Cohomology and Representations of Galois Groups

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1 Cohomology and Representations of Galois Groups Eric Stubley Discussed with Professor Frank Calegari February 28, 2017 Deformations of Galois representations saw their introduction in the late 1980s, and soon became a crucial component of many deep results including Wiles proof of Fermat s Last Theorem. This topic proposal is an exposition of both the deformation theory of Galois representations and Galois cohomology, one of the main computational tools making the study of deformations feasible. The first section is an introduction to the results of Galois cohomology; the second section uses these results to prove two classical theorems. Deformations of Galois representations are introduced in the third section. The fourth section is an exposition of some applications of deformations of Galois representations. 1 Galois Cohomology: Definitions and Results 1.1 Group Cohomology Throughout this section, let G = Gal(L/K) for a Galois extension of fields L/K. Definition 1.1. A G-module is an abelian group A, equipped with a continuous action of G where A is given the discrete topology. Example 1.2. Two examples of G-modules we will see occur throughout are A = Z/nZ (equipped with trivial action of G) and A = µ n (L), the group of n-th roots of unity contained in L equipped with their natural action of G. This section focuses on the group cohomology of G-modules. The zeroth cohomology of A is defined as the fixed points under the action of G, H 0 (G, A) = A G. Since the fixed point functor is left-exact, we can define the higher cohomology groups H i (G, A) as the right derived functors of the fixed point functor; this perspective is useful in that if we have a short exact sequence 0 A B C 0 of G-modules, there is a connecting homomorphism δ such that taking fixed points results in a long exact sequence 0 A G B G C G δ H 1 (G, A) H 1 (G, B) H 1 δ (G, C) H 2 (G, A).... However, we eschew this derived functor perspective in favour of a more concrete description of H 1. Definition 1.3. A 1-cocycle for A is a continuous map ϕ : G A such that ϕ(gh) = ϕ(g) + gϕ(h) for all g, h G. A 1-coboundary for A is a 1-cocycle such that there exists an a A with ϕ(g) = g(a) a for all g G. 1-cocycles and 1-coboundaries both form groups with the operation induced from that of A. 1

2 Although these definitions may seem mysterious at the moment, we will see 1-cocycles arise naturally in our study of deformations of Galois representations. The first cohomology of A can be defined as H 1 (G, A) = {1-cocycles for A} {1-coboundaries for A}. Remark 1.4. If the action of G on A is trivial, a 1-cocycle is simply a homomorphism, and a 1-coboundary is just the zero homomorphism. Thus in this case we have that H 1 (G, A) = Hom(G, A). There is a similarly concrete definition of the higher cohomology with cocycles and coboundaries, but we won t use this construction. We end this discussion of group cohomology by mentioning a fundamental result specific to the situation of G being a Galois group. Theorem 1.5 (Hilbert s Theorem 90). Let L/K be a (possibly infinite) Galois extension of fields. Then H 1 (Gal(L/K), L ) and H 1 (Gal(L/K), L) are both equal to 0. Proof. This is proposition 1 of 1.2 of [4]. 1.2 Local Results We collect here results pertaining to the situation when K is a finite extension of Q p, and A a finite G K - module. Denote by k the residue field of K, hence G k = Ẑ is the quotient of G K by the inertia group I K. Definition 1.6. Say that A is unramified if I K acts trivially. If A is unramified, we may consider it as a G k -module. Since A I K is always unramified, we define the unramified cohomology H i ur(g K, A) = H i (G k, A I K ). Note that H 0 ur(g K, A) = H 0 (G K, A), H 1 ur(g K, A) is a subgroup of H 1 (G K, A) by the inflation-restriction sequence, and H i ur(g K, A) = 0 for i 2. Definition 1.7. The dual module of A is A = Hom(A, µ), where µ is the group of roots of unity in K. The action of G K on A is (gφ)(a) = gφ(g 1 a) for g G K, φ A, and a A. Example 1.8. The dual module of Z/nZ is µ n. Note that if µ n K, the action of G K on µ n is trivial, so µ n is isomorphic (albeit non-canonically) to Z/nZ. The relationship between the cohomology of a G K -module and its dual is captured by the following result. Theorem 1.9 (Local Tate Duality). Let A be a finite G K -module, and let A be the dual module. (1) The groups H i (G K, A) are finite for all i 0, H i (G K, A) = 0 for i 3. (2) For i = 0, 1, 2, the evaluation pairing A A µ induces a perfect pairing (the cup product) H i (G K, A) H 2 i (G K, A ) H 2 (G K, µ) = Q/Z. (3) If #A is coprime to p, the residue characteristic of K, then the unramified subgroups H i ur(g K, A) and H i ur(g K, A ) are each other s annihilators in the cup product pairing. Proof. This statement combines Theorem 2 and Proposition 19 of II.5 of [4]. Knowing that the cohomology groups H i (G K, A) are finite, and in fact 0 for i 3, we can define the Euler characteristic as an alternating product of sizes and know that it will be finite. This next result shows that this Euler characteristic is independent of the action of G K on A, depending only on #A. 2

3 Theorem 1.10 (Euler Characteristic Formula). Proof. Theorem 5 of II.5.7 of [4]. #H 0 (G K, A)#H 2 (G K, A) #H 1 (G K, A) = p vp(#a)[k:qp]. Remark We will most often be interested in calculating the size of first cohomology groups. Using that H 2 (G K, A) is the dual of H 0 (G K, A ) under the cup product pairing, we deduce the formula #H 1 (G K, A) = #H 0 (G K, A)#H 0 (G K, A )p vp(#a)[k:qp] which will prove quite effective, as calculating the H 0 terms is usually feasible. Note also that if A is an F l vector space the Euler characteristic formula can be rephrased in terms of the dimensions h i (G K, A) = dim Fl (H i (G K, A)), yielding the more familiar Euler characteristic { h 0 (G K, A) h 1 (G K, A) + h 2 v p (#A)[K : Q p ] l = p (G K, A) = 0 l p. 1.3 Global Results Let F be a number field, and denote by Σ F the set of all places of F. The first result we mention is a global analogue of the Euler characteristic formula of the previous section. Theorem Let A be a finite G F,S -module, where S is a finite set of places containing all the places dividing #A. Denote by S S the infinite places belonging to S. Then the cohomology groups H i (G F,S, A) are finite, and Proof. Theorem 5.1 of [3]. #H 0 (G F,S, A)#H 2 (G F,S, A) #H 1 (G F,S, A) = (#A) [F :Q] v S #H 0 (G Fv, A). For each place v of F, the inclusion G Fv G F induces a restriction map res v : H i (G F, A) H i (G Fv, A). In this way we can demand local properties of our global cohomology classes; the framework in which we do this is that of Selmer groups. Definition A set of local conditions L = {L v : v Σ F } is a collection of subgroups L v H 1 (G Fv, A) where L v = H 1 ur(g Fv, A) for almost all v. Associated to a set of local conditions L we define the L-Selmer group HL(G 1 F, A) = {x H 1 (G F, A) : res v (x) L v for all v} ( = ker H 1 (G F, A) ) H 1 (G Fv, A). L v v Any set of local conditions has a dual set of conditions, defined by L = {L v : v Σ F }, where L v is the annihilator of L v under the cup product pairing. For any finite S of places containing all of the infinite places, we have a Selmer group HS 1(G F, A), where the local condition at v is { H 1 (G Fv, A) v S L v = Hur(G 1 Fv, A) v S. The following proposition relates these Selmer groups to the cohomology of G F,S -modules. 3

4 Proposition For any finite set of places S and finite G F,S -module A, we have that Proof. See 8 of [6]. H 1 S(G F, A) = H 1 (G F,S, A). Remark This result in fact proves that all Selmer groups are finite. If L and L are two sets of local conditions with L v L v for every place, then we have an exact sequence 0 H 1 L(G F, A) H 1 L (G F, A) v L v/l v. By this result every Selmer group embeds in one of the groups H 1 S (G F, A), which are finite by the previous theorem and proposition. We conclude our discussion of cohomology with another formula for computing orders of cohomology groups, this time for Selmer groups. Theorem 1.16 (Greenberg-Wiles Selmer Group Formula). Let F be a number field, and A a finite group with a continuous action of G F. Suppose that L = {L v } is a set of local conditions for G F and A, and let L be the dual set of conditions. Then #HL 1(G F, A) #HL 1 (G F, A ) = #H0 (G F, A) #L v #H 0 (G F, A ) #H 0 (G v Fv, A). Proof. See 8 of [6] for the proof in the case of F = Q. 2 Two Classical Results In this section we use the tools of Galois cohomology to prove two classical results. For each, we start with a single (surjective) map, and deduce our results from the long exact sequence in cohomology associated to the short exact sequence of the map. 2.1 Kummer Theory Let F be a perfect field, or a field of characteristic prime to n. The n-th power map F F is surjective, hence we have a short exact sequence Taking the long exact sequence in cohomology yields 1 µ n F F 1. 1 µ n (F ) F F δ H 1 (G F, µ n ) H 1 (G F, F ).... We note that the connecting homomorphism δ has an explicit description: given x F, choose any n-th root y of x in F ; δ(x) is the class of the 1-cocycle g g(y) y µ n. This is well-defined, as choosing different n-th roots of x yields 1-cocycles which differ by 1-coboundaries for µ n. Truncating the sequence at H 1 (G F, µ n ) and noting that H 1 (G F, F ) = 0 by Hilbert s Theorem 90, we deduce that δ furnishes an isomorphism δ : F /(F ) n = H 1 (G F, µ n ). If we assume that µ n F, we have that µ n Z/nZ as G F -modules, and hence H 1 (G F, µ n ) = Hom(G F, Z/nZ). The following theorem is now a straightforward consequence of the above isomorphism and the description of the map δ. Theorem 2.1 (Kummer). Let F be any field containing the group µ n of n-th power roots of unity. Then there is a bijection between subgroups of F /(F ) n and abelian extensions of F of exponent dividing n. Further, given a subgroup of F /(F ) n, the extension corresponding to it is F ( 1/n ). 4

5 2.2 Weak Mordell-Weil Theorem Let E be an elliptic curve defined over a number field F. In this section we will prove the weak Mordell-Weil Theorem, which states that the group E(F )/ne(f ) is finite for n 2; from this and a descent argument using height functions one can conclude that the F -rational points E(F ) are a finitely generated abelian group. The multiplication by n map is a surjective morphism E(F ) E(F ), commuting with the action of G F. Thus we have a short exact sequence of G F -modules 0 E[n] E(F ) E(F ) 0. Taking the long exact sequence and truncating at the first H 1 term produces 0 E(F )/ne(f ) H 1 (G F, E[n]) H 1 (G F, E(F ))[n] 0. If we knew that the central term H 1 (G F, E[n]) was finite, we could immediately conclude our desired result. Unfortunately this is not the case in general. What we will do instead is show that the group E(F )/ne(f ) lands inside a certain Selmer group H 1 L (G F, E[n]) H 1 (G F, E[n]), which gives us finiteness as Selmer groups are finite. For each place v Σ F, we have (by the same construction as above) a short exact sequence of G Fv - modules 0 E(F v )/ne(f v ) H 1 (G Fv, E[n]) H 1 (G Fv, E(F v ))[n] 0 and these sequences are compatible with the restriction maps induced from G Fv conditions L v by G F. Define the local L v = E(F v )/ne(f v ). For finite places v where E has good reduction and the residue characteristic of v is coprime with n, we do in fact have that L v = H 1 ur(g Fv, E[n]); so these L v are actually a system of local conditions. A simple diagram chase yields that the image of the connecting homomorphism E(F )/ne(f ) H 1 (G F, E[n]) lands in the Selmer group H 1 L (G F, E[n]). Since this embeds E(F )/ne(f ) into a finite group, we have proven the following theorem. Theorem 2.2 (Weak Mordell-Weil Theorem). Let E be an elliptic curve defined over a number field F, and n 2 an integer. Then the group E(F )/ne(f ) is finite. 3 Deformations of Galois Representations We follow the presentation of this material as given in [1]. For simplicity we will only discuss residual representations with values in F p, and our coefficient algebras are taken solely to be Z p -algebras. 3.1 Deformations We work throughout with a residual representation ρ : G GL n (F p ), where G is either G K for K a local field or G F,S for F a number field and S a finite set of places of F. Definition 3.1. A coefficient algebra A is a complete noetherian local Z p -algebra with residue field A/m = F p. A deformation of ρ to a coefficient algebra A is an equivalence class of liftings, where a lifting is a continuous homomorphisms ρ : G GL n (A) such that the following diagram commutes G ρ ρ GL n (A) GL n (F p ). Two liftings are equivalent if they are conjugate by a matrix in GL n (A) reducing to the identity mod m. 5

6 Example 3.2. Deformations of Galois representation arise naturally, for example in the case of elliptic curves: the p-power torsion E[p n ] and Tate module T p (E) can be thought of as Z/p n Z and Z p valued deformations of the p-torsion E[p]. A main result that makes the study of deformations feasible is that the deformation functor, sending a coefficient algebra A to the A-valued deformations of ρ, is usually representable. Given the types of groups we re allowing, the only condition that we need to guarantee representability is that End(ρ) = F p, where End(ρ) = {M M n (F p ) : ρ(g)mρ(g) 1 = M for all g G} = H 0 (G, Ad(ρ)). Theorem 3.3 (Mazur, Ramakrishna). If End(ρ) = F p, there exists a universal deformation ring R which represents the deformations of ρ. More precisely, there is a deformation ρ univ : G GL n (R) such that for any other deformation ρ : G GL n (A) there exists a unique homomorphism f ρ : R A such that ρ = f ρ ρ univ. Proof. Theorem 3.3, Lecture 3 of [1]. Example 3.4. One example where the deformation ring can be completely determined is when ρ is a character G F p. Since every character has a canonical lift to Z p using the Teichmüller lifts, it suffices to look at deformations of the trivial character. A deformation of the trivial character to a coefficient algebra A is equivalently a homomorphism G 1 + m. Since 1 + m is an abelian pro-p group, any such homomorphism factors through G ab,(p), the pro-p completion of the abelianization of G. The universal deformation ring is thus the completed group ring Z p [[G ab,(p) ]]. The universal deformation is the natural map G G ab,(p) Z p [[G ab,(p) ]] multiplied by the Teichmüller lift of the original character. 3.2 Relationship with Cohomology There are two connections that we give here between the cohomology of Ad(ρ) and deformations of ρ. First we see that infinitesimal deformations are parametrized by the first cohomology of Ad(ρ). Proposition 3.5. There is a bijection between deformations of ρ to the algebra F p [ɛ] and the group H 1 (G, Ad(ρ). Proof. See 4 of [1]. Remark 3.6. Every deformation to F p [ɛ] is equivalently a map R F p [ɛ]. These maps are in turn equivalently maps m/(m 2, p) F p. This justifies calling H 1 (G, Ad(ρ)) the Zariski cotangent space of R. The ring R is actually constructed as a quotient of a Z p power series ring in h 1 (G, Ad(ρ)) = dim Fp (H 1 (G, Ad(ρ)) many variables. Next we see that we can use the second cohomology of Ad(ρ) to measure obstructions to lifting a representation. Proposition 3.7. For each coefficient algebra A with maximal ideal m satisfying pm = 0, there is a welldefined class in H 2 (G, Ad(ρ) m) that is 0 iff there exists a deformation a of ρ to A. Proof. See 4 of [1]. The first and second cohomology groups of Ad(ρ) in fact provide a great deal of information about the structure of R, as seen in the following theorem. Theorem 3.8. Suppose that End(ρ) = F p. Then if R is the universal deformation ring of ρ, we have that Krull dim(r/pr) h 1 (G, Ad(ρ)) h 2 (G, Ad(ρ)). Further, if h 2 (G, Ad(ρ)) = 0, then the above inequality is an equality and we have that R is isomorphic to a power series ring in h 1 (G, Ad(ρ)) many free variables over Z p. Proof. Theorem 4.2 of [1]. 6

7 Example 3.9. We can use this theorem to determine the Krull dimensions of deformation rings for 2- dimensional representations in many situations. Note that dim Fp (Ad(ρ)) = 4 for a 2-dimensional residual representation ρ. Using either the local or global Euler characteristic formula, and under the assumptions that p 2, End(ρ) = F p, and S p,, we can determine that Group Euler characteristic formula for h 1 h 2 Krull dimension bound G Ql, l p h 0 (G Ql, Ad(ρ)) 1 G Qp h 0 (G Qp, Ad(ρ)) + dim Fp (Ad(ρ)) 5 G Q,S, ρ even h 0 (G Q,S, Ad(ρ)) + dim Fp (Ad(ρ)) h 0 (G R, Ad(ρ)) 1 G Q,S, ρ odd h 0 (G Q,S, Ad(ρ)) + dim Fp (Ad(ρ)) h 0 (G R, Ad(ρ)) 3 It is natural to want to study only those deformations which satisfy a certain condition, for example we might want to study deformations having a fixed determinant χ : G Z p, or we might want only those deformations of G Q,S that are unramified at a certain prime p S. If the condition satisfies a certain set of axioms, then we are guaranteed that this restricted deformation problem is also representable. In lieu of giving the technical details of this, we simply discuss the relationship of deformation conditions and cotangent spaces. Suppose first that we are studying representations of a local Galois group. Given a ρ, there is a ring R representing all deformations of ρ, and if we have a deformation condition D there is a ring R D representing those deformations which satisfy D. In this case we have a natural surjective map R R D, which induces an injective map on cotangent spaces Hom(R D, F p [ɛ]) Hom(R, F p [ɛ]). Identifying this group with H 1 (G, Ad(ρ)), we get a subgroup HD 1 (G, Ad(ρ)) which can be identified with the cotangent space of R D. Suppose now that we are studying a representation of a global Galois group G F,S. Given a local condition D v at each place v S, say that a deformation is of type D if its restriction to each place v S satisfies the condition D v. The cotangent space for type D deformations is then described by the following proposition. Proposition In the above notation, the cotangent space of the universal deformation ring for deformation of type D is a Selmer group defined by the local conditions { HD 1 L v = v (G Fv, Ad(ρ)) v S Hur(G 1 Fv, Ad(ρ)) v S. Proof. Theorem 6.8 of [1]. 4 Applications of Deformations of Galois Representations This section comprises three applications of deformations of Galois representations. First, we give an exposition of a proof of the Kronecker-Weber theorem using deformations of characters of G Q. This is followed by a similar computation in the case of a real quadratic field. Last, we prove a general theorem about controlling ramification in deformations and use it to deduce some basic information about congruences of modular forms. 4.1 Kronecker-Weber by Deformations The Kronecker-Weber theorem states that every finite abelian extension of Q is contained in a cyclotomic extension. Thus every character with finite image should be a Dirichlet character, i.e. a character of the group (Z/nZ) = Gal(Q(ζ n )/Q). This section is an exposition of how the Kronecker-Weber theorem can be proven using a modularity lifting argument similar in nature to that of Wiles proof of modularity for semi-stable elliptic curves over Q; the main argument was extracted from notes available at [2]. The idea is to prove that all deformations of the trivial character satisfying a certain condition (S, j) are in fact Dirichlet characters. For a finite set of primes S and integer j 0 say that a deformation ρ : G Q A of the trivial character is of type (S, j) if 7

8 (1) ρ is unramified outside of S (so ρ is a map G Q,S A ) (2) ρ GQp factors through Gal(Q ur p (µ p j )/Q p ). Deformations of the trivial character of type (S, j) are represented by a ring R (S,j). Those deformations which are in addition Dirichlet characters (i.e. those which factor through the Galois group of a cyclotomic extension) are represented by a ring T (S,j), and there is a natural surjective map R (S,j) T (S,j). We want to prove that this map is an isomorphism for each (S, j), using the following isomorphism criterion. Theorem 4.1. Let φ R(S,j) = I/I 2, where I is the kernel of the map R (S,j) Z p corresponding to the trivial character; φ T(S,j) is defined analogously. Then #φ R(S,j) #φ T(S,j). Furthermore, if there is equality and #φ T(S,j) < then the map R (S,j) T (S,j) is an isomorphism. Proof. This is a simplified version of Criterion I from [5]. Remark 4.2. Under a suitable interpretation of these deformation rings as geometric objects, this theorem is stating that a closed subspace Y X whose tangent space is the same dimension as that of X must be equal to X. The first step of the proof is to reduce the computation of these relative cotangent spaces to a computation in Galois cohomology. For any finitely generated Z p module M we have that #M = lim k Hom Z p (M, Z/p k Z). In particular #φ R(S,j) can be computed if we know the #Hom Zp (φ R(S,j), Z/p k Z) for all k. It is this computation that we reduce to Galois cohomology. Consider the set of local conditions given by H 1 (Gal(Q ur p (µ p j )/Q p ), Z/p k Z) q = p L q = H 1 (G Qq, Z/p k Z) q p, q S Hur(G 1 Qq, Z/p k Z) q p, q S. Then we have that Hom Zp (φ R(S,j), Z/p k Z) = H 1 (S,j) (G Q, Z/p k Z) where the H(S,j) 1 is the Selmer group for the above local conditions. We now prove by induction that #φ R(S,j) #φ T(S,j) for all (S, j). The base case is an application of Minkowski s theorem: if S =, we get that R (S,j) = Z p as there are no deformations unramified everywhere. Hence φ R(S,j) = 0. There are two cases for the induction. Case 1: (S, j ) = (S {q}, j). The increase in local condition for S = S {q} yields an exact sequence 0 H 1 (S,j) (G Q, Z/p k Z) H 1 (S,j ) (G Q, Z/p k Z) H1 (G Qq, Z/p k Z) H 1 ur(g Qq, Z/p k Z). We calculate the third term, and hence an upper bound on the increase in size of Selmer groups, using the Euler characteristic formula. We have that Thus when we take limits as k we get that #H 1 (G Qq, Z/p k Z) #H 1 ur(g Qq, Z/p k Z) = #H0 (G Qq, Z/p k Z)#H 0 (G Qq, µ pk) #H 0 (G Fq, Z/p k Z) = #H 0 (G Qq, µ p k) = gcd(p k, q 1) #φ R(S,j ) #φ R(S,j) p vp(q 1). 8

9 Case 2: (S, j ) = (S, j + 1). Increasing the local condition here gives an exact sequence 0 H 1 (S,j) (G Q, Z/p k Z) H 1 (S,j ) (G Q, Z/p k Z) H 1 (Gal(Q ur p (µ p j+1)/q ur p (µ p j )), Z/p k Z). In this case we get our upper bound simply by computing the Galois group in the third term. { Gal(Q ur p (µ p j+1)/q ur (Z/pZ) j = 0 p (µ p j )) = Z/pZ j 1. Taking Hom to Z/p k Z and then a limit as k in the size computation we get { #φ R(S,j+1) 0 j = 0 #φ R(S,j) p j 1. The structure of the deformation rings T (S,j) can be completely determined by S, j. In particular, we can compute that #φ T(S p vp(q 1) S = S {q}, j = j,j ) = 0 S #φ T(S,j) = S, j = 0, j = 1 p S = S, j = j + 1, j 1. From this we conclude that #φ R(S,j ) #φ R(S,j) #φ T (S,j ) #φ T(S,j) which immediately gives our induction step after a simple rearrangement. Note that once we ve completed the induction proof, it actually implies that all of the above inequalities are in fact equalities. 4.2 Real Quadratic Field Example Let s examine what happens for deformations of the trivial character over a real quadratic field instead of Q. Let F = Q( 229). We ll study the behaviour of the deformation ring for characters G F F 3 as we move from unramified characters to characters having ramification at only one finite prime p. We need the following structural information about F. - F has class group Z/3Z; a generator is a, a prime of F dividing 3. - The cube of a (which is principal) can be generated by η = A fundamental unit for F is ɛ = Let R be the deformation ring for deformations of the trivial character G F F 3 which are unramified at all (finite) places. The tangent space to this deformation ring is the Selmer group H 1(G F, F 3 ), where the local conditions are unramified at all finite places and H 1 (G R, F 3 ) at each of the two infinite places. The Greenberg-Wiles Selmer group formula yields #H 1 (G F, F 3 ) #H 1 (G F, µ 3 ) = 1 3. Since H 1 (G F, F 3 ) = Hom(Cl F, F 3 ) is 1-dimensional as an F 3 vector space, we conclude that the dual group must be 2-dimensional as an F 3 vector space. If we relax the local conditions to L p = H 1 (G Fp, F 3 ) for a single prime p of norm congruent to 1 mod 3, the Selmer group formula now gives us 1. Thus the dimensions of H 1 p and its dual are equal, but we don t know whether they are both 1-dimensional or 2-dimensional. However, we will be able to determine which of these two cases holds depending on the behaviour of ɛ and η mod p. 9

10 Since the deformation ring for this relaxed local condition is R p = Z 3 [RCl F (p) 3 ], we just need to compute this ray class group (or rather, its 3-primary part). We have an exact sequence O F (O F /p) RCl F (p) Cl F 0. Clearly if the norm of p is not 1 mod 3, the 3-primary part of the ray class group won t increase. Similarly we must require that ɛ is a cube mod p, as otherwise the image of the global units will generate the 3-primary part of (O F /p). Under these conditions we see that RCl F (p) 3 is one of: Z/3 k+1 Z or Z/3Z Z/3 k Z (where 3 k is the order of the 3-primary part of (O F /p) /(ɛ)). If the ray class group is to be cyclic, a lift of a generator of Cl F should generate the whole thing; in particular its cube should generate the subgroup coming from (O F /p). The element η is the cube of such a lift; thus we get that RCl F (p) 3 is cyclic iff η is not a cube mod p. For a prime p of norm congruent to 1 mod 3, the following table displays the relationship between values of ɛ and η mod p, the structure of RCl F (p) 3, and the dimensions of the Selmer groups. ɛ mod p η mod p RCl F (p) 3 dim(hp) 1 dim(hp 1 ) Not a cube Not a cube = ClF 1 1 Not a cube Cube = ClF 1 1 Cube Not a cube Cyclic 1 1 Cube Cube Not cyclic Controlling Ramification in Deformations Theorem 4.3. Let ρ : G Q,S GL n (F p ) be a representation, unramified at a prime q S. Suppose that ρ(frob q ) is diagonalizable with eigenvalues λ i, and that λ i λ 1 j q, q 1 mod p for all i, j. Then all deformations of ρ are unramified at q. Proof. First off, we note that any deformation ρ : G Q,S GL n (A) is tamely ramified: the image of the inertia group at q lands in the kernel of the reduction map GL n (A) GL n (A/m), which is pro-p. Thus if we choose σ, a lift of Frobenius, and generator τ of the tame inertia group, the image of G Qq under ρ is topologically generated by ρ(σ) and ρ(τ), where ρ(τ) 1 + M n (m). These elements must satisfy the relation ρ(σ)ρ(τ)ρ(σ) 1 = ρ(τ) q. We prove that the only such ρ(τ) is 1, by proving this inductively in the rings A/m r. The base case is r = 1, where we know ρ(τ) = 1. Suppose we know this for r, we ll try to prove it for r + 1. Write ρ(τ) = 1 + T, where T M n (m r /m r+1 ). The tame equation that ρ(τ) satisfies implies that ρ(σ)t ρ(σ) 1 = qt in other words (since Ad(ρ) = Ad(ρ)(1) by the trace pairing) T H 0 (G Qq, Ad(ρ)(1) (m r /m r+1 )). However, this group is trivial: the action of Frob q in Ad(ρ)(1) is diagonal, with eigenvalues λ i λ 1 j q on the i, j coordinate. Since none of these are 1 by assumption, there are no fixed elements other than 0, so T = 0 when considered mod m r+1. Since A is the inverse limit of the A/m r, and deformations to each of the A/m r must be unramified at q, we conclude that ρ itself is unramified at q. Example 4.4. We can use this theorem to say something interesting about congruences of modular forms. Let f be the unique cusp form of weight 11 and level 2: f(z) = q (1 q n ) 2 (1 q 11n ) 2 n=1 = q 2q 2 q 3 + 2q 4 + q 5 + 2q Finding forms f of level 11q congruent to f mod 3 produces deformations of ρ f,3 that are ramified at q; thus we can prove that no such form exists whenever ρ f,3 (Frob q ) satisfies the conditions of the theorem. We know that the characteristic polynomial of ρ f,3 (Frob q ) (for q 3, 11) is x 2 a q x + q. A simple calculation shows that the condition a q ±(q + 1) implies that the eigenvalue conditions of the theorem are satisfied. Since a 5 = 1 and mod 3, we conclude that there are no newforms f of level 55 and weight 2 congruent to f mod 3. (Here the eigenvalues are defined only over F 9, but we can apply the theorem if we use this as our base field instead of F 3.) 10

11 References [1] Fernando Q. Gouvêa. Deformations of Galois representations. In: Arithmetic Algebraic Geometry (Park City, UT, 1999). Ed. by Brian Conrad and Karl Rubin. Vol. 9. IAS/Park City Math. Ser. Amer. Math. Soc., Providence, RI, 2001, pp [2] Emmanuel Kowalski. Kronecker-Weber by Deformation, or: another bad reference. July 14, url: weber- by- deformation- or- anotherbad-reference/ (visited on 02/23/2017). [3] J.S. Milne. Arithmetic Duality Theorems. Second Edition. BookSurge, LLC, 2006, pp. viii+339. [4] Jean-Pierre Serre. Galois cohomology. English Edition. Springer Monographs in Mathematics. Translated from the French by Patrick Ion and revised by the author. Springer-Verlag, Berlin, 2002, pp. x+210. [5] Bart de Smit, Karl Rubin, and René Schoof. Criteria for Complete Intersections. In: Modular Forms and Fermat s Last Theorem. Ed. by Gary Cornell, Joseph H. Silverman, and Glenn Stevens. Springer- Verlag, Chap. XI. [6] Lawrence C. Washington. Galois Cohomology. In: Modular Forms and Fermat s Last Theorem. Ed. by Gary Cornell, Joseph H. Silverman, and Glenn Stevens. Springer-Verlag, Chap. IV. 11