. 977, pp. 345–391. extends to an analytic function for (s) ≥ 1and [5] Clozel L, Harris M and Taylor R 2008 Auto- morphy of some -adic lifts of automorphic mod does not vanish there. This proves the required Galois representations; Publ. Math. IHES 108(1) assertion for m odd. For m even, we proceed as 1–182. before, by induction to obtain the desired result. [6] Cogdell J, Kim H and Murty R 2006 Lectures on auto- This completes the proof of theorem 1. morphic L-functions; Fields Institute Lecture Notes, Am. Math. Soc., Providence. [7] Cogdell J and Michel P 2004 On the complex moments 5. Concluding remarks of symmetric power L-functions at s =1;Int. Math. Res. Notices 31 1561–1617. [8] Deligne P 1974 La conjecture de Weil, I; Inst. Hautes We conclude this section with an alternate argu- Etudes´ Sci. Publ. Math. 43 273–307. ment in the case that M/K is a nilpotent Galois [9] Deligne P 1980 La conjecture de Weil, II; Inst. Hautes extension which is simpler. In future variations, Etudes´ Sci. Publ. Math. 52 138–252. this alternate argument may be useful. [10] Deuring M 1941 Die Typen der Multiplikatorenringe By standard Tauberian theory, as discussed in elliptischer Funktionenkörper; Abh. Math. Zem. Han- [29], we need to show that for any irreducible sischen Univ. 14 197–272. representation τ of Gal(K/K¯ ), with nilpotent [11] Frei G and Roquette P 2008 Emil Artin and Hel- image, the L-function mut Hasse, Die Korrespondenz 1923–1934;Univer- sitätsverlag Göttingen. m [12] Frey G 1989 Links between solutions of A − B = C L(s, τ ⊗ Sym ρ) and elliptic curves; Lecture Notes Math. 1380 31–62. is analytic and non-vanishing in the region [13] Grothendieck A 1958 The cohomology theory of s ≥ abstract algebraic varieties; Proc. Int. Congr. Math. ( ) 1. Since any irreducible representation of a Edinburgh 103–118. finite nilpotent group is induced from an abelian [14] Hasse H 1933 Beweis des Analogons der Rie- character χ of a subgroup H,so mannschen Vermutung für die Artinschen und F K Shmidtschen Kongruenzzetafunktionen in gewis- m G m sen zykischen Fällen, Vorläufige Mitteilung; Nachr. τ ⊗ Sym (ρ)=IndH (χ ⊗ Sym (ρ)|LH ). Ges. Wiss. Göttingen I. Math.-Phys. Kl. Fachgr. I Math. Nr. 42 253–262. m [15] Hecke E 1918 Eine neue Art von Zetafunktionen und χ⊗ ρ | H By Arthur–Clozel, Sym ( ) L is automorphic. ihre Beziehungen zur Verteilung der Primzalhen, I; We now complete the proof as before. Math. Zeit. 1 357–376. It is clear from the preceding arguments that [16] Hecke E 1920 Eine neue Art von Zetafunktionen und if one had the automorphic induction of Hecke ihre Beziehungen zur Verteilung der Primzahlen, II; characters, the proof would go through for any Math. Zeit. 6 11–51. Galois setting and not just in the nilpotent or [17] Jacquet H and Shalika J 1976–77 A non-vanishing theorem for zeta functions of GLn; Inventiones Math. solvable setting. Future advances in the Langlands 38(1) 1–16. program should translate into general theorems of [18] Khare C 2006 Serre’s modularity conjecture: The level Chebotarev–Sato–Tate type. one case; Duke Math. J. 134(3) 557–589. [19] Khare C and Wintenberger J-P, Serre’s modularity conjecture I (to appear). Acknowledgements [20] Lang S 1987 Elliptic Functions Graduate Texts in Mathematics, Second Edition (New York: Springer- We would like to thank Fernando Gouvêa, Verlag). [21] Langlands R P 1970 Problems in the theory of auto- Sanoli Gun, Chandrashekhar Khare, Dipen- morphic forms, in Lectures in Modern Analysis and dra Prasad and the referee for helpful remarks on Applications; Lecture Notes in Math. 170 Springer- an earlier version of this paper. Verlag, 18–86. 646 M RAM MURTY AND V KUMAR MURTY

[22] Mazur B 2006 Controlling our errors; Nature 443 [29] Serre J-P 1998 Abelian -adic representations and 38–40. elliptic curves; Research Notes in Mathematics 7 [23] Mazur B 2008 Finding meaning in error terms; Bull. (Massachusetts: A K Peters, Wellesley). Am. Math. Soc. 45 185–228. [30] Serre J-P 1987 Sur les répresentations modulaires de [24] Ram Murty M 2002 Recent developments in the degré2deGal(Q¯/Q); Duke Math. J. 54 179–230. Langlands program; Comptes Rendus Math. Rep. Sci. [31] Shahidi F 1981 On certain L-functions; Am. J. Math. Canada 24(2) 33–54. 103(2) 297–355. [25] Ram Murty M 1983 Oscillations of Fourier coef- [32] Tate J 1965 Algebraic cycles and poles of zeta ficients of modular forms; Math. Annalen 262 functions; in Arithmetic Algebraic Geometry (ed.) 431–446. Schilling F G, 93–110 (New York: Harper and Row). [26] Kumar Murty V 1981 On the Sato–Tate conjecture; in [33] Taylor R 2008 Automorphy of some -adic lifts of auto- Number theory related to Fermat’s last theorem (Cam- morphic mod representations, II; Publ. Math. IHES bridge, Mass.); 1982 Progress in Math. 26 195–205 108(1) 183–239. (Boston: Birkhäuser). [34] Yoshida H 1973 An analogue of the Sato–Tate conjec- [27] Ribet K 1990 From the Taniyama–Shimura conjecture ture; Inv. Math. 19 261–277. to Fermat’s last theorem; Annales de la facultédes [35] Weil A 1949 Number of solutions of equations in finite sciences de Toulouse, Ser. 5 11(1) 116–139. fields; Bull. Am. Math. Soc. 55 497–508. [28] Sarnak P 2004 Non-vanishing of L-functions on (s)= [36] Weil A 1971 Dirichlet series and automorphic forms; 1; in Contributions to automorphic forms, geometry, Springer Lecture Notes 189. and number theory 719–732 (Baltimore: Johns Hopkins [37] Wiles A 1995 Modular elliptic curves and Fermat’s last University Press). theorem; Ann. Math. 141(2, 3) 443–551.