Mod p Descent for Hilbert Modular Forms

Preliminary version

Chandrashekhar Khare

July 13, 1999

1 Introduction

Let F be a totally real extension of Q and

ρ_{F} : Gal(\overline{F} /F) ⟶ GL_{2}(k)

be an (absolutely) irreducible, continuous and odd representation, with k a finite field of characteristic p ≥ 7, and where Gal(\overline{F} /F) is the absolute Galois group of F (that we will henceforth abbreviate to G_{F}).

By an odd representation we mean det(ρ_{F} (c)) = −1, where c runs through the conjugacy classes of complex conjugation with respect to each real place of F. In this situation, the natural extension of Serre’s conjecture (cf. [S]) would say that ρ_{F} arises from a (holomorphic) Hilbert modular form of some infinity type and level. We assume this to be the case.

Now assume that ρ_{F} extends to a representation ρ_{Q} of the absolute Galois group G_{Q} of Q, i.e.,

ρ_{Q} ⟶ GL_{2}(k′)

where k′ is a finite extension of k and ρ_{Q}|_{G_{F}} is isomorphic to ρ_{F} (in GL_{2}(k) or GL_{2}(k′): the choice of k or k′ does not matter as, for instance, the Brauer group of a finite field is trivial). This extension need not be unique, but any such extension is odd and (absolutely) irreducible.

Under these circumstances the following question is natural:

Question 1 Is the extension ρ_{Q} modular?

Assuming Serre’s conjecture (cf. [S]) the answer to this question is in the affirmative as ρ_{Q} is odd and (absolutely) irreducible. It is the purpose of this appendix (to [R]) to provide an unconditional answer to this question, under some particular circumstances, using the work of [F], [L] and [R]. The simple argument used here is reminiscent of [C] (and [HM]), i.e., we use the result of [R] on lifting Galois representations, and then go via results of [F] to the automorphic or Hecke side. Namely we use:

• The identification of Hecke and deformation rings proved in [F] for totally real fields in many cases.

• The lifting result of [R].

• The characterisation of the image of base change for cyclic, prime degree extensions in [L].

2 The assumptions and the theorem

The particular circumstances are the following:

1. Assume that ρ_{F} is (absolutely) irreducible, odd and modular and that it in fact has a modular lift of the “minimal level”.

2. Assume that F/Q is a solvable extension, p does not divide the class number of F, and is unramified in F. We may have to assume further either

(i) a parity condition on the degree [F : Q] or

(ii) that ρ_{F} has a prime q at which it is ramified and semistable,

if Fujiwara needs such assumptions to go from the minimal to the nonminimal case (i.e., to raise levels), or to drop the hypothesis of the minimality of the modular lift (i.e., lower levels).

3. Assume that ρ_{F} restricted to F (\sqrt{p^{∗}}) is absolutely irreducible where p^{∗} = (−1)^{p−1/2}p.

4. Assume that the order of the projective image of ρ_{Q} is divisible by p, that the restriction ρ_{Q} to I_{p}, the inertia group at p, acts via fundamental characters of level one, is ordinary, and that we are not in an excluded case of Theorem 1, part (b) of [R]. Note that once I_{p} acts via fundamental characters of level one the ordinary hypothesis is harmless, as it can be ensured by twisting.

Then we have:

Theorem 1 Assuming the conditions above, the extended representation ρ_{Q} is modular.

3 Proof of theorem

First we quote for ease of reference a result stated in [R] that we use.

Theorem 2 Suppose ρ_{Q} : G_{Q,S} → GL_{2}(k′) is odd and Ad^{0}(ρ_{Q}) is absolutely irreducible. Suppose further inertia at p acts via fundamental characters of level one, is ordinary, that we are not in an excluded case of Theorem 1, part b) of [R], and the order of the projective image of ρ_{Q} is divisible by p. Then there is a finite set of primes R containing S and an ordinary at p deformation ρ̃_{Q} : G_{Q,R} → GL_{2}(W (k′)) of ρ_{Q}.

Remark. Note that the lift ρ̃_{Q} need not be “minimally ramified”.

The assumption on the absolute irreducibility of Ad^{0}(ρ_{Q}) is equivalent to assuming that ρ_{Q} is not dihedral. If ρ_{Q} is dihedral (and odd), the modularity of ρ_{Q}, and hence Theorem 1 follows from a classical result of Hecke. Thus we will assume that we are in the case when Ad^{0}(ρ_{Q}) is absolutely irreducible.

By the methods of [R] we can also ensure that the ordinary at p lifting ρ̃_{Q} : G_{Q} → GL_{2}(W (k′)), that exists by Theorem 2, has a given determinant of the form χ^{k−1}ϵ, where χ is the p-adic cyclotomic character and ϵ is some finite order character of G_{Q} (for some integer k ≥ 2), lifting the determinant of ρ_{Q}.

Below we will consider isomorphism classes of representations such as ρ̃_{Q} above, when considered as representations into GL_{2}(W (k′)⊗Q_{p}) rather than into GL_{2}(W (k′)). Though we are interested in the reduction mod p of such representations, as we are assuming that the mod p representations considered are irreducible, after the theorem of Brauer-Nesbitt that reduction mod p is well-defined upto semisimplification, this weaker notion of isomorphism is good enough for us.

Now we quote the result of [F] needed below:

Theorem 3 The representation ρ̃_{Q}|_{G_{F}} is associated to a (unique) Hilbert modular newform f over F of weight (k, ⋯, k).

Remark. By the word “associated” we mean that ρ̃_{Q}|_{G_{F}} is the p-adic representation attached to f by results of Taylor, Blasius-Rogawski et al.

Proof. This is hopefully proved in [F]: the manuscript is still not available to the public, but under assumptions 1 to 4, and the fact the lift ρ̃_{Q}|_{G_{F}} is ordinary at all places above p, this should be proven in [F] (see the end of the introduction of the version 1.2 dated July 9,1996 of [F] that the author has seen: the assumptions in [F] on semistability at primes other than those above p can probably be removed by the methods of [D]).

As ρ̃_{Q}|_{G_{F}} is invariant under the conjugation action of Gal(F/Q), we deduce that f^{σ} = f for all σ ∈ Gal(F/Q).

Let

1 = G^{(i)} ≤ G^{(i−1)} ⋯ ≤ G^{(0)} = Gal(F/Q)

be a composition series for Gal(F/Q) with G^{(j)} the commutator subgroup of G^{(j−1)} for 1 ≤ j ≤ i: thus the successive quotients G^{(j)}/G^{(j+1)} are abelian. Denote the subfields of F that are the fixed fields of G^{(j)} by F_{i−j} (j = 0, ⋯, i): thus F_{i} = Q and F_{0} = F. Note that these subfields are Galois extensions of Q.

Let F_{0,k} (0 ≤ k ≤ r) be the subextensions of the abelian extension F_{0}/F_{1}, such that F_{0,0} = F, F_{0,r} = F_{1}, and F_{0,s}/F_{0,s+1} is a cyclic extension of prime degree for 0 ≤ s ≤ r−1. By the results of [L], the automorphic representation π_{f_{0}} of GL_{2}(A_{F}) that corresponds to f_{0} := f arises by “base change” with respect to the prime, cyclic extension F_{0}/F_{0,1} from a Hilbert modular form f_{0,1} of weight (k, ⋯, k) (k ≥ 2), or equivalently an automorphic representation π_{f_{0,1}} of suitable infinity type of GL_{2}(A_{F_{0,1}}).

This means that the Galois representation ρ_{1} of G_{F_{0,1}} associated to f_{0,1} when restricted to G_{F_{0}} is isomorphic to the restriction of ρ̃_{Q} to G_{F_{0}}. But as we have assumed that ρ_{F_{0}} is (absolutely) irreducible, and F_{0}/F_{0,1} is cyclic, it follows that any representation of G_{F_{0,1}} which restricts to G_{F_{0}} to give ρ̃_{Q}|_{G_{F_{0}}} is (isomorphic as a W (k′) ⊗ Q_{p} representation to) a twist, by a character of Gal(F_{0}/F_{0,1}), of ρ̃_{Q} restricted to G_{F_{0,1}}. Thus we deduce that ρ̃_{Q}|_{G_{F_{0,1}}} is itself modular. Noting that ρ̃_{Q}|_{G_{F_{0,1}}} is invariant under the action of Gal(F_{0,1}/F_{0,2}), we argue as before, using the results of [L] applied to the cyclic prime degree extension F_{0,1}/F_{0,2}, to show that ρ̃_{Q}|_{G_{F_{0,2}}} is modular. Iterating the argument we conclude that ρ̃_{Q}|_{G_{F_{1}}} is modular.

Continuing in this way we deduce successively that ρ̃_{Q}|_{G_{F_{j}}} is modular for j = 0, ⋯, i − 1 and hence finally that ρ̃_{Q} is modular of weight k ≥ 2. Thus ρ_{Q}, the mod p reduction of ρ̃_{Q}, is itself modular and this finishes the proof of Theorem 1.

Remark. Note that the results of [L] are for prime degree, cyclic extensions, while above we have availed of Galois representations to argue in the solvable case.

4 Descent for cyclic extensions

In the case F/Q is a cyclic extension of prime degree we can in many cases start with descent data on ρ_{F}, rather than conditions on ρ_{Q}, just as in the classical case (cf. [L]). We note the following consequence of the proof above.

Theorem 4 Let F/Q be a cyclic extension of prime degree, coprime to the cardinality of k^{∗} with k a finite field of characteristic p, p unramified in F, and ρ_{F} : G_{F} → GL_{2}(k) an irreducible, modular, mod p representation that is invariant under the action of Gal(F/Q), such that the order of its projective image is divisible by p, with determinant the mod p cyclotomic character, and satisfying the assumptions 1 to 3 above.

Then it extends to a representation ρ_{Q} : G_{Q} → GL_{2}(k′) for some finite extension k′ of k. Further if we suppose that ρ_{F} restricted to the decomposition group at any (one) place ℘ of F above p (and hence at all places of F above p) is ordinary, then the extended representation ρ_{Q} is modular.

Proof. That ρ_{F} as in the theorem extends to a representation ρ_{Q} follows from cohomological arguments as in Proposition 1.1 of [C]. Namely, the obstruction lies in H^{2}(G_{Q}, k^{∗}), and it vanishes on restriction to H^{2}(G_{F} , k^{∗}).

But as F/Q is coprime to the order of k^{∗}, we deduce that the representation extends to G_{Q}. This extension, as noted above, is unique upto twisting by a character of Gal(F/Q).

Under the further hypothesis that the restriction of ρ_{F} to the decomposition group of a place of F above p is ordinary, we claim that ρ_{Q}|_{D_{p}} is again ordinary. This follows as, if the semisimplification of ρ_{F} restricted to the decomposition group is the sum of two characters ψ and ψ′, with say ψ′ unramified, then the determinant being ramified at p implies ψ and ψ′ extend to G_{Q}. As p is unramified in F, any extension of ψ′ to G_{Q} is unramified. This justifies the claim.

The assumption that F/Q is unramified at p, together with that on the determinant character of ρ_{F} implies that ρ_{Q} is not in the excluded case of Theorem 1 part (b) of [R]. After this we are done by the same argument as above, applied to the step F_{0}/F_{0,1} in the notation of the proof of Theorem 1.

5 Rationality of lifts

5.1 Modular compatibility of finite characteristic representations

Let ρ : G_{Q} → GL_{2}(k) be an odd, irreducible representation with k a finite field of characteristic p > 3. Fix an auxiliary representation ρ_{3} : G_{Q} → GL_{2}(Z/3Z) that is ordinary at 3, irreducible on restriction to Q(\sqrt{−3}). By Wiles, any ordinary lifting of ρ_{3}, such that its determinant character is “arithmetic”, i.e., a positive integral power of the 3-adic cyclotomic character upto a finite order character, is modular: to see this note that, by the Langlands-Tunnell theorem, ρ_{3} is modular. Thus if we could find a lifting of ρ that is “compatible” with an ordinary 3-adic lifting of ρ_{3} with “arithmetic” determinant character, we would have proven Serre’s conjecture: note that after Wiles the latter lifting is algebraic (i.e., the traces of frobenii are in a number field) and hence it makes sense to talk of compatibility. The results of [R] do not seem to give algebriac liftings. Notice that we have a lot of freedom in the choice of ρ_{3} and thus could choose it so that the corresponding deformation rings are unobstructed, though it is not clear if this will be helpful. On the other hand note that the choice of ρ_{3} cannot be unconstrained if we hope to succeed, as ρ_{3} should arise from newforms of level divisible by the conductor of ρ. These considerations give rise to the following 2 natural questions:

Question 2 Let p and q be distinct primes, ρ_{p} : G_{Q} → GL_{2}(\overline{F_{p}}), ρ_{q} : G_{Q} → GL_{2}(\overline{F_{q}}) be irreducible, continuous representations. Suppose ρ_{p} (resp., ρ_{q}) is semistable at all primes different from p (resp., q) and finite, flat at p (resp., q), and det(ρ_{p}) (resp., det(ρ_{q})) is the mod p (resp., mod q) cyclotomic character. Suppose further that ρ_{p} and ρ_{q} have the same prime to residue characteristic Artin conductor. Then do there exist compatible lifts of ρ_{p} and ρ_{q}?

Question 3 Let p and q be distinct primes, ρ_{p} : G_{Q} → GL_{2}(\overline{F_{p}}), ρ_{q} : G_{Q} → GL_{2}(\overline{F_{q}}) irreducible representations that satisfy Serre’s conjecture, i.e., which are irreducible and arise from modular forms. Suppose ρ_{p} (resp., ρ_{q}) is semistable at all primes different from p (resp., q) and finite, flat at p (resp., q), and det(ρ_{p}) (resp., det(ρ_{q})) is the mod p (resp., mod q) cyclotomic character. Suppose further that ρ_{p} and ρ_{q} have the same prime to residue characteristic Artin conductor. Then does there exist a newform of some level which gives rise to ρ_{p} and ρ_{q} simultaneously?

The local conditions are just to ensure that there are no local constraints in answering the question affirmatively, and may be refined further (see also [St]). Note that the conditions on ρ_{p} (resp., ρ_{q}) imply that it’s determinant is the mod p (resp., mod q) cyclotomic character. By mod p and mod q representations below we mean those that arise after having fixed a prime above p and q.

It is easy to see that the Question 3 has a negative answer if we fix levels: i.e., we claim that if S_{2}(Γ_{0}(R)) has dimension greater than 1, for a prime R, then there are primes p and q (≠ R) and representations ρ_{p} and ρ_{q} as above which arise individually from S_{2}(Γ_{0}(R)), but do not arise compatibly from S_{2}(Γ_{0}(R)). To justify the claim, if f_{1}, f_{2}, ⋯, f_{i} ∈ S_{2}(Γ_{0}(R)) are the distinct newforms, then pick large enough primes p and q so that the mod p (resp., mod q) representations arising from the f_{i}’s are mutually distinct and irreducible. Pick ρ_{p} to be the mod p representation arising from f_{1}, and ρ_{q} to be the mod q representation arising from f_{2}. Then ρ_{p} and ρ_{q} do not arise simultaneously from any of the f_{i}’s, justifying the claim. B. Mazur and K. Ribet have considered a question similar to Question 3 above, and there are some computations of William Stein related to it (see [St]).

For 1-dimensional characters the analogous questions have affirmative answers if and only if there are no local constraints. Namely, we claim that if we have characters η_{p} : G_{Q} → \overline{F}_{p}^{∗} (resp., η_{q} : G_{Q} → \overline{F}_{q}^{∗}) then there exists a character η : G_{Q} → μ_{∞}, where μ_{∞} is the torsion subgroup of \overline{Q}^{∗}, which reduces modulo a chosen place above p (resp., above q) to η_{p} (resp, η_{q}) if and only if this can be done locally at all primes. This follows because a necessary and sufficient condition for the existence of η is that if η̃_{p} (resp., η̃_{q}) is a lifting of η_{p} (resp., η_{q}) with the same order (a Teichmuller lifting) then an η of the desired sort exists if and only if the order of η̃_{p}η̃_{q}^{−1} is of the form p^{α}q^{β}. This together with the Cebotarev density theorem justifies the claim.

5.2 Descent for non-solvable extensions

In the situation of the present paper if we could produce a lift of ρ_{Q} such that on restriction to G_{F} it is modular, for F a totally real Galois extension of Q, and the corresponding Hilbert modular form is rational over a field in which there is no residue field extension at 3, and is ordinary at places dividing 3, we could apply results in [C] to prove mod p descent for certain Galois, non-solvable, totally real extensions.

6 Remarks

• The results of [F] seem to be in a state of flux: thus it may either be necessary to augment the conditions 1 to 4 above, or it may turn out that many of them are superfluous. Especially the condition of minimal level modular lift may be superfluous at least assuming condition 2 (ii) above. One might just have to assume that ρ_{F} has a modular lift which is of course indispensable to the method used here.

• The method should work for any totally real base field. Here we have taken it to be Q, as the lifting results of [R] have that hypothesis.

• There are many instances in which one knows ordinary at p liftings of ρ_{Q} exist simply because the corresponding ordinary (at p) deformation ring (with respect to some ramification data) is unobstructed (see [M] and [BM] for examples of unobstructed ordinary deformation rings).

• In the situation of Theorem 4, if F/Q is a quadratic extension of discriminant a prime ℓ, and if ρ_{F} arises from a form of level 1, then ρ_{Q} arises from a form f ∈ S_{k}(Γ_{0}(ℓ), χ_{ℓ}) or ∈ S_{k}(SL_{2}(Z)), where χ_{ℓ} is the quadratic character mod ℓ. It seems difficult to a priori distinguish between these two mutually exclusive possibilities.

• It would be of interest to extend the mod p descent criterion of this paper to general number fields, or even prove it in the present context using more “cohomological” methods.

7 Conclusion and acknowledgements

The author had observed in the summer of 1997 that results about p-adic semistable (at p) lifting of representations such as ρ_{Q} above, results of the type proved in [F], together with the classical base change theory of [L] would prove certain cases of mod p descent for Hilbert modular forms. Because of [R], this strategy can now be implemented, resulting in the present work.

The author would like to thank the National University of Singapore for its hospitality and financial support during the period in which this work was carried out.

8 Bibliography

[BM] N. Boston, B. Mazur, Explicit universal deformations of Galois representations, Adv. Studies in Pure Math. 17 (1989), 1–21.

[C] L. Clozel, Sur la théorie de Wiles et le changement de base nonabélien, IMRN (1995), no. 9.

[D] F. Diamond, On deformation rings and Hecke rings, Ann. of Math. (2) 144 (1996), no. 1, 137–166.

[F] K. Fujiwara, Deformation rings and Hecke algebras in the totally real case, preprint.

[HM] H. Hida, Y. Maeda, Non abelian base change for totally real fields, Pacific J. of Math. 1998, 189–217.

[L] R. P. Langlands, Base change for GL_{2}, Princeton University Press, 1980.

[M] B. Mazur, Deforming Galois representations, in Galois Groups over Q, eds. Y. Ihara, K. Ribet, J-P. Serre, Springer (1989), 385–437.

[R] R. Ramakrishna, Deforming Galois representations and the conjectures of Serre and Fontaine-Mazur, same journal.

[S] J-P. Serre, Sur les représentations modulaires de degré 2 de Gal(\overline{Q}/Q), Duke Math. J. 54 (1987), 179–230.

[St] W. Stein, Serre’s conjecture mod pq?, available at http://shimura.math.berkeley.edu/˜ was/Tables/serremodpq/ School of Mathematics, TIFR, Homi Bhabha Road, Mumbai 400 005, INDIA. e-mail: shekhar@math.tifr.res.in