1 Introduction

Lifting Galois representations to ramified coefficient fields

Abstract.

Let be a prime integer and a finite ramified extension with ring of integers and uniformizer . Let be a positive integer and be a continuous Galois representation. In this article we prove that under some technical hypotheses the representation can be lifted to a representation . Furthermore, we can pick the lift restriction to inertia at any finite set of primes (at the cost of allowing some extra ramification) and get a deformation problem whose universal ring is isomorphic to . The lifts constructed are “nearly ordinary” (not necessarily Hodge-Tate) but we can prove the existence of ordinary modular points (up to twist).

Key words and phrases:
Galois Representations; Modular forms.
2010 Mathematics Subject Classification:
11F33; 11F80
The author was partially supported by a CONICET doctoral fellowship

1. Introduction

The present article is a continuation of the work done in [arxiv], where we constructed, for a finite field , lifts of representations to . Here we prove how to extend the results to finite ramified extensions of ramification degree .

The method used in [arxiv] followed the ideas of [ravi2] and [ravi3], adapted to the modulo setting. As noticed in [arxiv] (see the remark before Proposition 5.9) these methods do not generalize to representations where is the ring of integers of .

The obstacle is that the modulo reduction of (which we denote ) fixes the same field extension of as an element (where isthe adjoint representation of the reduction mod of ). This implies that whenever we define a local condition for deformations containing at a prime it automatically contains and therefore lies inside its tangent space. On this way, no matter which set of primes we choose, the morphism of Theorem of [arxvv]

will always have non-trivial kernel and therefore will never be an isomorphism (as required in [arxiv]).

The key innovation of this work is to relax local conditions so that the morphism (1) is no longer an isomorphism, but a surjective map with one dimensional kernel. This will be enough for the lifting purpose, since it allows to find global elements that make the required the local adjustments at each step. We cannot relax conditions at primes , since the local deformation ring of should not have a smooth quotient of dimension bigger than . Therefore, we need to impose a different local condition at the prime . The condition we impose is the same as in [CM].

Definition.

We say that a deformation is “nearly ordinary” if its restriction to the inertia subgroup is upper-triangular and its semisimplification is not scalar, i.e. if

 ρ|Ip=(ψ1∗0ψ2).

with .

Using this local condition at , we are able to derive a slightly weaker version of the following theorem (see Theorem 1 for the precise statement), which is one of the main results of this work.

Theorem.

Let be a continuous representation which is odd and nearly ordinary at . Assume that contains if and otherwise. Let be a set of primes of containing the ramification set of . For each fix a local deformation that lifts . Then there exists a continuous representation and a finite set of primes such that:

• lifts , i.e. .

• is unramified outside .

• For every , .

• is nearly ordinary at .

• All the primes of , except possibly one, are not congruent to modulo .

In fact, the method provides us not only a lift of to but a family of lifts to characteristic rings, parametrized by a lift to the coefficient ring (see Theorem 13). The downside is that this family of representations is not ordinary but nearly ordinary, which implies that most points are not Hodge-Tate (in particular not modular). However, the freedom of the coefficient ring allows us to prove the existence of modular points, which is the second main result of the present article (see Theorem 1 for the precise statement).

Theorem.

Let be a prime, the ring of integers of a finite extension with ramification degree and its local uniformizer. Let be a continuous representation satisfying

• is odd.

• contains if and otherwise.

• is ordinary at .

Let be a set of primes containing the ramification set of , and for each pick a local deformation lifting . Then there exists a finite set of primes and a continuous representation such that

• lifts , i.e. .

• is modular.

• For every , .

• For every is unipotent and all but possibly one prime of satisfy .

• is ordinary at .

The strategy to prove both theorems is similar to the one in [ravi3] and [arxiv]. We will construct, for each prime , a set of deformations of to which contains and a subspace preserving its reductions in the sense of Proposition 1. Also, for , we take to be the set of nearly ordinary deformations, which will give a larger subspace and therefore a smaller codomain for the morphism (1). Given this local setting, and after some manipulation of the groups appearing in (1) (and its analogue for ) we are able to make such map surjective (with a -dimensional kernel) and the corresponding map for injective. This implies that, with enoguh local conditions, the problem of lifting is unobstructed, which gives Theorem 13.

The tricky part here is that, differently from what happens in [arxiv], in some cases the subspaces preserve the reductions modulo of the elements of , not for all but for bigger than a certain integer . To overcome this situation we will, following the ideas of [ravilarsen], lift by adding one set of auxilliary primes for each power of , until we reach the lift modulo for the main method to work. In this way we will get Theorem 1.

Theorem 1 follows from studying the possible modular points appearing in the universal ring provided by Theorem 13. Notice that we will obtain a modular lift of each time we find a nearly ordinary lift such that the characters appearing in the diagonal of can be written as an integral power of the cyclotomic character times a character of finite order.

We would like to remark that the method employed for the proof of Theorem 1 seems to generalize well to base fields other than . However, the more interesting question of getting an analog to Theorem 1 seems to be of higher difficulty. We currently have some work in progress towards this direction. As a final remark, the results obtained in this work have overlap with the ones in [KR] (in particular Theorem 1 is analog to Theorem of [KR]). Both works were independent in their first versions and in the present one we applied some of their results in order to remove hyphoteses. Also, the methods of section of [KR] allow one to get the stronger Corollary 2.

The article is organized as follows: Section 2 concerns with the construction of the sets and subspaces for primes . In Section 3 we do the same for the nearly ordinary condition at . Section 4 treats global arguments for lifting and proves the first main theorem of this work. It is divided into two subsections, one for exponents at which we have and for every and one for the ones at which we do not. In Section 5 we prove the other main theorem which concerns modularity.

1.1. Notation and conventions

In this article will denote a rational prime, the ring of integers of a ramified finite extension , its local uniformizer and its ramification degree. We will denote the residual field by . For a prime we denote by the local absolute Galois group and and stand for a Frobenius element and a generator of the tame inertia group of . We denote

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which form a basis for the space of matrices with trace . Also, given a prime , will stand for for . The cyclotomic character will be denoted by . All our deformations will have fixed determinant. By we denote a continuous Galois representation, denotes its residual representation, and by we denote a representation with image in . Finally stands for the valuation in satisfying .

For the definitions and main results of deformation theory, we refer to [mazur].

1.2. Acknowledgments

We would like to thank Ariel Pacetti for many useful conversations, and to Ravi Ramakrishna for many remarks and suggestions in a draft version of the present article.

2. Local deformation theory at q≠p

Let a prime and let be a continuous representation. We denote by its reduction mod . We know that the elements of act on the deformations with coefficients in by . Recall the notion of an element of preserving a set of deformations.

Definition.

Let be a set of deformations and . We say that preserves if for any which is the reduction of a deformation of we have that is also the reduction of some deformation of .

We want to prove that for every there exists a set of deformations containing it that is preserved by a subspace of certain dimension. We can prove such set exists for almost all possible but some particular ones.

Definition.

We say that a representation is “bad” if

 ρ≃(ψ1ψ1−ψ2πr0ψ2)

over with and moreover the following holds:

• is unramified and is scalar.

• .

We currently do not know how to find the desired family of deformations in these cases. Observe that finding the families and subspaces accounts for proving that the Balancedness Assumption of [KR] holds (see Assumption before Definition 3). In that scenario, if we start with a mod deformation and a prime for which every lift to is bad, we do not know how to prove that the Assuption holds.

Proposition 1.

Let be a continuous representation. If is not bad then there always exists a positive integer , a set of deformations of the reduction of modulo to characteristic that contains (up to -isomorphism), and a subspace such that:

• All the elements of are isomorphic over when restricted to inertia.

• has codimension equal to .

• Every preserves the mod reductions of elements of for .

In other words, there exists a smooth deformation condition containing of dimension equal to if we start lifting from a big enough exponent.

In order to prove the proposition, let us recall the main results about types of deformations and types of reduction mod . Although they are mainly known, the results and its proofs are contained in section of [arxiv].

Proposition 2.

Let , be a prime number, with . Then every representation , up to twist by a character of finite order, belongs to one of the following three types:

• Principal Series:

• Steinberg: where and .

• Induced: , where is a quadratic extension and is a character not equal to its conjugate under the action of .

Here is a multiplicative character and is an unramified additive character.

Proposition 3.

Let be a continuous representation. Then up to twist (by a finite order character times and unramified character) and equivalence we have:

• Principal Series: , with satisfying or .

• Steinberg: , with .

• Induced: There exists a quadratic extension and a character not equal to its conjugate under the action of such that , where for a generator of and , the action is given by

 β(v1)=ξ(β)v1,β(v2)=ξα(β)v2,α(v1)=v2andα(v2)=ξ(α2)v1,

or

 ~ρ(β)=(ξ(β)ξ(β)−ξα(β)πn0ξα(β)) and ~ρ(α)=(−aξ(α2)−a2πnπna)

where is the character of defined by and . Observe that when is ramified we can take and to be a Frobenius element and a generator of the tame inertia respectively.

Proposition 4.

Let be as above and its mod reduction. We have the following types of reduction:

• If is Principal Series, then can be Principal Series or Steinberg, and the latter occurs only when .

• If is Steinberg, then can be Steinberg or Principal Series, and the latter occurs only when is unramified.

• If is Induced, then can be Induced, Steinberg or unramified Principal Series. For the last two cases we must have and ramified.

To prove Proposition 1 we consider all the possible pairs of and -equivalence classes for and (indexing them first by the class of ), and for each of them we define the corrsponding deformation class and cohomology subspace.

Case 1: is ramified Principal Series. Proposition 4 implies that a mod Principal Series can only come from a characteristic principal series. The full study of this case is done in Case 1 of Section 4 of [CP14]. The work there is done for unramified but the same applies in our situation.

Case 2: is Steinberg. When is Steinberg, it can be the reduction of any of the three characteristic ramified types:

Case 2.1: is Steinberg. The definition of and is essentially the same as [arxiv], Case 2 of Section 4. Although in that work only the case where is unramified is treated, the ramification of does not affect the results.

Case 2.2: is Principal Series. Proposition 4 implies that . Let . Without loss of generality, we can take . We have the following lemma:

Lemma 5.

A deformation which has the form:

 ~ρ(τ)=(ψ(τ)101)and~ρ(σ)=(αγ0β),

has a unique lift to characteristic zero of the same form if and only if , where is a fixed lift determinant.

Proof.

This Lemma is part of a computation made in Proposition 3.4 of [khare]. There, it is done for deformations with coefficients in unramified coefficient field and its mod reductions, but the same proof works in general. ∎

Let be the element defined by:

 j(σ)=e2,j(τ)=0,

and take to be the subspace it generates. Also let be the set of deformations to which have the form given in Lemma 5. Observe that any mod reduction of an element of satisfies the equation , and acting by on it does not affect this (as divides so adding a multiple of to does not change the equation modulo ). Then, Lemma 5 guarantees that and satisfy the property we are looking for.

Case 2.3: is Induced. Proposition 4 tells us that necessarily and by results in Section of [CP14] we have that . Therefore we can take and .

Case 3: is Induced. By Proposition 4, when is Induced, must be Induced as well. The choice of and in this case is explained in Case of Section of [CP14].

Case 4: is unramified. Being unramified, allows lifts to any type of deformation. We must treat each of them separately as they have many subcases. The case of being Steinberg is dealt with in Case 4 of Section 4 of [CP14]. There are two other cases left to study. In each of them we will distinguish between three types of equivalence classes for , acording to the image of Frobenius. This case is the one that includes bad primes, the calculations made here show where the badness condition appears.

Case 4.1: is Principal Series. In this case we have with . By Proposition 4 we necessarily have .

If with we have and so we are looking for a one-dimensional subspace . Observe that necessarily over as so is not an integer.

Let be the cocycle defined by and . We can take and the set of representations of the form for unramified. It is easily checked that these satisfy the desired properties.

If the corresponding dimensions are and . In this case we have , with . Now take the cocycle defined by and . We take and set and the set of deformations of such that for some and an unramified character congruent to modulo .

Lemma 6.

The set and subspace defined above satisfy that preserves mod for all such that is ramified mod .

Proof.

Assume that and . Our assumptions on not being bad tell us that . If we have then we change by (which is another Frobenius element). In this way, we can assume that .

We want to prove that if we have a deformation that sends

 ~ρ(σ)=(ac0b)and~ρ(τ)=(xz0y),

with , and is the reduction of some element of (i.e. and for some ), then is also the reduction of some element of . Recall that

 u⋅~ρ(σ)=(ac0b)andu⋅~ρ(τ)=(xz+πm−10y).

It is easily checked that the cocycle that sends to and to is a coboundary for any choice of . So can also be tought as

 Missing dimension or its units for \hskip

for any . Therefore, it is enough to find some such that

 x−yz+πm−1=a(1+λ1πm−1)−b(1+λ1πm−1)−1c+λ2πm−1.

given that

 x−yz=a−bc.

Expanding this equation we find out that it is equivalent to

 λ1(z+πm−1)(a+b(1+λ1πm−1)−1)+a−b=λ2(x−y).

which has a solution given that (we solve first for in order for both sides to have the same valuation, and then there is a that makes the equality true). ∎

Remark.

Observe that when the condition does not hold, the last equation of the proof does not have a solution, since no matter which we pick, the valuation of the left hand side is and the valuation of the right hand side is bigger or equal than . Moreover, following the same type of computations we can prove that for the chosen set there is no non-trivial such that the subspace preserves .

If we have and therefore we need to find a subspace of dimension . This case is a little more involved than the other two as there are non-trivial elements of that act trivially on modulo deformations for high powers of . We follow the same ideas as in the study of the Steinberg-reducing-to-unramified case (the spirit of these ideas is taken from the approach to trivial primes followed in [ravihamblen]).

Assume first that . We will take as the set of representations of the form

 ρ′≃(γϕ00γ−1)

with an unramified character, that lift the reduction mod of , with . Clearly, the set is preserved by the cocycle that sends and .

We will construct two more cocycles and that act trivially on reductions modulo of deformations on for . Let be the mod reduction of an element in . Let and . Let . To prove that acts trivially on we need to find a matrix such that and . One can find such matrix by taking and explicitly computing at and . In this way, one finds out that if then the cocycles and sending to and respectively and to act trivially on the reductions of elements of . The corresponding base change matrices that conjugate into are

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It remains to check what happens when . Observe that we can always assume that in this case, by simply changing for . In this case we can take sending to and to and sending to and to for (notice that this does not depend on ). Again, the action of both cocycles will be trivial and the base change matrices will be the same as before.

It remains to consider the case where for . Let and let be the set of deformations of the mod reduction of such that for some and an unramified character congruent to modulo . By doing the exact same calculation as in Lemma 6 it can be proved that the cocycle that sends to and to preserves the reductions of elements in , given that is not bad. We still need two more elements preserving . As in the previous case, we have two cocycles that act trivially on mod reductions of elements of for . Let is a mod reduction of some element in given by and .

If , we take sending to and to 0 and the one that sends to and to . We claim that these act trivially on if .

If otherwise we can assume that as in Lemma 6. Let

 λ=x−ya−b=zc.

Let be the cocycle that sends to and to and the one that sends to and to . These act trivially on if .

It can be checked that in both cases the base change matrices given by

 Missing dimension or its units for \hskip

serve to prove the trivialness of the action of and respectively. This concludes the case.

Case 4.2: is Induced. Proposition 4 says that whenever is induced and is unramified, has eigenvalues and (up to twist) and . In this case we have that and . We want to find a set and a subspace of dimension preserving it. Let . As in the Principal Series case, we will be able to find non trivial cocycles that act trivially on mod reductions of for big enough. We split into the two possible families of -equivalence classes of induced representations given by Proposition 3.

If and , it can be checked that the cocycle sending to and to is non trivial. The cocycle acts trivially modulo for all whenever . In this case the base change matrix is given by

 C=⎛⎜⎝1−πm−1x−y−πm−1x−y1⎞⎟⎠.

If we are in a case where then we can go back to a case where by twisting . Let be an unit such that and an unramified character mapping to . If we twist by then the deformation obtained is equivalent to sending

 ρ′(σ)=(0η2t10)andρ′(τ)=(x00y)

via the base change matrix

 C=(η001).

If and as in the previous case, it can be checked that the cocycle that sends to and to is non trivial. Again, the action of this cocycle in the reduction modulo of is trivial for . The base change matrix that works for this case is

 C=⎛⎜⎝1+πm−1zbπm−12azcπm−12az1⎞⎟⎠.

3. Local deformation theory at p

At the prime we will impose the deformation condition of being “nearly ordinary” (as in [CM]). This section is mainly about gathering previously done calculations, and all the deformations appearing are deformations of the local Galois group .

Definition.

We say that a deformation of is “nearly ordinary” if its restriction to the inertia subgruop is upper-triangular and its semisimplification is not scalar, i.e. if

 ρ|Ip=(ψ1∗0ψ2).

with .

We will prove the following theorem.

Theorem 1.

Let be a nearly ordinary deformation and its mod reduction. There is a family of nearly ordinary deformations to characteristic such that is the reduction of a member of and a subspace of codimension equal to preserving in the sense of Proposition 1.

Proof.

Let

 ¯¯¯ρ≃(ψ1∗0ψ2).

By twisting by we can assume that . Let be the set of upper-triangular matrices of trace . To prove the theorem we will construct for each possible , the corresponding set and subspace and verify that its dimensions satisfy the statement of the theorem. In most of the cases will consist on all nearly ordinary deformations of and will be the image of in .

In order to see this, we simply compute the dimension of the image of in and compare it with . We can reduce the computation of all these values to finding the dimensions of , , and by using local Tate duality and the formula for the Euler-Poincare characteristic (which is equal to for and for ). The kernel of the map induced by is contained in , which is equal to given that . With all these tools, the required dimensions are easily computed and we obtain that taking as the set of all nearly ordinary deformations and as the image of in works for all cases but the one in which is decomposable and is the cyclotomic character (recall we are assuming ).

In this case the universal ring for nearly ordinary deformations is not smooth and is not preserved by as there are some mod nearly ordinary deformations of that do not lift back to characteristic . We need to take a smaller set in this case. In order to solve this, we claim that the universal deformation ring for nearly ordinary lifts of with fixed determinant is isomorphic to the universal deformation ring for ordinary lifts of with arbitrary determinant. To see this, just observe that from any nearly ordinary deformation of to a ring we can obtain an ordinary lift of by twisting by inverse of the character appearing in the place . To go the other way round, if we have an ordinary deformation of to given by

 ~ρ=(ω1∗0ω2)

where is unramified and want to obtain a nearly ordinary deformation of with determinant , we need to twist by a square root of . This character has a square root by Hensel’s lemma, as its reduction modulo the maximal ideal is which has a square root. Given this identification, the wanted result follows from Proposition of [KR], where the same result is proven for the ordinary arbitrary determinant case. ∎

4. Global deformation theory

In this section we prove the one of the main results of this article.

Theorem 1.

Let be an integer and be a continuous representation which is odd and nearly ordinary at . Assume that contains