DEFORMATIONS OF GALOIS REPRESENTATIONS AND THE THEOREMS OF SATO-TATE AND LANG-TROTTER. 1. Introduction

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1 DEFORMATIONS OF GALOIS REPRESENTATIONS AND THE THEOREMS OF SATO-TATE AND LANG-TROTTER AFTAB PANDE Abstract. We construct innitely ramied Galois representations ρ such that the a l (ρ)'s have distributions in contrast to the statements of Sato-Tate, Lang- Trotter and others. Using similar methods we deform a residual Galois representation for number elds and obtain an innitely ramied representation with very large image, generalising a result of Ramakrishna. 1. Introduction If E is an elliptic curve over Q, then by the Mordell-Weil theorm we know that the set of rational points E(Q) is a nitely generated abelian group. For l a prime, let F l be the prime eld and dene a l (E) := l + 1 #E(F l ). The integers a l (E) provide us a lot of information about the connections that exist between elliptic curves, modular forms and Galois representations. If f is a classical modular form then it has a fourier expansion f(τ) = n a n(f)q n, where τ h and q = e 2πiτ. The celebrated Shimura-Taniyama conjecture which said that all elliptic curves are modular could be rephrased as a l (E) = a l (f). To each elliptic curve E we can also associate a Galois representation ρ E : G GL 2 (Z p ), ramied at nitely many primes, where G = Gal(Q/Q). If we de- ne a l (ρ E ) := T r(ρ E (σ l )), where σ l is the Frobenius at l, then we know that a l (ρ E ) = a l (E). Serre conjectured [20] (now a theorem due to Khare and Wintenberger) that every odd, irreducible Galois representation ρ is modular, which could be rephrased as a l (ρ) = a l (f) for a certain modular form f of weight 2. The a l (E) (for simplicity we denote them a l ) have also been studied asymptotically as l varies. Hasse showed that a l 2 l and in this paper we examine the following statements from the point of view of innitely ramied Galois representations: 0 Aftab Pande, Universidade Federal do Rio de Janeiro, RJ, Brazil, aftab.pande@gmail.com 1

2 2 AFTAB PANDE (Sato-Tate) If α l = a l 2, then the probability distribution function for the l α l is given by P (l A α l B) = 2 B π A 1 x2 dx. (Lang-Trotter) #{l < x a l = D} = O( x log(x) ), where D Z is a xed integer. (Serre and Elkies) #{l < x a l = 0} = O(x 3/4 ). (Serre) #{l < x a l = 1} = O( (logx) ), for any ɛ < 1 1+ɛ 3. x The Sato-Tate [21] conjecture, assuming multiplicative reduction at some prime, was proved recently by Taylor, Clozel, Harris and Shepherd-Barron ([22], [5] and for an elementary introduction about the signicance of the conjecture read [11]). There is also a recent preprint ([2] by Barnet-Lamb, Geraghty, Harris and Taylor) which claims to have proved the full version of the Sato-Tate conjecture. In the case that a l = 0 (also known as supersingular primes), the Lang-Trotter [8] conjecture assumes that E has no complex multiplication. Serre [18] showed supersingular primes have density zero using the Cebotarev Density theorem and then proved [19] the above estimate assuming the generalised Riemann Hypothesis. Elkies [3] showed that there are innitely many supersingular primes and later proved [4] Serre's estimate without assuming the GRH. When a l = 1, Mazur [10] refers to the corresponding l's as anomalous primes, and the estimate was established by Serre [19]. Our goal is to construct representations ρ : G GL 2 (Z p ), ramied at innitely many primes, such that the a l (ρ)'s have distributions in contrast with the 4 statements above. We obtain the following theorems in section 5: Theorem 1.1. For any ɛ 1, there exists a deformation ρ : G GL 2 (Z p ) ramied at innitely many primes, such that the set R = {l a l 2 ɛ} is of density one. l Theorem 1.2. If D Z is xed, we can nd a deformation ρ : G GL 2 (Z p ), such that the set R(x) = {l < x a l = D} O( x logx ). The rst theorem is in contrast with Sato-Tate and the second theorem contrasts with Lang-Trotter. The statements about supersingular and anomalous primes follow almost immediately from the second theorem. Serre had shown examples of representations ramied at all primes, but in his case the representations were reducible. Ramakrishna [15] showed the existence of

3 DEFORMATIONS OF GALOIS REPRESENTATIONS AND THE THEOREMS OF SATO-TATE AND LANG-TROTTER3 irreducible representations, ramied at innitely many primes. Innitely ramied representations do not arise from geometry, but as the limit of geometric representations. They have been useful in showing the independence of the conditions of potentially semistable and nite ramication (as in the Fontaine-Mazur conjecture), and also in constructing representations which are pure or rational as in [6]. To construct such representations, we make use of techniques which we develop in section 3. While using these techniques we obtain a generalisation of a theorem of Ramakrishna [17] `en passant' in section 4: Theorem 1.3. Let k be a nite eld of characteristic p, W (k) the ring of Witt vectors of k, F a number eld such that [F (µ p ) : F ] = p 1 and G F its absolute Galois group Gal(F /F ). If ρ : G F GL 2 (k) is a representation whose image contains SL 2 (k) then there exists a deformation of ρ, ramied at innitely many primes, ρ : G F GL 2 (W (k)[[t 1, T 2,.., T r,..., ]]), whose image is full i.e. contains SL 2 (W (k)[[t 1, T 2,.., T r,..., ]]). We should point out that the reader interested in the distribution results does not need to read propostions 3.11, 3.12 and section 4 which are required only for the above theorem. 2. Deformation Theory We briey recall some facts of deformation theory and refer the reader to Mazur [9] and Ramakrishna [14], [17] for more details. Let ρ : G GL 2 (k) be an absolutely irreducible representation of a pronite group G, and let R be a ring in the category C of Artinian local rings with residue eld k. If γ is a lift of ρ to GL 2 (R), then two lifts γ 1, γ 2 are equivalent if γ 1 = Aγ 2 A 1, where A is congruent to the identity matrix modulo m R, the unique maximal ideal of R. A deformation γ of ρ to R is an equivalence of lifts of ρ to R. We have the following theorem due to Mazur [9]: Theorem 2.1. There is a complete local Noetherian ring R un with residue eld k and a continuous deformation ρ : G GL 2 (R un ) such that: Reduction of ρ modulo the maximal ideal of R un gives ρ For any ring R in C and any deformation γ of ρ to GL 2 (R) there is a unique homomorphism φ : R un R in C such that φ ρ = γ as deformations.

4 4 AFTAB PANDE R un is called the universal deformation ring if ρ is irreducible (or if the centraliser of the image of ρ is exactly the scalars), and the versal ring otherwise. In this paper we x the determinants of all deformations of ρ, which means we study the cohomology of Ad 0 ρ (the 2 2 matrices of trace zero, with G acting via conjugation by ρ) and not Adρ. It is known that R un is a quotient of W (k)[[t 1, T 2,..., T r ]], where r = dim k H 1 (G, Ad 0 ρ). For the rest of the paper, we let X denote Ad 0 ρ. Let G F be the Galois group of a number eld F, ρ n : G F GL 2 (W (k)/p n ) be a deformation of ρ, and suppose we want to deform ρ n to ρ n+1 : G F GL 2 (W (k)/p n+1 ). When we x determinants, the obstructions to deformation lie in H 2 (G F, X). For a given ρ and a nite set of primes S containing p and the ramied primes of ρ, dene X i S (X) to be the kernel of the map Hi (G F,S, X) q S H i (G q, X), where G F,S is the Galois group of the maximal extension of F unramied outside S, and G q = Gal(F q /F q ) is the decomposition group at q. If X 2 S (X) = 0, then all the obstructions can be detected locally. If the obstruction is trivial, and ρ n+1 is a deformation of ρ n, then H 1 (G F, X) acts on the set of deformations. The action is given by (f.ρ n+1 )(σ) = (I + p n f(σ))(ρ n+1 (σ)), where f H 1 (G F, X). When f is a coboundary, then (I + p n f(σ))ρ n+1 (σ) is the same deformation. If the image of ρ is exactly the scalars, then H 1 (G, X) acts on the deformations of ρ n to GL 2 (W (k)/p n+1 ) as a principal homogeneous space. Deformations of residual Galois representations exist due to the work of Diamond, Taylor, Ramakrishna [14], [16] and recently, Manoharmayum ([12]) for number elds. 3. The setup In this section we generalise some denitions and propositions from [17] and [6] for a number eld F, such that [F (µ p ) : F ] = p 1. Since they are required for both sections, we sketch the proofs and refer the reader to the respective sources for more details. Denition. Let T be a nite set of primes and X be the Cartier dual of X. If L q H 1 (G q, X) is a subspace with annihilator L q H 1 (G q, X ) (under the local pairing), for q T, then we dene HL 1(G F,T, X) and H 1 L (G F,T, X ) to be, respectively, the kernels of the restriction maps;

5 DEFORMATIONS OF GALOIS REPRESENTATIONS AND THE THEOREMS OF SATO-TATE AND LANG-TROTTER5 H 1 (G F,T, X) q T H 1 (G q, X)/L q H 1 (G F,T, X ) q T H 1 (G q, X )/L q These cohomology groups are also known as Selmer and dual Selmer groups, and the next proposition about them is due to Wiles [23] (see also [13]). We let T be a nite of primes, which contains all primes of F above rational primes dividing #X. Proposition 3.1. dim H 1 L (G F,T, X) dim H 1 L (G F,T, X ) = dim H 0 (G F,T, X) dim H 0 (G F,T, X) + q T (dim L q dim H 0 (G q, X)). Denition. Let q be a rational prime which splits completely in F such that q ±1 mod p, and q be a prime lying above it in F. We say that q is ρ-nice if; ρ is unramied at q the ratio of the eigenvalues of ρ(σ q ) is q, where σ q is Frobenius at q. Denition. Let R be a complete local Noetherian ring with residue eld k, and let J be an ideal of nite index in R. If ρ R/J is a deformation of ρ to GL 2 (R/J), then q is ρ R/J -nice if q is nice for ρ ρ R/J is unramied at q and the ratio of the eigenvalues of ρ R/J (σ q ) is q. ρ R/J (σ q ) has the same (prime to p) order as ρ(σ q ). Nice primes have useful properties, which are exhibited in the following propositions. Proposition 3.2. If q is a nice prime, then the dimensions of H i (G q, X) are 1, 2, 1 for i = 0, 1, 2, respectively. Proof. If q is a nice prime, we see that as G q -modules X = k k(1) k( 1), and X = k(1) k k(2), where k(r) accounts for the twist by the r-th power of the cyclotomic character. We assumed that F doesn't contain the p-th roots of unity and q ±1 mod p, so H i (G q, k(r)) = 0 for r 0, 1 by local duality. The rest follows by using the local Euler-Poincare characteristic. See Lemma 2 of [6] for more details. Proposition 3.3. a) Nice primes have positive density. b) We can nd a nite set of nice primes Q such that X i S Q (X) = 0 for i = 1, 2.

6 6 AFTAB PANDE c) Assume that X 2 T (X) = 0. For any nice prime β / T, the ination map H 1 (G F,T, X) H 1 (G F,T {β}, X) has one dimensional cokernel. Proof. a) We follow the argument as in Theorem 2 of [14].Let F (ρ) be the xed eld of the kernel of ρ, F (X) be the eld xed by the action of G F on X = Ad 0 ρ, K be the composite of F (X) and F (µ p ), and D be the intersection of the two elds. To show that nice primes have positive density, we need to use Cebotarev's theorem and nd the right conjugacy class associated to nice primes in Gal(K/F ). Note that Gal(F (X)/D) Gal(F (µ p )/D) = Gal(K/D) Gal(K/F ), so we will dene C to be the conjugacy class of a b Gal(F (X)/D) Gal(F (µ p )/D), where a will correspond to nice primes. Claim: [D : F ] = 1 or 2. As D/F is abelian, the commutator subgroup of Gal(F (µ p )/F ) which is simply SL 2 (k), is contained in Gal(F (ρ)/d). Since D F (X), the xed eld associated to ρ, we see that Gal(F (ρ)/d) contains the scalar matrices Z in Imρ, so Z.SL 2 (k) is a normal subgroup of Gal(F (ρ)/d). As Imρ/SL 2 (k) is cyclic, we see that [Imρ : Z.SL 2 (k)] = 1 or 2. By denition Gal(F (ρ)/f ) = Imρ, and Z.SL 2 (k) Gal(F (ρ)/d) so [D : F ] = 1 or 2, which proves the claim. Assume p 7, and then choose x F p such at x 2 ±1. Consider x 2 in Gal(F (µ p )/F ). As [D : F ] = 1 or 2, we can see that x 2 Gal(F (µ p )/D). x 2 will correspond to b in our desired conjugacy class C. Let x 0 Gal(F (ρ)/d) 0 x 1 and consider its projection a in Gal(F (X)/D). Then a corresponds to an element in the image of ρ whose eigenvalues have ratio t ±1 F p Dene α = a b Gal(F (X)/D) Gal(F (µ p )/D) = Gal(K/D) Gal(K/F ) and let C be the conjugacy class of α. By Cebotarev we see that nice primes have positive density. b) For the second part, we need to nd a set of nice primes Q = {q 1,.., q m } such that f i Gqi 0 and g j Gqj 0 and 0 otherwise, for any set of linearly indepedent {f 1,..., f r } X 1 S (X) and {g 1,.., g s } X 1 S (X ). Let K fi and K gj be the xed elds of the kernels of the restrictions of f i, g j to Gal(K/K). Both these elds are linearly disjoint over K, so to nd a nice prime q which satises the required properties we need to make sure that the primes lying above it do not split completely from K to

7 DEFORMATIONS OF GALOIS REPRESENTATIONS AND THE THEOREMS OF SATO-TATE AND LANG-TROTTER7 K fi, but split completely from K to K fj for i j (similarly for K gj ). Such primes can be found because of the linear disjointedness over K of K fi and K gj. Now we choose a basis f i i I of X 1 S (X) and g j j J of X 1 S (X ) and sets of primes {q i } i I and {p j } j J as above, so that f i (σ qi ) 0 and 0 otherwise (similarly for the g j 's). Let T = {q i, p j } i I,j J, and we see that X 1 S T (X) = 0 and X1 S T (X ) = 0. By duality, X 2 S T (X) = 0. c) Let L q = H 1 (G q, X) for all q T β. Using Proposition 3.1, we see that the LHS is simply H 1 (G F, T ), and the RHS changes by dim H 1 (G q, X) dim H 0 (G q, X) = dim H 2 (G q, X). Using Proposition 3.2, we see that for a nice prime q, dim H 2 (G q, X) = 1. Now we dene some deformation conditions. Denition. Let N q = H 1 (G q, k(1)) H 1 (G q, X), and its annihilator N q = H 1 (G q, k(1)) H 1 (G q, X ). We let C q be the class of deformations of ρ which are preserved by the action of N q. The class of deformations C q correspond to the behaviour of the ρ's when restricted to nice primes. This class of deformations behaves nicely and can be lifted to the next level easily. N q is called the set of null cohomology classes. When we act on the deformations by cohomology classes, the set N q is the set of classes which when restricted to nice primes leave the deformation untouched, as there are no obstructions to lifting at those primes. If there are obstructions, we will choose a cohomology class which is not in N q to overcome the obstructions. See [16] for more details on both C q and N q. Assume that T is large enough so that X 2 T (X) = 0. Consider the restriction maps ψ T : H 1 (G F,T, X) q T H 1 (G q, X) and ψ T : H1 (G F,T, X ) q T H 1 (G q, X ). By global duality, we know that the images of ψ and ψ are exact annihilators of each other under the local pairing. annihilation property. This is also called the Denition. (Local condition property) Assume that T is large enough so that X 2 T (X) = 0. Let (z q) q T q T H 1 (G q, X), such that (z q ) q T / Imψ T. For β a nice prime, h β H 1 (G F,T {β}, X) is a solution to the local condition problem if h β Gq = z q for all q T.

8 8 AFTAB PANDE Remark: It is not hard to see (Prop 3.2 of [17]) that if (z q ) q ζ H 1 (G F,T, X ) such that ψ T (ζ) does not annihilate (z q) q T. / Imψ T, then Proposition 3.4. Let ρ R/J be given with (z q ) q T, and ζ be as above. Choose a basis ζ 1,..., ζ s of ψ 1 T (Ann(z q ) q T ). Let Q be the set of nice primes q such that ζ i Gq = 0. ζ Gq 0 f H 1 (G F,T, X) f Gq = 0 Then, for any β Q h β H 1 (G F,T {β}, X) which solves the local condition property. Proof. Following Proposition 3.4 of [17], we see that H 1 L (G F,T, X) = H 1 L (G F,T {β}, X), where L q = 0 for q T and L β = H 1 (G β, X) for β Q. From Proposition 3.3c), we see that the ination map from H 1 (G F,T, X) H 1 (G F,T {β}, X) has a cokernel of dimension one, so g H 1 (G F,T {β}, X) such that (g Gq ) q T / Imψ T. We will modify this g to get our required h β. Using the global reciprocity law, we have that q T {β} inv q(ζ i g) = 0 for any i. As ζ i Gβ = 0, so q T inv q(ζ i g) = 0. We also chose ζ i to be the annihilators of z q, so q T inv q(ζ i z q ) = 0. Thus q T inv q(ζ i (g cz q )) = 0, for any scalar c. By the remark preceding this proposition, we know that q T inv q(ζ z q ) = b 0. Now we need to show that q T inv q(ζ g) = a 0. If it was zero, the annihilation property implies that (g Gq ) q T Imψ T, which contradicts the choice of g. So, q T inv q(ζ (g a b z q)) = 0, which means that ζ annihilates (g a b z q) q T. If we choose c = a/b, then every element of ζ 1,.., ζ s, ζ (which is a basis of H 1 (G F,T, X )) annihilates (g a b z q) q T k H 1 (G F,T, X) such that ψ T (k) = (g a b z q) q T. Finally, set h β = b a (g k) which solves the local condition property. The following proposition is critical for the distribution results in section 5 and we briey summarise the ingredients in the hypothesis for readers interested in those theorems. ρ R/J is a deformation of a given ρ to GL 2 (R/J), where J is an ideal of nite index in R, a complete local Noetherian ring.

9 DEFORMATIONS OF GALOIS REPRESENTATIONS AND THE THEOREMS OF SATO-TATE AND LANG-TROTTER9 Q is a set of nice primes (Denition 3) which also satises the hypothesis of Proposition 3.4. N q = H 1 (G q, k(1)) H 1 (G q, X) is a subspace whose action preserves deformations (Denition 3). T is a set of primes such that X 2 T (X) = 0, (z q) q T q T H 1 (G q, X) and h β H 1 (G F,T {β} such that h β Gq = z q for all q T ( Denition 3).. Proposition 3.5. Let ρ R/J, Q, h β be as above. Then there exists a β Q such that h β Gβ N β or there exist β 1, β 2 Q and h = α 1 h β1 + α 2 h β2 such that h βi N βi for i = 1, 2 and h solves the local condition property. Proof. From Proposition 3.2, we know that H 1 (G β, X) = H 1 (G β, k) H 1 (G β, k(1)). Recall that we dened N β = H 1 (G β, k(1)), so if h β N β for some β Q, we're done. If not, we need to nd β 1, β 2 as stated. Let h β (σ β ) denote the evaluation of the projection of h β (σ β ) at H 1 (G β, k) and consider the matrix consisting of the values of h βi (σ βj ) for i, j = 1, 2: σ β1 σ β2 h β1 a b h β2 c d Since we are assuming that one prime β doesn't exist as required, a, d / N βi (we say that a, d 0 for this condition). To prove the theorem we need to show that we can nd a linear combination h = α 1 h β1 + α 2 h β2 such that h βi N βi for i = 1, 2 and h solves the local condition property. As h β1, h β2 solve the local condition property, to show that h solves the local condition property we need α 1 + α 2 = 1. Thus, we need to prove that the above matrix has unequal rows and zero determinant. Let y be the value of h β (σ β ) that occurs most often. Then the set Y = {β Q h β (σ β ) = y} has positive upper density (Q as noted earlier is a Cebotarev set). For a nice prime β choose η β H 1 (G β, X ) such that inv β (η β h β ) 0. Let z be the value that occurs most often. Then Z = {β Y inv β (η β h β ) = z} also has positive upper density.

10 10 AFTAB PANDE Choose any β 1 Z. Then, we need to nd a β 2 Z such that ad bc = 0, and b, c y. To choose b(= h β1 (σ β2 )) to be whatever we want is basically a Cebotarev condition on β 2, corresponding to h β1. Now we need to choose c(= h β2 (σ β1 )) appropriately. By Proposition 3.6 of [17], we see that hβ 2 (σ β1 ) y. 1 p = 1 p ξβ1 (σ β2 )z, where ξ β1 HP 1 (G F,T {β}, X ), P q = 0 for q T and P q = N q. Thus, we need to choose ξ β1 (σ β2 ) appropriately. When we consider h β1 and ξ β1, we see that due to the linear disjointedness of the eld extensions associated to them (as in the proof of Proposition 3.3), they both give independent Cebotarev conditions. Suppose there is no β 2 Z for which h β1 (σ β2 ) and ξ β1 (σ β2 ) can be what we want. Then the set Z \ {β 1 } is in a Cebotarev class which is complementary to the Cebotarev conditions imposed on σ β2 due to the choice of h β1 (σ β2 ) = x 0, y. Let D be the density of Q. Then the two complementary Cebotarev classes have density Dγ where γ < 1. If we replace β 1 by a sequence of primes i Z which don't allow us to choose the appropriate second prime, then Z \ { i } lies in a Cebotarev class complementary to the Cebotarev class associated to h i and ξ i, which are all independent of each other. By imposing n such conditions, we see that the density of these complementary Cebotarev classes is Dγ n. As n gets larger and larger, we see that the set Z \ { 1,..., n } lies inside sets of arbitrarily small density, which is a contradiction. Thus we can always nd 2 primes β 1, β 2 which satisfy the hypothesis of the proposition. The next two propositions are used in proving the large image theorem in section 4 and are not required for the distribution results. Proposition 3.6. Let B be a set of one or two nice primes such that H 1 N (G S n, X) H 1 N (G F,S n B, X) has cokernel of dimension 1. Let A be the set of one or two nice primes from the previous proposition and let (z q ) q Sn B / q Sn BN q ψ Sn B(H 1 (G F,Sn B, X)). Then, H 1 N (G F,S n B A, X) = H 1 N (G F,S n B, X).

11 DEFORMATIONS OF GALOIS REPRESENTATIONS AND THE THEOREMS OF SATO-TATE AND LANG-TROTTER 11 Proof. In the hypothesis of Proposition 3.4, f H 1 (G F,T, X) f Gq = 0. As N q = 0 for q S n B and f N q for q A, we see that H 1 N (G F,S n B, X) H 1 N (G F,S n B A, X). For the other containment we consider the 2 cases for A: 1) A has one prime β: Any element of H 1 N (G F,S n B A, X) \ H 1 N (G F,S n B, X) looks like f + αh β, where f H 1 (G F,Sn B, X), α 0 and h β solves the local condition property i.e. h β Gq = z q for q S n B. Thus, (f + αh β ) Gq = f + αz q N q, for all q S n B A. But we assumed that z q / N q Im(ψ Sn B) α = 0, a contradiction. 2) A has 2 primes β 1, β 2 : Any element of H 1 N (G F,S n B A, X) \ H 1 N (G F,S n B, X) looks like f + α 1 h β1 + α 2 h β2, where f H 1 (G F,Sn B, X). Using a similar argument as in case 1, we see that α 1 + α 2 = 0. We know that α 1 (h β1 h β2 ) Gq = 0 for all q S n B. This means that f Gq N q for all q S n B f HN 1 (G F,S n B, X). We also know that f Gβi = 0 for i = 1, 2, so f H 1 N (G F,S n B A, X). Thus, α 1 (h β1 h β2 ) H 1 N (G F,S n B A, X) α 1 (h β1 h β2 ) Gβi N βi, for i = 1, 2. By the construction of β i in the previous proposition, we see that α 1 = 0 f H 1 N (G F,S n B, X), which is a contradiction. Proposition 3.7. There exists a set B of one or two ρ n -nice primes such that the map H 1 N (G F,S n, X) H 1 N (G F,S n B, X) has one dimensional cokernel. Proof. By a straight generalisation of Proposition 10 of [6] we can nd a set B = {β 1, β 2 } of one or two ρ n -nice primes for which there is a linear combination f = α 1 f β1 + α 2 f β2 such that f(σ β1 ) = 0 for i = 1, 2 and f Gq = 0 for q S n. As β 1, β 2 are nice primes, we see that N βi = 0 for i = 1, 2, which means that f is in the kernel of the map H 1 (G F,Sn B, X) q Sn BH 1 (G q, X)/N q. 4. Large image We want to construct ρ : G F GL 2 (W (k)[[t 1,..., T r,...]]) such that Imρ SL 2 (W (k)[[t 1,..., T r,...]]). We start with ρ : G F GL 2 (k) and lift it successively to ρ n : G F,Sn GL 2 (W (k)[[t 1,..., T n ]]/(p, T 1,..., T n ) n ), and dene ρ = lim n ρ n such that at each stage n, Imρ n SL 2 (W (k)[[t 1,..., T n ]]/(p, T 1,..., T n ) n ).

12 12 AFTAB PANDE If R n is the deformation ring of ρ n with m Rn the maximal ideal, then we see that R n /m n R n = (W (k)[[t 1,..., T n ]]/(p, T 1,..., T n ) n ). We will add more primes of ramication to S n and get a new set of primes S n+1, such that the deformation ring associated to S n+1 has R n+1 /m n+1 R n+1 as a quotient. This gives us a surjection from R n+1 /m n+1 R n+1 R n /m n R n, which allows us to get the inverse limit R = lim n R n /m n R n. We assume that there exists ρ n : G F,Sn GL 2 (R n /m n R n ), and dim H 1 N (G F,S n, X) = n. By Proposition 3.7 we can nd a set B of ρ n -nice primes such that dim H 1 N (G F,S n B, X) = n + 1. Let U be the deformation ring associated to the augmented set S n B, with the deformation conditions (N q, C q ). As B consists of ρ n -nice primes, we have a surjection φ : U R n /m n R n which means that for some I 1, we U/I 1 = R n /m n R n. As dim H 1 N (G F,S n B, X) = n + 1, we see that as a ring U consists of power series of (n + 1) variables. Thus, for some I 2, U/I 2 = k[[t 1,..., T n+1 ]]/(T 1,..., T n+1 ) 2. Let I = I 1 I 2, and dene U 0 = U/I. Our goal is to get a deformation ring which has R n+1 /m n+1 R n+1 U 0 is such a deformation ring, we're done. If not, we get a sequence R n+1 /m n+1 R n+1... U 1 U 0, as a quotient. If where the kernel at each stage has order p. Then we add more primes of ramication to S n B so that the augmented deformation ring has U 1 as a quotient. We then keep iterating to get our required deformation ring. As U 0 is a quotient of U, we let ρ U0 be the induced deformation. Since B consists of nice primes and all global obstructions can be detected locally (X 2 S n (X) = 0 even as S n gets bigger) we see that ρ U0 can be lifted to U 1, and we call the deformation ρ U1. If there are no local obstructions to lifting ρ U1, we're done. If there are local obstructions, then we choose a set of cohomology classes (z q ) q Sn B such that the action of z q on ρ U1 zq overcomes the local obstructions at q S n B. By Proposition 3.5 we can nd a set A and a cohomology class h such that: q is ρ U1 -nice, for q A (no new obstructions at A) h Gq N q for q A (h preserves the class of deformations) h Gq = z q, for q S n B (h overcomes local obstructions at S n B) We now dene ρ U1 = (I +h) ρ U1. There are no obstructions to lifting ρ U1 and the augmented deformation ring has U 1 as a quotient. By Proposition 3.6 we know that dim H 1 N (G F,S n B, X) = dim H 1 N (G F,S n B A, X) = n + 1, so we can keep iterating

13 DEFORMATIONS OF GALOIS REPRESENTATIONS AND THE THEOREMS OF SATO-TATE AND LANG-TROTTER 13 in a similar way by adding more primes, and get a set S n+1 which has R n+1 /m n+1 R n+1 as a quotient. It remains to show that at each stage n, Imρ n SL 2 (R n /m n R n ). We recall a proposition due to Boston [1]. Proposition 4.1. Let R be a complete noetherian local ring with maximal ideal m, with R/m nite, charr/m = p 2. Let H be a closed subgroup of SL n (R) projecting onto SL n (R/m 2 ). Then H = SL n (R). For our purpose, we let H = Imρ n SL 2 (R n /m n R n ). To show that Imρ n SL 2 (R n /m n R n ), all we need to show is that Imρ n /m 2 R n SL 2 (R n /m 2 R n ). Consider the exact sequence SL 2 (m Rm /m 2 R m ) θ SL 2 (R n /m 2 R n ) φ SL 2 (R n /m Rn ). As a k[imρ]-module, Kerφ has (n + 1) copies of X. To see this, note that Ker{SL 2 (W (k)/p 2 ) SL 2 (k)} consists of one copy of X (R n /m Rn = k). R n /m 2 R n consists of power series like a 0 +a 1 T a n T n, where a 0 W (k)/p 2 corresponds to one copy of X, and the remaining n a i 's ( k) account for the other n copies. Now we look at the projection Imρ n /m 2 R n Imρ. Recall that we assumed Imρ SL 2 (k), so we need to obtain (n + 1) copies of X in the kernel to show that Imρ n /m 2 R n is big enough. We know there is an equivalence between Ext 1 (X, X) (killed by p), H 1 (G F, X) and lifts of ρ to the dual numbers k[ɛ]. Since ρ n is a deformation of ρ and we assumed that dim H 1 N (G F, X) = n, we get n copies of X. These give split extensions as we are considering lifts to the dual numbers and k embeds in k[ɛ]. To get the last copy of X, we see that for n 2, the deformation of ρ to GL 2 (W (k)/p 2 ) gives us an extension of X. The exact sequence SL 2 (pw (k)/p 2 ) SL 2 (W (k)/p 2 ) SL 2 (W (k)/p) doesn't split, so this corresponds to a non-split extension and see that Imρ n /m 2 R n SL 2 (R n /m 2 R n ). Now we are in a position to state the nal theorem. Theorem 4.2. There exists a deformation of ρ, ramied at innitely many primes, ρ : G F GL 2 (W (k)[[t 1, T 2,.., T r,..., ]]), whose image contains SL 2 (W (k)[[t 1, T 2,.., T r,..., ]]). Proof. All that we need to show is that ρ = lim n ρ n is full i.e. given α SL 2 (R), g G F such that ρ(g) = α, where R = lim n R n /m n R n. At each stage n we know that Imρ n SL 2 (R n /m n R n ). So if α n R n /m n R n the image of α, then there exists a g n G F such that ρ n (g n ) = α n. Ideally we is

14 14 AFTAB PANDE would like to dene g = lim n g n, and take limits again but we need to modify g n a little bit. As G F is compact and Hausdor, there exists a subsequence g nm of g n, such that g nm has g as a limit point. For n m > n, we know that ρ nm is a deformation of ρ n. So when we reduce ρ nm mod m n R n, we see that ρ n (g nm ) = α n. Also, as ρ n is continuous we can switch limits and see that lim n ρ n (lim m g nm ) = lim n lim m ρ n (g nm ). Finally, ρ(g) = lim n ρ n (g) = lim n ρ n (lim m g nm ) = lim n lim m ρ n (g nm ) = lim n (lim m α n ) = lim α n = α. n 5. Distribution Statements We start with a residual representation ρ : G GL 2 (Z/p), and lift it successively to ρ n : G GL 2 (Z/p n ). At each stage we will add more primes Q n to the ramication set S n, and x the characteristic polynomials for a nite, but large, set of unramied primes R n. To overcome local obstructions to lifting at the primes that we keep adding we will need to add some more primes A n. This will allow the lifts to be unobstructed at the ramied primes S n and keep the characteristic polynomials unchanged at the unramied primes R n. As we keep lifting ρ n to ρ n+1, we will control the cardinality at each stage for R n so that after taking limits the cardinality is large enough to get a representation ρ : G GL 2 (Z p ) for which the distribution of the a l 's is in contrast to the distribution statements in the rst section. Let ρ : G GL 2 (Z/p) be a surjective Galois representation, with p 5 that arises from S 2 (Γ 0 (N)) for some (N, p) = 1 and N square-free. By results of Serre, we know that such representations exist. If we take a semistable elliptic curve E over Q, then for all but nitely many primes the corresponding mod p representation satises the required conditions. Let S be a nite set of primes containing p and the primes of ramication of ρ. The key lemma that allows us do this is Lemma 8 from [6]:

15 DEFORMATIONS OF GALOIS REPRESENTATIONS AND THE THEOREMS OF SATO-TATE AND LANG-TROTTER 15 Lemma 5.1. Let ρ n be a deformation of ρ to GL 2 (Z/p n ), unramied outside a set S, and assume X 1 S (X) and X2 S (X) are trivial. Let R be any nite collection of unramied primes of ρ disjoint from S. Then, there is a nite set Q = {q 1, q 2,..., q m } of ρ n -nice primes disjoint from R S such that we get the following isomorphisms: H 1 (G S R Q, X ) v Q H 1 (G v, X ) H 1 (G S Q, X) ( s S H 1 (G s, X)) ( r R H 1 nr(g r, X)) (H 1 nr(g v, X) = Im(H 1 (G v /I v, X Iv ) H 1 (G v, X)), where I v G v is the inertia group) The above lemma allows us to nd a global cohomology class such that its action on a deformation overcomes obstructions at primes of S and allows us to choose the characteristic polynomials at R as we want. Now we might have added local obstructions at the primes of Q. To overcome these obstructions we use Proposition 3.5 to add a set A of one or two nice primes and a global cohomology class h which overcomes local obstructions at primes of Q, leaves the characteristic polynomials at R unchanged, and doesn't add any new obstructions at A. We now prove the rst theorem (based on [6]) about the distribution statements (Sato-Tate). The subtle dierence here is the use of the local condition property to nd the set of primes A and one global cohomology class h, rather than working individually with each local condition. Theorem 5.2. For any ɛ 1, there exists a deformation ρ : G GL 2 (Z p ) ramied at innitely many primes, such that the set R = {l a l 2 ɛ} is of density one. l Proof. We assume that S is large enough so that X 1 S (X) and X2 S (X) are trivial. We now proceed to work inductively and construct ρ = lim n ρ n as desired. n = 2 : Let S 2 = S. As X 2 S (X) = 0 and there are no local obstructions, we can lift ρ to GL 2 (Z/p 2 ), so we let ρ 2 : G S GL 2 (Z/p 2 ) be any deformation of ρ. We choose any deformation of ρ Gv to GL 2 (Z p ) for v S 2 and call it ρ S2. Let c be an arbitrary positive constant and let R 2 = {l ( c 2 p2 ) 2 < l < ( c 2 p3 ) 2 } be a nite set of primes not in S 2 for which we choose the characteristic polynomials such that a 2 l 4 c 2 l < 0 consistent with those of ρ. As the a l 's are determined mod p 2

16 16 AFTAB PANDE and ( c 2 p2 ) 2 < l p 2 < 4 c 2 l we see that it is possible to choose the characteristic polynomials such that a 2 l 4 c 2 l < 0. By Lemma 5.1, we can nd a set Q 2 for which there is a unique f 2 H 1 (G S2 Q 2, X), such that (I + pf 2 )ρ 2 Gv ρ S2 Gv mod p 2. As f 2 Gr H 1 nr(g r, X) for r R 2, we can see that the characteristic polynomials of this new representation remain the same for the primes in R 2. By adding the set Q 2 we may have some new obstructions. To overcome the obstructions we use Proposition 3.5 to add a set A 2 to the ramication set and choose h 2 H 1 (G S2 Q 2 R 2 A 2 ) such that (I + p(f 2 + h 2 ))ρ 2 is unobstructed at the new primes, and still has the same characteristic polynomials for R 2. n = 3 : Since adding primes to the set of ramication doesn't add new global obstructions (X 2 S 2 Q 2 = 0), we can lift (I + p(f 2 + h 2 ))ρ 2 to a representation ρ 3 : G S3 GL 2 (Z p ), where S 3 = S 2 Q 2 A 2 and let R 3 = {l / S 3 ( c 2 p3 ) 2 < l < ( c 2 p4 ) 2 } R 2. For the set S 3 \S 2, choose a deformation ρ S3\S2 to GL 2 (Z p ), and also choose characteristic polynomials for the set R 3 \R 2 such that a 2 l 4 c 2 l < 0. We can choose such characteristic polynomials as the primes l's are suitably large. As in the case n=2, we can add two sets of primes Q 3, A 3 to the ramication set and get f 3 H 1 (G S3 Q 3, X) and h 3 H 1 (G S3 Q 3 R 3 A 3, X), such that (I + p 2 (f 3 + h 3 ))ρ 3 ρ mod p 3, when restricted to the primes v S 3. Also, (I +p 2 (f 3 +h 3 ))ρ 3 is unobstructed at the new primes we have added and has the same characteristic polynomials at primes in R 3. n = 4 Since there are no obstructions to lifting, we lift (I + p 2 (f 3 + h 3 ))ρ 3 to ρ 4, and continue the same argument by choosing the sets R n, S n appropriately. By this induction process we get representations ρ n : G Sn GL 2 (Z/p n ), which are unramied away from S n, and have characteristic polynomials of Frobenius as chosen at all but nitely many of the unramied primes. Let ρ = lim n ρ n, and R = lim n R n. By Theorem 1 of Khare-Rajan [7] we know that the density of ramied primes of a continuous semisimple representation is of density zero. As our ρ is irreducible, we see that the set R of unramied primes is of density one. The Sato- Tate formula tells us that a l 2 lies in the interval [ 1, 1], and they follow a particular l

17 DEFORMATIONS OF GALOIS REPRESENTATIONS AND THE THEOREMS OF SATO-TATE AND LANG-TROTTER 17 distribution. By choosing c > 1, we are shrinking the distribution between [ 1 c, 1 c ]. Now, let ɛ = 1 c. As we choose our c to be bigger (or ɛ to be smaller) the set of unramied primes R n also gets bigger but stays nite, which allows us to use the methods above to get the set R = lim n R n. Hence we get a probability distribution for the α l 's which is contrary to the Sato-Tate conjecture. The next theorem concerns the Lang-Trotter conjecture and the proof is similar to the theorem above and the main dierence is our choice of the set R n. Theorem 5.3. If D Z is xed, we can nd a deformation ρ : G GL 2 (Z p ), such that the set R(x) = {l < x a l = D} O( x logx ). Proof. We follow the same inductive process as in the previous theorem. We dene R n = {l < x n a l D mod p n }, where we choose x n to be large enough such that #R n > x1 2/n n log(x. As each R n) n is a Cebotarev set, we can always nd a large enough x n. At the (n + 1)th stage, the x n+1 will be substantially bigger than x n, but these sets have positive density so we can keep choosing the x n 's to be large enough. When we take limits R = {l a l = D} = lim n R n is a set of zero density but #R(x) = O( x1 ɛ logx ), where 0 < ɛ < 1. This ρ gives us an asymptotic formula contrary to the Lang-Trotter conjecture. By the above theorem, it follows almost immediately that we can get distributions in contrast to the bounds of Serre-Elkies and Serre as mentioned in the introduction. For the Serre-Elkies bound for supersingular primes(a l = 0) we let ɛ < 1/4 and for the bound on anomalous primes (a l = 1), we only have to choose the cardinality of the set R n > x n log(x n) 1+1/2n 6. Concluding remarks It would be interesting to know if it is possible to manipulate the deformations so that we can get any distribution we want for the a l 's. The main problem we encountered in using the above methods was trying to control innitely many values for the a l 's with only nitely many cocycles at each stage.

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