ON LIFTING AND MODULARITY OF REDUCIBLE RESIDUAL GALOIS REPRESENTATIONS OVER IMAGINARY QUADRATIC FIELDS

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1 ON LIFTING AND MODULARITY OF REDUCIBLE RESIDUAL GALOIS REPRESENTATIONS OVER IMAGINARY QUADRATIC FIELDS TOBIAS BERGER 1 AND KRZYSZTOF KLOSIN 2 Abstract. In this paper we study deformations of mod p Galois representations (over an imaginary quadratic field F ) of dimension 2 whose semisimplification is the direct sum of two characters 1 and 2. As opposed to [BK13] we do not impose any restrictions on the dimension of the crystalline Selmer group H 1 Σ (F, Hom( 2, 1 )) Ext 1 ( 2, 1 ). We establish that there exists a basis B of H 1 Σ (F, Hom( 2, 1 )) arising from automorphic representations over F (Theorem 8.1). Assuming among other things that the elements of B admit only finitely many crystalline characteristic 0 deformations we prove a modularity lifting theorem asserting that if itself is modular then so is its every crystalline characteristic zero deformation (Theorems 8.2 and 8.4). 1. Introduction Let p be an odd prime. Let F be a number field, Σ a finite set of primes of F (containing all primes p of F lying over p) and G Σ the Galois group of the maximal extension of F unramified outside Σ. Let E be a finite extension of Q p with ring of integers O and residue field F. Let 1, 2 : G Σ GL ni (F) be two absolutely irreducible mutually non-isomorphic representations with n 1 + n 2 = n, which we assume lift uniquely to crystalline representations i : G Σ GL ni (O). The aim of this article is to study deformations of non-semi-simple continuous crystalline representations : G Σ GL n (F) whose semi-simplification is 1 2 in the case n = 2 and F is an imaginary quadratic field. We analyzed this deformation problem in [BK13] under the additional assumption that HΣ 1 (F, Hom( 2, 1 )) is onedimensional (which is equivalent to saying that there exists only one such up to isomorphism). Here HΣ 1 denotes the subgroup of H1 consisting of classes unramified outside Σ and crystalline at all p p. In this paper we do not make any assumption on this dimension. Disposing of the dim=1 assumption is more than a technicality as in the general case one can no longer expect to be able to identify the universal deformation ring with a Hecke algebra. This question was studied by Skinner and Wiles for n = 2 and totally real fields F in the seminal paper [SW99]. In that paper the authors analyze primes q of the (ordinary) universal deformation ring R of and prove that they are pro-modular in the sense that the trace of the deformation corresponding to R R /q occurs in the Hecke algebra T. In particular no direct identification of R and T is made. In this article we take a different approach and work with the reduced universal deformation ring R red and its ideal of reducibility. In the dim=1 -case, the authors Date: 2 April Mathematics Subject Classification. 11F80, 11F55. Key words and phrases. Galois deformations, automorphic forms. 1

2 2 TOBIAS BERGER 1 AND KRZYSZTOF KLOSIN 2 proved (as a consequence of an R = T -theorem - Theorem 9.14 in [BK13]) that R red is a finitely generated Z p -module. In contrast, if dim HΣ 1 (F, Hom( 2, 1 )) > 1, while there are only finitely many automorphic representations whose associated Galois representations are deformations of, the ring R red may potentially be infinite over Z p (Remark 2.17). This is a direct consequence of the existence of linearly independent cohomology classes inside the Selmer group which can be used to construct non-trivial lifts to GL 2 (F[[X]]). The resulting (potentially large) characteristic p components of R red do not arise from automorphic representations and in this paper we will ignore them by considering the image of R red we will denote by R 0 ) instead of R red inside R red Zp Q p (which itself. It is however possible that by doing so we are excluding some characteristic p deformations whose traces may be modular in the sense of [CMar] (i.e. arise from torsion Betti cohomology classes). On the other hand, as opposed to the situation studied in [SW99], over an imaginary quadratic field there are no reducible deformations to characteristic zero which in turn is a consequence of the finiteness of the Bloch-Kato Selmer group HΣ 1 (F, Hom( 2, 1 ) Q p /Z p ) (Lemma 2.20), where 1, 2 are (unique) lifts to characteristic zero of 1 and 2 respectively. While each may possess non-modular reducible characteristic p deformations, the situation is complicated further by the fact that in general many s do not admit any modular deformations at all (this phenomenon does not arise in the dim=1 case). Indeed, first note that two extensions in Ext 1 G Σ ( 2, 1 ) define the same representation of G Σ if and only if they are (non-zero) scalar multiples of each other. In particular, if dim F Ext 1 G Σ ( 2, 1 ) = 1, then there is a unique nonsemi-simple representation of G Σ with semi-simplification 1 2. (Similarly, if dim F HΣ 1 (F, Hom( 2, 1 )) = 1 then there exists a unique crystalline such representation.) However, if dim F Ext 1 G Σ ( 2, 1 ) = m, then there are qm 1 q 1 non-isomorphic such representations where q = #F. This demonstrates that in general not all reducible representations can be modular (of a particular level and weight), as the number of such characteristic zero automorphic forms is fixed (in particular it is independent of making a residue field extension). Nevertheless, we are able to prove (see Corollary 4.8) that there exists an F- basis B of HΣ 1 (F, Hom( 2, 1 )) arising from modular forms. For this we combine a congruence ideal bound for a Hecke algebra with the upper bound on the Selmer group of Hom( 2, 1 ) predicted by the Bloch-Kato conjectures. Let R tr,0 be the image in R 0 of the subalgebra generated by traces of R red for arising from a modular form. As pointed out we can extend the set consisting of to a modular basis B := { 1 =, 2,..., s } of HΣ 1 (F, Hom( 2, 1 )). Our ultimate goal is to show that it is possible to identify R tr,0 with the quotient T of a Hecke algebra T. Here the quotient T corresponds to automorphic forms for which there exists a lattice in the associated Galois representation with respect to which the mod p reduction equals. To prove our main modularity lifting theorem (Theorem 8.2) we work under the following two assumptions. On the one hand we assume that the modular basis B is unique in the sense that any other such consists of scalar multiples of the elements of B. On the other hand we assume that R 0 is a finitely generated Z p - module for all B. The first assumption can be replaced with the assumption that the Bloch-Kato Selmer group HΣ 1 (F, Hom( 2, 1 ) Q p /Z p ) is annihilated by p (Theorem 8.4). This second result is in a sense orthogonal to the main results of

3 ON MODULARITY OF REDUCIBLE RESIDUAL GALOIS REPRESENTATIONS 3 [BK11] and [BK13], where the same Selmer group is assumed to be cyclic, but of arbitrary finite order. Our approach relies on simultaneously considering all the deformation problems for representations i (i = 1, 2,..., s). As in [BK13] we first study reducible deformations via the quotients R tr,0 /I tr,0 i for the reducibility ideal I tr,0 i of the trace i of the universal deformation into GL 2 (R 0 ) as defined by Bellaïche and Chenevier. These ideals are the analogues of Eisenstein ideals J i on the Hecke algebra side. To relate # i Rtr,0 /I tr,0 i to the order of a Bloch-Kato Selmer group we make use i of a lattice construction of Urban (Theorem 1.1 of [Urb01], see Theorem 4.1 in this paper). In fact it is a repeated application of Urban s theorem (on the Hecke side and on the deformation side) that allows us to prove a modularity lifting theorem. We show that when the upper bound on the Selmer group and the lower bound on the congruence ideal agree (which in many cases is a consequence of the Bloch-Kato conjecture), this implies that every reducible deformation which lifts to characteristic zero of every i is modular (cf. section 6). It is here that we make use of the assumption on the uniqueness of B to be able to use a result of Kenneth Kramer and the authors [BKK13] on the distribution of Eisenstein-type congruences among various residual isomorphism classes of Galois representations (cf. Section 5). Yet another application of Urban s Theorem allows us to prove the existence of a deformation to GL 2 (R tr,0 ) and as a consequence to identify R tr,0 with R 0 (Theorem 6.2). Using the fact that the ideal of reducibility of R red and hence also of R 0 is principal (Proposition 7.1) and applying the commutative algebra criterion (Theorem 4.1 in [BK13]) we are finally able to obtain an isomorphism R red = T and thus a modularity lifting theorem (Theorems 8.2 and 8.4). Throughout the paper we work in a slightly greater generality than necessary for the imaginary quadratic case to stress that our results apply in a more general context if one assumes some standard conjectures. However, in section 8 we gather all the assumptions in the imaginary quadratic case as well as the statements of the main theorems (Theorems 8.1, 8.2 and 8.4) in this context. Hence the reader may refer directly to that section for the precise (self-contained) statements of the main results of the paper in that case. 2. Deformation rings Let F be a number field and p > 2 a prime with p # Cl F and p unramified in F/Q. Let Σ be a finite set of finite places of F containing all the places lying over p. Let G Σ denote the Galois group Gal(F Σ /F ), where F Σ is the maximal extension of F unramified outside Σ. For every prime q of F we fix compatible embeddings F F q C and write D q and I q for the corresponding decomposition and inertia subgroups of G F (and also their images in G Σ by a slight abuse of notation). Let E be a (sufficiently large) finite extension of Q p with ring of integers O and residue field F. We fix a choice of a uniformizer ϖ Deformations. Denote the category of local complete Noetherian O-algebras with residue field F by LCN(E). Let m be any positive integer. Suppose r : G Σ GL m (F) is a continuous homomorphism. We recall from [CHT08] p. 35 the definition of a crystalline representation: Let p p and A be a complete Noetherian Z p -algebra. A representation ρ : D p

4 4 TOBIAS BERGER 1 AND KRZYSZTOF KLOSIN 2 GL n (A) is crystalline if for each Artinian quotient A of A, ρ A lies in the essential image of the Fontaine-Lafaille functor G (for its definition see e.g. [BK13] Section 5.2.1). We also call a continuous finite-dimensional G Σ -representation V over Q p (short) crystalline if, for all primes p p, Fil 0 D = D and Fil p 1 D = (0) for the filtered vector space D = (B crys Qp V ) Dp defined by Fontaine (for details see [BK13] Section 5.2.1). A (crystalline) O-deformation of r is a pair consisting of A LCN(E) and a strict equivalence class of continuous representations r : G Σ GL m (A) that are crystalline at the primes dividing p and such that r = r (mod m A ), where m A is the maximal ideal of A. (So, in particular we do not impose on our lifts any conditions at primes in Σ \ Σ p ). Later we assume that if q Σ, then #k q 1 (mod p), which means that all deformations we consider will trivially be Σ-minimal. As is customary we will denote a deformation by a single member of its strict equivalence class. If r has a scalar centralizer then the deformation functor is representable by R r LCN(E) since crystallinity is a deformation condition in the sense of [Maz97]. We denote the universal crystalline O-deformation by ρ r : G Σ GL m (R r ). Then for every A LCN(E) there is a one-to-one correspondence between the set of O-algebra maps R r A and the set of crystalline deformations r : G Σ GL m (A) of r. For j {1, 2} let j : G Σ GL nj (F) be an absolutely irreducible continuous representation. Assume that 1 = 2. Consider the set of isomorphism classes of n-dimensional residual (crystalline at all primes p p) representations of the form: 1 (2.1) = : G Σ GL n (F), 2 which are non-semi-simple (n = n 1 + n 2 ). From now on assume p n!. Lemma 2.1. Every representation of the form (2.1) has scalar centralizer. Proof. This is easy. We write R red for the quotient of R by its nilradical and ρ red for the correspond-. We further generated by the set ing universal deformation, i.e. the composite of ρ with R R red write R tr R red for the closed O-subalgebra of R red S := {tr ρ (Frob q ) q Σ}. Lemma 2.2. Write m for the maximal ideal of R red with respect to its maximal ideal m tr := m R tr. The O-algebra R tr and local. Moreover, R tr is complete /m tr = F. Proof. First note that m tr is a maximal ideal of R tr. Indeed, as a contraction of a prime ideal it is clearly prime, so R tr /m tr is a domain which injects into R red /m. Since the latter is a finite field, R tr /m tr must also be a finite field. Moreover, by Theorem 8.1 in [Mat89] we know that R tr is complete with respect to m tr. Thus by [Eis95], p. 183, R tr must be local. The last assertion follows from the fact that R tr is an O-algebra Pseudo-representations and pseudo-deformations. We next recall the notion of a pseudo-representation (or pseudo-character) and pseudo-deformations (from [BC09] Section and [Böc11] Definition 2.2.2).

5 ON MODULARITY OF REDUCIBLE RESIDUAL GALOIS REPRESENTATIONS 5 Definition 2.3. Let A be a topological ring and R and A-algebra. A (continuous) A-valued pseudo-representation on R of dimension d, for some d N >0, is a continuous function T : R A such that (i) T (1) = d and d! is a non-zero divisor of A; (ii) T is central, i.e. such that T (xy) = T (yx) for all x, y R; (iii) d is minimal such that S d+1 (T )(x) = 0, where, for every integer N 1, S N (T ) : R N A is given by S N (T )(x) = σ S N ɛ(σ)t σ (x), where for a cycle σ = (j 1,... j m ) we define T σ ((x 1,... x d+1 )) = T (x j1 x jm ), and for a general permutation σ with cycle decomposition r i=1 σ i we let T σ (x) = r i=1 T σi (x). In the case when R = A[G Σ ] we will also call the restriction of T to G Σ a pseudorepresentation. We note that if ρ : A[G Σ ] M n (A) is a morphism of A-algebras then tr ρ is a pseudo-representation of dimension n (see [BC09] Section 1.2.2). According to [BC09] Section 1.2.1, if T : R A is a pseudo-representation of dimension d and A an A-algebra, then T A : R A A is again a pseudorepresentation of dimension d. Following [SW99] (see also [Böc11] Section 2.3) we define a pseudo-deformation of tr 1 + tr 2 to be a pair (T, A) consisting of A LCN(E) and a continuous pseudo-representation T : G Σ A such that T = tr 1 + tr 2 (mod m A ), where m A is the maximal ideal of A. By [SW99] Lemma 2.10 (see also [Böc11] Proposition 2.3.1) there exists a universal pseudo-deformation ring R ps LCN(E) and we write T ps : G Σ R ps for the universal pseudo-deformation. For every A LCN(E) there is a one-to-one correspondence between the set of O-algebra maps R ps A (inducing identity on F) and the set of pseudo-deformations T : G Σ A of tr 1 +tr 2. Any deformation of a representation as in (2.1) gives rise (via its trace) to a pseudo-deformation of tr 1 + tr 2, so there exists a unique O-algebra map R ps R such that the trace of the deformation equals the composition of T ps with R ps R. Lemma 2.4. The image of R ps R red is R tr. Proof. This is clear (cf. [CV03] Theorem 3.11) since R ps is topologically generated by T (Frob p ) (and R tr is closed). Corollary 2.5. The ring R tr Proof. As a quotient of R ps the ring R tr Lemma 2.2. is an object in the category LCN(E). is Noetherian. So, the claim follows from 2.3. Selmer groups. For a p-adic G Σ -module M (finitely generated or cofinitely generated over O - for precise definitions cf. [BK13], section 5) we define the Selmer group H 1 Σ (F, M) to be the subgroup of H1 cont(f Σ, M) consisting of cohomology classes which are crystalline at all primes p of F dividing p. Note that we place no restrictions at the primes in Σ that do not lie over p. For more details cf. [loc.cit.]. We are now going to state our assumptions. The role of the first one is to rigidify the problem of deforming the representations j appearing on the diagonal of the

6 6 TOBIAS BERGER 1 AND KRZYSZTOF KLOSIN 2 residual representations. The role of the second is to rule out characteristic zero upper triangular deformations. Assumption 2.6. Assume that R j = O and denote by j the unique lifts of j to GL nj (O). Assumption 2.7 ( Bloch-Kato conjecture ). One has the following bound: for some non-zero L O. #H 1 Σ(F, Hom O ( 2, 1 ) O E/O) #O/L, Remark 2.8. In applications the constant L will be the special L-value at zero of the Galois representation Hom O ( 2, 1 ) divided by an appropriate period. For the remainder of this section we will work under the above two assumptions Ideal of reducibility. Let A be a Noetherian Henselian local (commutative) ring with maximal ideal m A and residue field F and let R be an A-algebra. We recall from [BC09] Proposition the definition of the ideal of reducibility of a (residually multiplicity free) pseudo-representation T : R A of dimension n, for which we assume that T = tr 1 + tr 2 mod m A Definition 2.9 ([BC09] Definition 1.5.2). The ideal of reducibility of T is the smallest ideal I of A such that T mod I is the sum of two pseudo-characters T 1, T 2 with T i = tr i mod m A. We will denote it by I T. Definition We will write I ps R ps for the ideal of reducibility of the universal pseudo-deformation T ps : R ps [G Σ ] R ps, I R for the ideal of reducibility of tr ρ : R [G Σ ] R, I red R red for the ideal of reducibility of tr ρ red : R red [G Σ ] R red and I tr for the ideal of reducibility of tr ρ red : R tr [G Σ ] R tr. Lemma Let I 0 be the smallest closed ideal of R ps containing the set {T ps (Frob v ) tr 1 (Frob v ) tr 2 (Frob v ) v Σ}. Then I 0 equals the ideal of reducibility I ps R ps. Proof. By the Chebotarev density theorem we get T ps = tr 1 +tr 2 (mod I 0 ), hence I 0 I ps. Conversely, we know from the definition of the ideal of reducibility that T ps (mod I ps ) is given by the sum of two pseudo-characters reducing to tr i. By assumption 2.6 and Theorem 7.6 of [BK13] (see also [Böc11] Theorem 2.4.1) these two pseudo-characters must equal tr i mod(i). This shows that I ps I 0. Corollary The quotient R ps /I ps is cyclic. Remark Combined with Lemma 7.11 of [BK13] this shows that for any pseudo-deformation T : A[G Σ ] A of tr 1 + tr 2 with ideal of reducibility I T for which the corresponding map R ps A is surjective, the quotient A/I T is cyclic. Proposition The module R /I is a torsion O-module. Proof. Let R + R /I be the image of the structure map O R /I. Fix σ Z + and consider the quotient R := R /(I + ϖ σ R ). Suppose that R /I is not torsion. This implies that R + = O and A = O/ϖ σ R, where A is the image of the structure map O R. We begin with the following lemma.

7 ON MODULARITY OF REDUCIBLE RESIDUAL GALOIS REPRESENTATIONS 7 Lemma There exists an O-submodule B R such that as O-modules. R = A B Proof. This follows from the following result. Lemma 2.16 (Lemma 6.8(ii), p.222 in [Hun80]). Let A be a module over a PID R such that p n A = 0 and p n 1 A 0 for some prime p R and a positive integer n. Let a be an element of A of order p n. Then there is a submodule C of A such that A = R a C. Apply Lemma 2.16 for R = O, A = R, p = ϖ, n = σ, a = ψ(1). Then R a = A. We now finish the proof of Proposition Let e be an O-module generator of A. Write ρ I : G GL n (R) for the deformation corresponding to the canonical map R R. Then we can write ] [ 1 αe + β ρ I =, 2 where α : G M n1 n 2 (O) and β : G M n1 n 2 (B) are maps (here we identify j with its composition with O R). Define ] [ 1 ρ + I : G GL (g) α(g)e n(a) g. 2 (g) We must check that ρ + I is a homomorphism. This follows easily from the fact that ρ I is a homomorphism and the fact that A is a direct summand of R. Moreover, note that the image of α is not contained in M n1 n 2 (ϖo) because ρ I reduces to which is not semi-simple. Note that ρ + I is an upper-triangular deformation into GL n(o/ϖ σ ). Moreover, since ρ + I reduces to, it gives rise to an element in H1 Σ (F, Hom O( 2, 1 ) E/O) which generates an O-submodule isomorphic to O/ϖ σ. Since σ was arbitrary we conclude that HΣ 1 (F, Hom O( 2, 1 ) E/O) must be infinite which contradicts Assumption 2.7. This concludes the proof of Proposition Remark If dim F HΣ 1 (F, Hom( 2, 1 )) = 1 then R /I and R red /I red are cyclic O-modules by Corollary 7.12 in [BK13] which combined with Proposition 2.14 implies finiteness of R /I and R red /I red. On the other hand if dim F HΣ 1 (F, Hom( 2, 1 )) > 1 it is easy to construct an upper-triangular (not necessarily crystalline) lift of to F[[X]] which would suggest that in general R /I, and even R red /I red (since F[[X]] is reduced), may have positive Krull dimension. Indeed, to see this, let f be a cohomology class corresponding to and let g be a cohomology class linearly independent from f. Then the representation 1 ρ = 2 (f + gx) 0 2 is a non-trivial lift of to GL n (F[[X]]). In particular there is no guarantee that R red is a finitely generated O-module. Since our method of proving modularity relies on that property we will restrict in the following section to the characteristic zero part of R red of which we will demand that it is finite over O.

8 8 TOBIAS BERGER 1 AND KRZYSZTOF KLOSIN The ring R 0. Write R 0 for the image of R inside R O E. Equivalently (since E is sufficiently large ), R 0 is the image of R in p P() O, where P() := {p Spec(R ) R /p = O}. It is clear that R 0 is an object in LCN(E). Note that the canonical surjection R R 0 factors through R red. Write ρ 0 for the composition of ρ with the map ϕ : R R 0. Write I 0 for the ideal of reducibility of tr ρ 0. By [BK13], Lemma 7.11, we have ϕ (I ) I 0 (in fact equality holds since the opposite inclusion is obvious) and thus ϕ induces a surjection R /I R 0 /I 0. Lemma If R 0 is finitely generated as an O-module, then R 0 /I 0 is finite. Proof. This follows immediately from Proposition 2.14 and the surjectivity of R /I R 0 /I 0. Define R tr,0 R 0 to be the closed O-subalgebra generated by the set S := {tr ρ 0 (Frob q ) q Σ}. Lemma Write m for the maximal ideal of R 0. The ring R tr,0 with respect to its maximal ideal m 0 := m R tr,0 R tr,0 = ϕ (R tr ). Thus R tr,0 is complete and local. Moreover, one has is an object in the category LCN(E). Proof. The first part and the fact that the residue field of R tr,0 as Lemma 2.2. For the second part, note that it is clear that R tr,0 is F is proved exactly ϕ (R tr ). On. the other hand S ϕ (R tr ), so the equality holds because S is dense in R tr,0 We will write I tr,0 R tr,0 for the ideal of reducibility of tr ρ 0. By Lemma 2.19 and Lemma 7.11 in [BK13] we get that ϕ(i tr ) I tr,0 (in fact equality holds) and thus ϕ induces a surjection R tr /I tr R tr,0 /I tr,0. By Remark 2.13 the quotient is a cyclic O-module. R tr,0 /I tr, Generic irreducibility of ρ 0. Lemma For any as in (2.1) if R 0 is finitely generated as a module over O, then ρ 0 R 0 F is irreducible. Here F is any of the fields F s in F 0 = s F s, where F 0 is the total ring of fractions of R 0. Proof. First note that since R 0 is a finitely generated O-module and since E is assumed to be sufficiently large we may assume that all of the fields F s are equal to E. If any of the representations ρ 0 R 0 F is reducible write ρ = s j=1 ρ j for its semi-simplification with each ρ j irreducible, j = 1, 2,..., s. Then by compactness of G Σ for each 1 j s there exists a G Σ -stable O-lattice inside the representation space of ρ j. This implies that tr ρ j (σ) O for all σ G Σ and all 1 j s. Hence tr ρ splits over O into the sum of traces of ρ j. Since ρ 0 is a deformation of we easily conclude that ρ = ρ 1 ρ 2 with ρ j (with respect to some lattice) being a deformation of j (j = 1, 2). Using the fact that ρ 0 is a deformation of we now deduce that there is an O-lattice inside the space of ρ 0 R 0 F with respect to which ρ 0 R 0 F is block-upper-triangular (with correct dimensions) and non-semi-simple. When we reduce it modulo ϖ m, the upper-right shoulder will give rise to an element of order ϖ m in HΣ 1 (F, Hom O( 2, 1 ) O E/O). Since m is arbitrary this contradicts Assumption 2.7.

9 ON MODULARITY OF REDUCIBLE RESIDUAL GALOIS REPRESENTATIONS 9 3. The rings T Let us now define the rings T that will correspond to R 0 on the Hecke side. Proposition 3.1. If ρ : G Σ GL n (E) is irreducible and satisfies (3.1) ρ ss = 1 2 then there exists a lattice inside E n so that with respect to that lattice the mod ϖ reduction ρ of ρ has the form 1 ρ = 0 2 and is non-semi-simple. Proof. This is a special case of [Urb01], Theorem 1.1, where the ring B in [loc.cit.] is a discrete valuation ring = O. For each representation as in (2.1) let Φ be the set of (inequivalent) characteristic zero deformations of, i.e. crystalline at p p Galois representations ρ : G Σ GL n (O) whose reduction equals. Also, let Φ,E be the set of (inequivalent) crystalline at p p Galois representations ρ : G Σ GL n (E) such that there exists a G Σ -stable lattice L in the space of ρ so that the mod ϖ-reduction of ρ L equals. The following is a higher-dimensional analogue of Lemma 2.13(ii) from [SW99]: Proposition 3.2. One has Φ,E Φ,E = if =. Proof. Let ρ : G Σ GL n (E) be a representation such that ρ ss = 1 2 and let T equal its trace. Suppose there exist two lattices L i in the representation space of ρ such that the reductions of the corresponding representations ρ Li are given by and with = as in (2.1). We now consider the classes c Li of the cocycles corresponding to ρ Li in Ext 1 F[G Σ]/ ker T ( 2, 1 ). Using Assumption 2.7 above and Corollary 7.8 in [BK13] we conclude that the quotient O/I T is finite. Thus arguing as in the proof of Proposition in [BC09] but using Proposition 3.1 in [BK13] instead of generic irreducibility of T to conclude that ker T = ker ρ (see [BC09], Proof of Proposition 1.7.2, on how this equality - which follows from Proposition in [loc.cit.] in the generically irreducible case - is used) we obtain that the existence of ρ Li with trace T and non-split reduction as in (2.1) implies that Ext 1 X( 2, 1 ) is 1-dimensional, where X := (O[G Σ ]/ ker T )/ϖ(o[g Σ ]/ ker T ). First note that X = O[G Σ ]/(ϖo[g Σ ] + ker T ). Secondly one clearly has that ker(o[g Σ ] F[G Σ ]) = ϖo[g Σ ]. These two facts imply that the map O[G Σ ] X factors through O[G Σ ] F[G Σ ] and that the kernel of the resulting surjection F[G Σ ] X equals (ker T )F[G Σ ]. Thus we have X = F[G Σ ]/(ker T )F[G Σ ], so by the above we conclude that Ext 1 F[G Σ]/ ker T ( 2, 1 ) is one-dimensional. This means the corresponding representations of F[G Σ ]/ ker T are isomorphic. Since ker T = ker ρ (as noted above) the reductions both factor through this quotient of F[G Σ ], and so they are isomorphic as representation of F[G Σ ], in contradiction to our assumption. The following notation will remain in force throughout the paper. Notation 3.3. Write T for the set of isomorphism classes of residual representations of the form (2.1). Set Φ = T Φ.

10 10 TOBIAS BERGER 1 AND KRZYSZTOF KLOSIN 2 Remark 3.4. The condition that R 0 be a finitely generated O-module is equivalent to assuming that the set Φ is a finite set. We now fix subsets Π Φ and Π Φ of deformations. In our later application these will be taken to correspond to all the modular deformations corresponding to cuspforms of a particular weight and level which are congruent to a fixed Eisenstein series. In particular Π may be empty. Whenever Π we obtain an O-algebra map R ρ Π O. This induces a map from (3.2) R tr ρ Π O. Definition 3.5. We (suggestively) write T for the image of the map (3.2) - note that this also depends on the choice of the set Π - and denote the resulting surjective O-algebra map R tr T by φ. Also we will write T for the image of φ : R ps ρ Π O, where φ is induced from the traces of the deformations ρ π. Finally we will write J T for the ideal of reducibility of the pseudo-representation T R tr tr ρ,φ : T [G Σ ] T and J T for the ideal of reducibility of the pseudo-representation T ps R ps,φ T : T[G Σ ] T. Lemma 3.6. The maps R tr T and R ps T factor through R tr,0 image of R ps inside R ps O E respectively. and the Proof. Let P() = {p Spec(R ) R /p = O} be as before. Then clearly the kernel of R ρ Π O contains p P() p. Thus the map R ρ Π O factors through R 0. Then the claim follows since ϕ (R tr ) = R tr,0 by Lemma Lemma 3.7. The quotient T /J is cyclic and one has J = φ (I tr ). Proof. The first part is a consequence of Lemma 2.11 and was already mentioned in Remark By Lemma 7.11 in [BK13] we know that J φ (I tr ). For the opposite inclusion we argue as follows. We need to show that φ tr ρ Ψ 1 + Ψ 2 mod φ (I tr ) for Ψ 1, Ψ 2 pseudorepresentation. Put B = T, A = R tr and write ϕ for φ : R tr T and T B for T R tr tr ρ,φ. Let x B[G Σ ]. Since ϕ is surjective there exists y A[G Σ ] such that ϕ(y) = x. Then by definition of T B we have T B (x) = ϕ T (y) = ϕ(ψ 1 (y) + Ψ 2 (y) + i) for some pseudo-representations Ψ 1, Ψ 2 and i I tr. Now set Ψ j (x) := ϕ Ψ j(y) for j = 1, 2. Corollary 3.8. One has J = φ (I tr,0 ). Proof. By Lemma 3.6 the map φ factors through R tr,0. By abuse of notation we will denote the induced map also by φ as in the statement of the Corollary. Then since ϕ (I tr ) = I tr,0 we get the corollary. 4. The lattice L and modular extensions We will make a frequent use of the following result that is due to Urban [Urb01]. Let B be a Henselian and reduced local commutative algebra that is a finitely

11 ON MODULARITY OF REDUCIBLE RESIDUAL GALOIS REPRESENTATIONS 11 generated O-module. Since O is assumed to be sufficiently large and B is reduced we have B ˆB s s = O E = F B, i=1 where ˆB stands for the normalization of B and F B for its total ring of fractions. Write m B for the maximal ideal of B. For any finitely generated free F B -module M, any B-submodule N M which is finitely generated as a B-module and has the property that N B F B = M will be called a B-lattice. Theorem 4.1 ([Urb01] Theorem 1.1). Let R be a B-algebra, and let ρ be an absolutely irreducible representation of R on FB n such that there exist two representations ρ i for i = 1, 2 in M ni (B) and I a proper ideal of B such that (i) the coefficients of the characteristic polynomial of ρ belong to B; (ii) the characteristic polynomials of ρ and ρ 1 ρ 2 are congruent modulo I; (iii) ρ 1 := ρ 1 mod m B and ρ 2 := ρ 2 mod m B are absolutely irreducible; (iv) ρ 1 = ρ2. Then there exist an R-stable B-lattice L in FB n and a B-lattice T of F B such that we have the following exact sequence of R-modules: i=1 0 ρ 1 B T /IT L B B/I ρ 2 B B/I 0 which splits as a sequence of B-modules. Moreover, L has no quotient isomorphic to ρ 1. Since we will not only use Theorem 4.1 itself but also the construction of the lattice L let us briefly summarize how L is built (for details cf. [loc.cit.], p ). Let ρ i be the composition of the representation ρ with the projection B O onto the ith component of ˆB. Urban shows that we can always conjugate ρ i (over E) so that the mod ϖ -reduction of (the conjugate of ρ i which we will from now on denote by) ρ i has the form ρ1 (4.1). 0 ρ 2 In1 0 Set ρ ˆB := (ρ i ) i. It is also shown in [loc.cit.] that the matrices and are in the image of ρ 0 I n2 ˆB. One then defines the lattice L to be the B- submodule of ˆBn generated by ρ(r) [ t 0, 0,..., 0, 1 ], where r runs over R and set L 1 In := L and L := L. 0 I n2 Let T be as in Notation 3.3. Let T, Π Φ and Π Φ be as in section 3. Write T = π Π O for the normalization of T. Let ρ in Theorem 4.1 be ρ Π = T ρ π Π ρ π = π Π ρ π and ρ i = i, i = 1, 2, where i : G Σ GL ni (O) is a fixed crystalline deformation of i which we from now on assume exists. (If one works under Assumption 2.6, then the i s are unique, but we do not need this uniqueness for the arguments of this section.) Note that the reduction of ρ π already has the form (4.1), so we can take ρ ˆB = ρ Π and define lattices L, L 1 and L 2 as above (with B = T, R = T[G Σ ]). The G Σ -action on L is then via restriction of ρ Π to L. Write m = m T for the maximal ideal of the local ring T and let J as in section 3 be its reducibility ideal.

12 12 TOBIAS BERGER 1 AND KRZYSZTOF KLOSIN 2 By Theorem 4.1 (and Lemmas 1.1 and 1.5(ii) in [Urb01]) there exists a finitely generated free T-module T and a short exact sequence of T[G Σ ]-modules (which splits as a sequence of T-modules): (4.2) 0 L 1 T T/J L T T/J L 2 T T/J 0 with L 1 T T/J = 1 T T /JT and L 2 T T/J = 2 T T/J. Note that we have the following identification (4.3) Hom O ( 2, 1 ) T T /JT Hom T/J (L 2 T T/J, L 1 T T/J). Let s : L 2 T T/J L T T/J be a section of T/J-modules of (4.2). Using (4.3) as in [Klo09], p , we define a cohomology class c H 1 (F Σ, Hom O ( 2, 1 ) T T /JT ) by g (λ 2 t s(λ 2 t) g s(g 1 λ 2 t)). We also define a map ι J : Hom O (T /JT, E/O) H 1 (F Σ, Hom O ( 2, 1 ) O E/O), f (1 f)(c). Let us just briefly remark that ι J is independent of the choice of the section s. From now on we will make the following assumption on the quotient T/J. Assumption 4.2. One has with L as in Assumption 2.7. #T/J #O/L Remark 4.3. In Section 7 we will describe a particular setup for n = 2 and F an imaginary quadratic field under which Assumptions 2.7 and 4.2 are satisfied. However, we expect that these conditions hold also for other CM fields (for n = 2), and have therefore presented the results of this and the following sections under these two general assumptions. Lemma 4.4. If Assumptions 2.7 and 4.2 hold, then the map ι J : Hom O (T /JT, E/O) H 1 (F Σ, Hom O ( 2, 1 ) O E/O) is injective and its image equals H 1 Σ (F, Hom O( 2, 1 ) O E/O)). Proof. For the injectivity of ι J one follows the strategy in [Ber05], p which was later spelled out in a higher dimensional case in [Klo09], Lemmas 9.25 and Using [Klo09], Lemma 9.21 (which is just a slightly expanded version of Theorem 4.1) we get Fitt T T = 0. Then arguing as in [Klo09], the last four lines before the proof of Lemma 9.25 on p. 161 we get that (4.4) val p (#T /JT ) val p (#T/J) (see also the arguments on p. 120 in [Ber05]). This, combined with Assumption 4.2 and Assumption 2.7 implies that ι J must in fact surject onto the Selmer group. Since (4.2) splits as a sequence of T/J-modules we can tensor it with T/J F and obtain an exact sequence of F[G Σ ]-modules 0 L 1 T F L T F L 2 T F 0 with L 1 T F = T T 1 and L 2 T F = 2.

13 ON MODULARITY OF REDUCIBLE RESIDUAL GALOIS REPRESENTATIONS 13 Arguing as above (with m T instead of J) we again obtain an injective map ι : Hom O (T /JT, F) H 1 (F Σ, Hom F ( 2, 1 )). Lemma 4.5. Suppose that Assumptions 2.7 and 4.2 hold. The map ι : Hom O (T /JT, F) H 1 (F Σ, Hom F ( 2, 1 )) is injective and its image equals H 1 Σ (F, Hom F( 2, 1 )). Proof. We have the following commutative diagram Hom O (T /JT, E/O) ι J H 1 (F Σ, Hom O ( 2, 1 ) O E/O) Hom O (T /JT, F) ι H 1 (F Σ, Hom F ( 2, 1 )) Denote the right vertical arrow by f. Lemma 4.4 implies that the image of f ι is contained in the ϖ-torsion of HΣ 1 (F, Hom O( 2, 1 ) O E/O). Moreover, by Proposition 5.8 in [BK13], we know that the ϖ-torsion in HΣ 1 (F, Hom O( 2, 1 ) O E/O) coincides with f(hσ 1 (F, Hom F( 2, 1 ))). Since f is injective, this implies that the image of ι is contained in the Selmer group. Hence it remains to show that ι J is an isomorphism on ϖ-torsion. But this is clear since ι J is an isomorphism by Lemma 4.4. Lemma 4.6. Suppose that Assumptions 2.7 and 4.2 hold. Write ρ Π for ρ π Π ρ π. The F[G Σ ]-module L T F coincides with the F-subspace of ρ π Π Fn generated by ρ Π (r)e n, where r runs over F[G Σ ], e n is a column matrix in F n whose last entry is 1 and all the other ones are zero. Proof. By definition of L, every element of L T F can be written as i t iρ Π (g i )e n a i with t i T, a i F and g i G Σ. Writing t i for the image of t i under the canonical map T F we can re-write the above sum as i ρ Π(g i )e n a i t i and a i t i F. It suffices to show now that for every g G Σ we get ρ Π (g)e n 1 = ρ Π (g)e n 1. Write a11 (g) + a ρ Π (g) = 11(g) a 12 (g) + a 12(g) a 21(g) a 22 (g) + a, 22(g) where a 11, a 11 are (n 1) (n 1)-matrices, a 22, a 22 are scalars and the other matrices have sizes determined by these two and the entries of a ij (g) lie in ϖo ϖo ϖo Thus, a12 (g) + a ρ Π (g)e n 1 = 12(g) a12 (g) a a 22 (g) + a 1 = (g)/ϖ 22(g) a 22 (g) a ϖ, 22(g)/ϖ and the latter tensor is zero. This proves the lemma. Let us now turn to the 2-dimensional situation, where every is (up to a twist) of the form 1 = 0 χ for a Galois character χ. Note that Hom O (T /JT, F) = Hom O (T T T/J, F) = Hom O (T T T/m, F) = Hom O (T T F, F). Proposition 4.7. Suppose that Assumptions 2.7 and 4.2 hold. The image of ι : Hom(T T F, F) H 1 Σ (F, χ 1 ) is spanned by extensions such that Π.

14 14 TOBIAS BERGER 1 AND KRZYSZTOF KLOSIN 2 Proof. Let Π be as above and Π be a subset Π consisting of representatives of distinct isomorphism classes of residual representations (i.e., one element from every non-empty Π ). By Lemma 4.6 the lattice L T F is generated by vectors [ (1, 1,..., 1) χ(g)α(g) 0 x =. (0, 0,..., 0) (χ(g), χ(g),..., χ(g))] 1 Let us explain the notation: There are r elements in Π (which we will denote by 1, 2,..., r ), σ := r i=1 s i elements in Π. Moreover, α is a σ-tuple of functions such that α(g) equals (α 1,1 f 1 (g),..., α 1,s1 f 1 (g), α 2,1 f 2 (g),..., α 2,s2 f 2 (g),..., α r,1 f r (g),... α r,sr f r (g)) F σ, where the α i,j are elements of F. We get that x equals (α1,1 f χ(g) 1 (g),..., α 1,s1 f 1 (g), α 2,1 f 2 (g),..., α 2,s2 f 2 (g),..., α r,1 f r (g),... α r,sr f r (g)). (1, 1,..., 1) Let α j be the jth entry of α. Then we conclude that L T F = T T V = V, where V is the subspace of (F F) #Π spanned over F by the set vectors of the form ( ) χ(g)α 1 (g) χ(g)α, 2 (g) χ(g)α,..., σ (g). χ(g) χ(g) χ(g) For j {1, 2,..., σ} define integers n(j) {1, 2,..., r} and m(j) {1, 2,..., s n(j) } by the equality α j (g) = α n(j),m(j) f n(j) (g). χ(g)α The G Σ -action on V is via ρ Π, hence h G Σ acts on j (g) via the n(j)th χ(g) residual representation in Π, i.e., by multiplication by n(j) (h). In particular all the vectors in V have the form (4.5) v = By definition we have (4.6) L 2 T F = ([ a1 a ], a2,..., a 0 0 L 0 1 T F = ) aσ. a 0 0 V 0 1 = F(χ) as F[G Σ ]-modules, where we write F(χ) for the one-dimensional F-vector space on which G Σ acts via χ. The surjective F[G Σ ]-module map V F(χ) is given by sending a vector v as[ in (4.5) ] to a. Write [ V ] for the kernel of this map. Identifying V with L 1 T F = L 0 0 T F = V provides us with a splitting (only 0 0 as F-vector spaces) of the short exact sequence of F[G Σ ]-modules 0 V V F(χ) 0. Since the G Σ -action on L 1 T F is trivial, we have V = L 1 T F = T T F. Clearly, we may assume that the vectors in V all have the form ( ) a1 a2 aσ (4.7) v 0 =,,..., Let φ j Hom F (V, F) = Hom(T T F, F) be the homomorphism sending v 0 as in (4.7) to a j.

15 ON MODULARITY OF REDUCIBLE RESIDUAL GALOIS REPRESENTATIONS 15 Then the [ map ι sends φ j to ] the cocycle α n(j),m(j) f n(j), i.e, to the residual representation n(j) 1 χαn(j),m(j) f, which is isomorphic to the residual representation 0 χ of the n(j)th element of Π. So, this proves that the image of ι is spanned by modular extensions. Corollary 4.8. If Assumptions 2.7 and 4.2 are satisfied then the space H 1 Σ (F, χ 1 ) has a basis consisting of extensions such that Π. Proof. This follows from Lemma 4.5 and Proposition 4.7. Remark 4.9. Corollary 4.8 does not imply that Π for all isomorphism classes T. In fact, if we replace F by its finite extension F of degree m, then the order of T increases (since it is given by #H 1 Σ (F, Hom( 2, 1 ))/(q 1), where q is the order of the residue field), while the number of modular forms, i.e., T #Π remains the same. 5. Bounding the size of i T i/φ(i i ) In this section we keep in force Assumptions 2.7 and 4.2. Moreover, we work in the two-dimensional setup and we set 1 = 1 and 2 = χ (which can always be achieved by twisting by a Galois character). Let B := {e 1,..., e s } be a basis of HΣ 1 (F, Hom( 2, 1 )) = HΣ 1 (F, χ 1 ) consisting of modular extensions, i.e., extensions such that Π (cf. Corollary 4.8) and write i for the corresponding residual representations. Let us write T i for T i. Similarly let us write J i and Π i for J i and Π i respectively. Write p i : T T i for the canonical projection. Consider the map T s i=1 T i. Let J T be as in section 3. Set J i = p i (J). Note that J i is an ideal because p i is surjective. Let us begin with a trivial observation that one expects that the canonical map s T/J i=1 T i /J i is neither injective nor surjective, because the source is cyclic, while the target is not. However, as we shall see below the orders of both sides are equals provided that the basis B is unique up to scaling and that all of the ideals J i are principal. Proposition 5.1. # s i=1 T i/j i #T/J. Proof. In this proof we follow mostly the notation of [Klo09], section 9. Let Π i be as before. As in section 4 we get a lattice L i π Π i ρ π and a finitely generated T i -module T i such that the following sequence of T i /J i [G Σ ]-modules is exact: (5.1) 0 (T i /J i T i ) 1 L i /J i (T i /J i ) 2 0. Fix i. As before we obtain an injective map (5.2) ι i : Hom(T i /J i T i, E/O) H 1 Σ(F, Hom O ( 2, 1 ) O E/O). By the structure theorem of finitely generated modules over PID we get that (5.3) HΣ(F, 1 Hom O ( 2, 1 ) O E/O)) = O/ϖ ri O. s j=1

16 16 TOBIAS BERGER 1 AND KRZYSZTOF KLOSIN 2 Write c j H 1 Σ (F, Hom O( 2, 1 ) O E/O) for a generator of the jth factor in (5.3). Since H 1 Σ(F, Hom O ( 2, 1 ) O E/O)[ϖ] = H 1 Σ(F, Hom F ( 2, 1 )) by Proposition 5.8 in [BK13], we conclude that s = s and that we can choose the c j s so that each c j corresponds to e j under the canonical projection O/ϖ ri O s j=1 s F = HΣ(F, 1 Hom F ( 2, 1 )). j=1 Lemma 5.2. For every i = 1, 2,..., s one has Im(ι i ) c i. Proof. Let f Hom(T i /J i T i, E/O). It is enough to show that the image of f in Hom(T i /J i T i, F) is sent by ι i F to a scalar multiple of e i HΣ 1 (F, χ 1 ). Arguing exactly as in the proof of Proposition 4.7 we see that the image of ι i F is one-dimensional and is spanned by the cohomology classes corresponding to the isomorphism class of i. Lemma 5.2 implies that Hom(T i /J i T i, E/O) and hence T i /J i T i is a cyclic O- module, and hence a cyclic T i /J i (= O/ϖ di )-module. Arguing as in section 4 we obtain an analog of inequality (4.4): val p (#T i /J i ) val p (#T i /J i T i ). This combined with the fact that T i /J i T i is a cyclic T i /J i -module implies that T i /J i = Ti /J i T i. In particular this implies that the lattice L i /J i = (Ti /J i ) 2 as T i -modules. Let ρ i : G GL 2 (T i /J i ) be the representation given by the short exact sequence 0 (T i /J i ) 1 L i /J i (T i /J i ) 2 0 (coming from the sequence (5.1) and the fact that T i /J i T i = Ti /J i ). One has T i /J i = O/ϖ di and since ρ i reduces to i we must have d i r i, where Oc i = O/ϖ r i O. So, in particular we get that i d i r i. Combining Assumptions 2.7 with 4.2 we obtain the claim of Proposition 5.1. Our goal is now to prove the opposite inequality, which under some additional assumption will follow from a more general commutative algebra result which was proved by the authors and Kenneth Kramer in [BKK13] and which we will now present. Let s Z + and let {n 1, n 2,..., n s } be a set of s positive integers. Set n = s i=1 n i. Let A i = O ni with i {1, 2,..., s}. Set A = s i=1 A i = O n. Let ϕ i : A A i be the canonical projection. Let T A be a (local complete) O- subalgebra which is of full rank as an O-submodule and let J T be an ideal of finite index. Set T i = ϕ i (T ) and J i = ϕ i (J). Note that each T i is also a (local complete) O-subalgebra and the projections ϕ i T are local homomorphisms. Then J i is also an ideal of finite index in T i. Theorem 5.3 ([BKK13], Theorem 2.1). If #F s 1 and each J i is principal, then # s i=1 T i/j i #T/J. Let V be a vector space and write P 1 (V ) for the set of all lines in V passing through the origin. There is a canonical map V \ {0} P 1 (V ) sending a vector v to the line spanned by v. Let S be the set of all modular bases of HΣ 1 (F, Hom( 2, 1 )), i.e., the set of bases B = {e 1, e 2,..., e s} having the property that Π i, where i is the residual

17 ON MODULARITY OF REDUCIBLE RESIDUAL GALOIS REPRESENTATIONS 17 representation corresponding to the extension represented by e i. The set S is nonempty as B S. Definition 5.4. We will say that H 1 Σ (F, Hom( 2, 1 )) has a projectively unique modular basis if the images of all the elements of S in P 1 (H 1 Σ (F, Hom( 2, 1 )) agree. In the case when H 1 Σ (F, Hom( 2, 1 )) has this property we will refer to any element of S as the projectively unique modular basis. Note that it is possible to find i 0 {1, 2,..., s} such that the set B := B {e }\ {e i0 } is still a basis of H 1 Σ (F, Hom( 2, 1 )) (and one still has that Π for all B ). Hence we can assume without loss of generality that B = {e 1, e 2,..., e s } with 1 =. In fact, if H 1 Σ (F, Hom( 2, 1 )) has a projectively unique modular basis, it follows that if B is another modular basis, then the isomorphism classes of the residual representations corresponding to the elements of B are the same as the isomorphism classes of the residual representations corresponding to the elements of B. Proposition 5.5. If H 1 Σ (F, Hom( 2, 1 )) has a projectively unique modular basis and each J i is a principal ideal of T i, then # s i=1 T i/j i #T/J. Proof. First note that our assumption that E be sufficiently large allows us to assume that #F satisfies the inequality in Theorem 5.3. Since T is a free O- module of finite rank we set n to be that rank and define n i to be the rank of T i. The assumption that B be projectively unique guarantees that every O-algebra homomorphism has a corresponding residual Galois representation isomorphic to i for some i. Hence n = s i=1 n i. The Proposition now follows from Theorem 5.3 by taking T = T. Remark 5.6. Theorem 5.3 also has consequences for congruences between modular forms. Suppose that T = T Σ is the cuspidal Hecke algebra (acting on the space of automorphic forms over imaginary quadratic fields of weight 2 right invariant under a certain compact subgroup K f ) localized at a maximal ideal corresponding to an Eisenstein series, say E. Let J = J Σ be the Eisenstein ideal corresponding to E (see section 7.3 for the details). Let N be the set of O-algebra homomorphisms T O, i.e. to cuspidal Hecke eigencharacters congruent to the eigencharacter λ 0 of E mod ϖ. For λ N write m λ for the largest positive integer such that λ(t ) λ 0 (T ) mod ϖ m λ for all T T. Let e be the ramification index of E over Q p. As a consequence of Theorem 5.3 we get the following inequality (cf. Proposition 4.3 in [BKK13] 1 e m λ val ϖ (#T/J). λ N For many more applications of Theorem 5.3 see [loc.cit.]. Corollary 5.7. Suppose that Assumptions 2.7 and 4.2 hold. If the modular basis B is projectively unique and each J i is a principal ideal of T i, then # s T i /J i = #T/J #O/L. i=1

18 18 TOBIAS BERGER 1 AND KRZYSZTOF KLOSIN 2 6. Urban s method applied to R tr,0 In this section we again set n = 2, 1 = 1 and 2 = χ and we fix a residual representation 1 χf : G Σ GL 2 (F), =. 0 χ Let Π be as in section 3. From now on we will assume that Π is non-empty. The surjection φ : R tr,0 T (cf. Definition 3.5 and Lemma 3.6) descends to a surjection (6.1) R tr,0 /I tr,0 T /J. (since by Lemma 3.8 φ(i tr,0 ) = J ). The main goal of this section is to prove that under certain assumptions the map in (6.1) is an isomorphism (Theorem 6.3). Before we state the theorem let us demonstrate several properties of R tr,0. In particular we will show that R tr,0 = R 0 (Theorem 6.2). In this section we assume that Assumptions 2.6, 2.7 and 4.2 are satisfied. Lemma 6.1. The O-rank of R 0 equals the O-rank of R tr,0. In particular the normalizations (and the total rings of fractions) of R 0 and R tr,0 coincide. Proof. Write ρ for ρ 0, i.e., ρ : G Σ GL 2 [(R 0 ). ] We claim that we can conjugate ρ a b so that for every g G Σ we have ρ(g) = with a, c, d R c d tr,0. Indeed, since the characters on the diagonal of are distinct mod ϖ, we can find σ G Σ on which they differ, so that the eigenvalues of (σ) lift by Hensel s lemma to distinct eigenvalues of ρ(σ) in R 0, and we can conjugate (over R 0 ) to have ρ(σ) be diagonal with these lifted eigenvalues. For a general element g G Σ, we then compare tr ρ(g) with tr ρ(σg) and use that the eigenvalues are distinct mod ϖ to see that the two diagonal entries of ρ(g) lie in R tr,0. Similarly we show (cf. the proof of Lemma 3.27 in [DDT97]) that the lower-left entry also lies in R tr,0. Note that since R 0 is a finitely generated O-module (and O is assumed to be sufficiently large) we get a canonical embedding R 0 ˆR 0 k = O, where ˆR 0 is the normalization of R 0. For i = 1,..., k, write ρ i for the composition of ρ with the projection onto the ith component of ˆR 0. Suppose that the O-rank of R tr,0 is strictly smaller than the O-rank of R 0. Then there exist two minimal primes (after possibly renumbering the minimal primes we will call them p 1, p 2 ) of R 0 which contract to the same minimal prime p of R tr,0. Hence we get the following commutative diagram: R tr,0 R 0 i=1 R 0 /p 1 R 0 /p 2 R tr,0 /p

19 ON MODULARITY OF REDUCIBLE RESIDUAL GALOIS REPRESENTATIONS 19 a1 b This implies that the corresponding two deformations (to O) ρ 1 = 1 and c 1 d 1 a2 b ρ 2 = 2 (since their a-, c- and d-entries (as functions) factor through R c 2 d tr,0 /p) 2 must satisfy a 1 = a 2 =: a, c 1 = c 2 =: c and d 1 = d 2 =: d. In particular their traces are equal. Using Lemma 2.20 we see that both ρ 1 O E and ρ 2 O E are irreducible and thus [ by the ] Brauer-Nesbitt Theorem we conclude that ρ 1 O E = ρ 2 O E. A B Let M = GL C D 2 (E) be such that (6.2) Mρ 1 = ρ 2 M. Then an easy matrix calculation shows that (6.3) Aa + Bc = aa + b 2 C and (6.4) Cb 1 + Dd = cb + dd from which we get that Cb 1 = Cb 2. Suppose for the moment that there exists g G Σ such that b 1 (g) b 2 (g). Since O is a domain we conclude that C = 0. Since the representations ρ 1 and ρ 2 are irreducible over E, the function c cannot be identically zero. Using (6.4) we conclude that B = 0. Finally, computing the lower-left entries on both sides of (6.2) we get Dc = ca, so again using the fact c is not identically zero and that O is a domain we get that A = D. Thus, M is a non-zero scalar matrix. Hence we get a contradiction to our assumption on the existence of g and we conclude that b 1 = b 2. In particular ρ 1 and ρ 2 are identical deformations of which correspond to distinct minimal primes of R 0. Hence ρ 1 and ρ 2 give rise to two different homomorphisms from R red to O. This contradicts the bijectivity of the correspondence Hom O alg (R red, O) {deformations of into O}/equivalence. Theorem 6.2. Suppose that R 0 is a finitely generated O-module. There exists a deformation ρ tr,0 : G Σ GL 2 (R tr,0 ) of. The resulting canonical map R red factors through R 0 and induces an isomorphism R 0 = R tr,0 R tr,0 Proof. We will (once again) apply Theorem 4.1 (due to Urban). In the notation of section 4 we will write F B = F to be total ring of fractions of B = R tr,0 F. Note that by Lemma 6.1, F is also the total ring of fractions of R 0. Moreover, we take R = R tr,0 [G Σ ], ρ = ρ B F : G Σ GL 2 (F) which induces a morphism ρ : R tr,0 [G Σ ] M 2 (F) of R tr,0 -algebras. As before, the representations denoted in Theorem 4.1 by ρ 1 and ρ 2 are our unique lifts 1, 2 : G Σ GL 2 (O) GL 2 (R tr,0 ) and we set I = I tr,0. Note that conditions (i) and (ii) of Theorem 4.1 are satisfied respectively by the definition of R tr,0 and of I tr,0 and conditions (iii) and (iv) are satisfied by our assumptions on 1 and 2. Finally, the condition of irreducibility of ρ is satisfied by Lemma Hence we conclude from Theorem 4.1 that there exists an R tr,0 [G Σ ]-stable lattice L F 2 and a finitely generated R tr,0 -module T F such that we have an exact sequence of R tr,0 [G Σ ]-modules: (6.5) 0 1 T /I tr,0 T L R tr,0 /I tr,0. 2 R tr,0 /I tr,0 0.