RESEARCH PROPOSAL: MODULARITY OF GEOMETRIC GALOIS REPRESENTATIONS

DAVID SAVITT

1. Aims and strategies

My research is part of the effort to answer the following fundamental question of arithmetic algebraic geometry: when can a representation of the absolute Galois group of Q be found in geometry? To make this question more precise, observe that for a projective algebraic variety V over Q, the l-adic étale cohomology groups of V are finite-dimensional l-adic vector spaces which carry a natural action of G_{Q} = Gal(\overline{Q}/Q); this is our geometric source of Galois representations. The elusive object of desire, then, is a conjecture due to Fontaine and Mazur [FM95]:

Conjecture A: Suppose that ρ : G_{Q} → GL_{n}(\overline{Q}_{l}) is a continuous irreducible representation such that

1. ρ is ramified at only finitely many primes, and

2. the restriction of ρ to a representation of G_{Q_{l}} = Gal(\overline{Q}_{l}/Q_{l}) is potentially

semi-stable. Then there exists a smooth projective variety V defined over Q such that ρ is a Tate twist of a subquotient of the l-adic étale cohomology of V.

The converse to this conjecture is known: any Galois representation arising from geometry in this fashion is indeed ramified at only finitely many primes, and its restriction to G_{Q_{l}} is potentially semi-stable. (I will define potential semi-stability in the next section.)

The case n = 1 in the above conjecture is known, and amounts to class field theory. Recently, Richard Taylor [Tay00] proved a number of cases of the conjecture for n = 2. Until Taylor’s result, all of the known cases of Conjecture A were verifications of the following (stronger) conjecture, which combines the Fontaine-Mazur conjecture with the expectations of the Langlands program:

Conjecture B: Let ρ : G_{Q} → GL_{2}(\overline{Q}_{l}) be a continuous irreducible representation satisfying

1. ρ is ramified at only finitely many primes,

2. the restriction of ρ to a representation of G_{Q_{l}} = Gal(\overline{Q}_{l}/Q_{l}) is potentially semi-stable, and

3. det ρ(c) = −1, where c is any choice of complex conjugation. Then some Tate twist of ρ is modular; in other words, ρ has a Tate twist which is the representation associated to a modular form via the methods of [Shi71, Del71, DS74].

The work of Wiles [Wil95] and Taylor-Wiles [TW95] in proving Fermat’s Last Theorem, and the continuation of this work by Breuil, Conrad, Diamond, and Taylor [Dia96, CDT99, BCDT] to prove the Shimura-Taniyama conjecture, all involve demonstrating special cases of Conjecture B. My research is geared towards proving additional cases of this conjecture.

The aforementioned authors have proved cases of Conjecture B by employing a strategy that can be divided roughly into three steps:

1. Show that \overline{ρ}, the mod l reduction of ρ, is modular.

2. Prove results of the form “Under certain hypotheses, the modularity of \overline{ρ} implies the modularity of ρ”.

3. Verify these “certain hypotheses” for the cases in question.

For example, Langlands [Lan80] and Tunnell [Tun81] proved results of the form (1), and further results of this form are known for l = 3 and l = 5. Wiles, Taylor-Wiles, Conrad-Diamond-Taylor, and Skinner-Wiles [SW97] demonstrated results of the form (2). The difficulty of step (3) should not be underestimated; for example, it comprises the bulk of the work in [BCDT].

Indeed, it is on this sort of verification that my current work is focused. Very roughly, this entails showing that certain Galois deformation rings have one-dim-ensional tangent spaces. Since the Galois representations under consideration arise from finite flat group schemes, understanding these deformation rings involves extensive calculation with such group schemes. After providing the necessary background material, I will give a detailed description of this work and related potential areas of research.

2. Background: anatomy of l-adic Galois representations

2.1. l-adic representations. Fontaine (see [Fon94] or [FI93]) has constructed period rings B_{HT}, B_{dR}, B_{st}, and B_{cris} (respectively, the Hodge-Tate, de Rham, semi-stable, and crystalline period rings) in order to prove comparison theorems on cohomology. For each of these rings, he defines a functor on the category of l-adic representations of G_{K} = Gal(\overline{Q}_{l}/K) for any finite extension K of Q_{l}:

D_{X}(ρ) = (B_{X} ⊗_{Q_{l}} ρ)^{G_{K}}

for each X ∈ {HT, dR, st, cris}. Let K_{X} be the field of invariants (B_{X})^{G_{K}}; in other words, K_{X} is the image under D_{X} of the trivial representation. If K_{0} denotes the maximal unramified subfield of K, then K_{HT} = K_{DR} = K and K_{st} = K_{cris} = K_{0}. The map

α_{X} : B_{X} ⊗_{K_{X}} D_{X}(V) → B_{X} ⊗_{Q_{l}} V

is always injective. We say that the representation V is Hodge-Tate, de Rham, semistable, or crystalline if the corresponding map α_{X} is an isomorphism, i.e., if there is an equality

dim_{K_{X}} D_{X}(ρ) = dim_{Q_{l}} ρ.

Finally, to say that a representation of G_{K} is potentially semi-stable or crystalline is to say that there is a finite extension L/K such that the restriction to a representation of G_{L} is, respectively, semi-stable or crystalline. These properties fit into the following hierarchy:

pot. crystalline ⟹ pot. semi-stable ⟹ de Rham ⟹ Hodge-Tate.

Geometrically, the l-adic étale cohomology of a variety possessing a semi-stable model over the ring of integers O_{K} is always a semi-stable representation. Crystalline representations should be thought of as coming from varieties having smooth models.

The advantage of working with representations having these properties is that the period rings B_{X} have structure which carry over to D_{X}(ρ); via the isomorphism α_{X}, the structure on D_{X}(ρ) yields information about ρ. Moreover, the D_{X}(ρ) are essentially linear algebra objects, and hence easier to compute with than the Galois representations ρ. For example, B_{HT} is graded, so for any two-dimensional Hodge-Tate representation ρ, the two-dimensional vector space D_{HT} (ρ) is also graded. Hence we can assign to ρ a pair of integers called the Hodge-Tate type of ρ: the degrees of the two (possibly equal) nonzero graded pieces of D_{HT} (ρ). If ρ is potentially semi-stable, hence Hodge-Tate, its Hodge-Tate numbers (r, s) have the following significance: in Conjecture B, one would expect the modular form from which ρ arises to have weight s−r+1. (In Conjecture B, representations with lesser Hodge-Tate number equal to 0 are themselves expected to be modular. Giving ρ a Tate twist by n translates both Hodge-Tate numbers of ρ by n, and so indeed we see that any semi-stable representation should be a Tate twist of a modular one.)

2.2. Galois types and deformation problems. The ring B_{st} carries a monodromy operator N and a Frobenius-semilinear endomorphism ϕ; if ρ is semi-stable and the monodromy operator N vanishes on V_{st}(ρ), then ρ is crystalline. If ρ is a two-dimensional Galois representation which becomes semi-stable over the field K, we use the additional structure on V_{st}(ρ) to construct a new representation W D(ρ) : W_{l} → GL_{2}(\overline{Q}_{l}), where W_{l} is the Weil subgroup of G_{l}. The isomorphism class of the representation

τ(ρ) = W D(ρ) |_{I_{l}}

is called the Galois type of ρ.

Let \overline{ρ} : G_{Q_{l}} → GL_{2}(k) be a representation over a finite field k of characteristic l, and assume that the only matrices which commute with the image of \overline{ρ} are scalar matrices. Fix a positive integer k and a Galois type τ. We are interested in lifts ρ : G_{l} → GL_{2}(\overline{Q}_{l}) of \overline{ρ} with the following properties, which can be thought of as properties coming from modular forms:

1. ρ is potentially semi-stable with Hodge-Tate weights (0, k − 1),

2. τ(ρ) is isomorphic to τ, and

3. det(ρ) = ω^{k−1}χ, where ω denotes the l-adic cyclotomic character on G_{Q_{l}} and χ is a character of finite order prime to l.

Denote by R(\overline{ρ}, k, τ) the quotient of the universal deformation ring of \overline{ρ} which parametrizes representations of the above type. This deformation ring is of particular interest because [CDT99] and [BCDT] use the methods of [Wil95] and [TW95] to prove results of roughly the following form: if ρ is a potentially semi-stable l-adic representation of G_{Q} of Galois type τ and Hodge-Tate weights (0, k − 1) and such that \overline{ρ} is modular, and if there is a surjection W (k)[[X]] ↠ R(\overline{ρ}, k, τ), then ρ is modular. (W (k) is the Witt ring of k.)

We record the following facts: if ρ is potentially semi-stable and τ(ρ) is not scalar, then the monodromy operator N must vanish and so ρ is potentially crystalline. Moreover, ρ is potentially crystalline with Hodge-Tate weights (0, 1) if and only if ρ is potentially Barsotti-Tate, i.e., some restriction ρ |_{G_{K}} is the representation on the Tate module of an l-divisible group over the ring of integers O_{K}.

3. Current and Potential Research

3.1. Conjectures of Conrad, Diamond, and Taylor. Let ω̃ and ω̃_{2} denote, respectively, the Teichmüller lifts of the fundamental characters of level 1 and 2 of the inertia group I_{l}. For Galois types τ of the form ω̃^{i} ⊕ ω̃^{j} and ω̃_{2}^{m} ⊕ ω̃_{2}^{ml}, Conrad, Diamond, and Taylor ([CDT99], Conjectures 1.2.2 and 1.2.3) give a precise list of those \overline{ρ} for which they expect there to exist a surjection W (k)[[X]] ↠ R(\overline{ρ}, 2, τ); in other words, for these τ they list the residual representations \overline{ρ} for which the Taylor-Wiles method currently succeeds. I am currently working on verifying these conjectures, using methods akin to the computations carried out by Conrad and Diamond in [BCDT]; I expect results in this direction to form the core of my doctoral dissertation. For each such result, I will have obtained proof of new examples of Conjecture B, and therefore of the Fontaine-Mazur conjecture, conditional on the modularity of \overline{ρ} (which is known in a great number of cases for l = 3 and l = 5).

The idea is to replace R(\overline{ρ}, 2, τ) with a deformation ring R′ which surjects onto R(\overline{ρ}, 2, τ) but whose tangent space can be found more easily, and then to tighten R′ until the ring is as small as needed. Suppose τ is non-scalar. Since we have taken k = 2, we are considering deformations ρ which are potentially Barsotti-Tate. The phenomenal recent work of C. Breuil [Bre99b] gives an equivalence of categories between the category of finite flat groups schemes over O_{K} killed by l, and a certain category of linear-algebraic objects which have become known as Breuil modules; specifically, it is now tractable to perform computations involving finite flat group schemes over rings with arbitrary ramification index. Such computations are wellsuited to finding tangent spaces of deformation rings. We would therefore prefer to work with deformations ρ having constraints involving finite flat group schemes instead of l-divisible groups.

To accomplish the shift to working with finite flat group schemes instead of ldivisible groups, suppose ρ |_{G_{K}} is Barsotti-Tate, so that ρ |_{G_{K}} is the representation on the Tate module of an l-divisible group G over O_{K}. Then the l^{n}-torsion G[l^{n}] of G is a finite flat group scheme over O_{K} whose generic fibre, via the equivalence between finite group schemes over K and representations of G_{K}, corresponds to a Galois representation filtered by \overline{ρ} |_{G_{K}}; moreover, due to the restriction of ρ from G_{Q_{l}} to G_{K}, the integral group scheme G[l^{n}] obtains descent data. Hence, instead of working with potentially Barsotti-Tate deformations, we may concern ourselves instead with the larger collection of deformations ρ such that the Artinian quotients of ρ |_{G_{K}} have finite flat group scheme (integral) models with the above properties: namely, a filtration by suitable models for \overline{ρ} |_{G_{K}}, plus descent data. We then impose additional constraints on our enlarged deformation ring in order to limit ourselves to deformations with our fixed Galois type τ and satisfying the determinant condition.

The task is therefore to identify the aforementioned suitable integral models for \overline{ρ} |_{G_{l}}, plus descent data, and to compute the dimension of the tangent space of the enlarged deformation ring R′ using the corresponding Breuil modules. We wish to show that the dimension of the tangent space is at most 1. This is exactly the strategy which was successful in [BCDT].

One difficulty is that in almost all cases the functor from group schemes to Breuil modules is sufficiently involved that one does not compute it directly. Moreover, the functor depends heavily on a choice of uniformizer, meaning that descent data (which is essentially change-of-uniformizer data) can be extremely complicated. For example, there are situations in which, given the Breuil module of a group scheme G over O_{K} with descent data, I need to know to which representation of G_{Q_{l}} the generic fibre of G descends. Instead of working directly with the Breuil module functor, I observe that the descent data can be studied on the closed fibre without serious loss of information. Therefore, I am able carry out the desired calculation using a careful examination of the descent data on the relevant Dieudonné modules.

At the moment, I am performing the desired computations for a special case of the conjectures of [CDT99]: namely, for τ = ω̃_{2}^{m} ⊕ ω̃_{2}^{lm} and \overline{ρ} : G_{l} → GL_{2}(F_{l}) with

\overline{ρ} = (\begin{matrix}ω^{i}χ_{1} & ∗ \\ 0 & ω^{j}χ_{2} \end{matrix}) ,

where ω denotes the mod l reduction of the cyclotomic character of G_{Q_{l}}, χ_{1} and χ_{2} are unramified characters, and i and j are integers. The reason for restricting to this special case is that to perform the desired computations for residual representations over extensions F_{l^{n}}, one should probably work with Breuil modules with coefficients, i.e., with a theory of Breuil modules suited for working specifically with F_{l^{n}}-vector space schemes. Such a theory has been developed in [BCDT] for the simpler situation when F_{l^{n}} and the residue field of K are linearly disjoint, which unfortunately would not be the case in the necessary computations (since both would contain F_{l^{2}}). I intend to develop this theory in general, in order to enable a proof of the full conjectures of [CDT99].

3.2. Conjectures of Breuil and Mézard. Earlier this year, Breuil and Mézard [BM00] proposed an extension of the conjectures of Conrad, Diamond, and Taylor. Using a result of Henniart associating to each Galois type τ a unique \overline{Q}_{l}- representation σ(τ) of GL_{2}(Z_{l}), Breuil and Mézard perform computations using σ which allow them to make specific predictions for R(\overline{ρ}, k, τ) for 2 ≤ k ≤ l − 1. Moreover, they provide a proof of their conjecture in case τ is scalar and k is even. (Notice that the conjectures from [CDT99] are a portion of the k = 2, τ non-scalar case of the Breuil-Mézard conjectures.)

Breuil and Mézard use methods which are somewhat different from those of [BCDT], computing with Fontaine’s weakly admissible filtered modules (the category of which is equivalent to the category of semi-stable representations [CF00]) as well as with Breuil’s strongly divisible modules [Bre99a]. Moreover, they indicate that their methods should yield a proof of Conjecture 1.2.2 of [CDT99]. I would like to try applying their techniques to Conjecture 1.2.3 of [CDT99], as well as to higher-weight cases of the Breuil-Mézard conjectures.

3.3. Extending the Taylor-Wiles argument. Earlier, we noted that at present the Taylor-Wiles argument only succeeds in giving results of the type “\overline{ρ} modular implies ρ modular” when there is a surjection W (k)[[X]] ↠ R(\overline{ρ}, k, τ). Wiles has made a suggestion which, if it succeeds, could lead to similar results for cases when W (k)[[X, Y]]/(XY − l) ↠ R(\overline{ρ}, k, τ). I intend to explore Wiles’ suggestion, in the hope of proving such a result.

For example, one case of Conjecture 1.2.2 of [CDT99] is of this type, and so the proof of that conjecture combined with an extension of the Taylor-Wiles argument would yield new modularity theorems. Moreover, the Breuil-Mézard conjectures indicate additional examples for which there should be a surjection W (k)[[X, Y]]/(XY − l) ↠ R(\overline{ρ}, k, τ), and so these are the portions of the Breuil-Mézard conjectures which I will attempt to resolve next.

References

[BCDT] C. Breuil, B. Conrad, F. Diamond, and R. Taylor, On the modularity of elliptic curves over Q, preprint.

[BM00] C. Breuil and A. Mézard, Multiplicités modulaires et représentations de GL_{2}(ℤ_{p}) et de Gal(\overline{Q}_{p}/Q_{p}) en l = p, preprint, 2000.

[Bre99a] C. Breuil, Représentations semi-stables et modules fortement divisibles, Invent. math. 136 (1999), no. 1, 89–122.

[Bre99b] C. Breuil, Schémas en groupe et modules filtrés, C. R. Acad. Sci. Paris. Sér. I Math. 328 (1999), no. 2, 93–97.

[CDT99] B. Conrad, F. Diamond, and R. Taylor, Modularity of certain potentially Barsotti-Tate Galois representations, J.A.M.S. 12 (1999), 521–567.

[CF00] P. Colmez and J.-M. Fontaine, Construction des représentations p-adiques semi-stables, Invent. math. 140 (2000), no. 1, 1–43.

[Del71] P. Deligne, Formes modulaires et représentations l-adiques, Séminaire Bourbaki, Exposés 347–363, Lecture Notes in Mathematics, vol. 179, Springer-Verlag, 1971, pp. 139– 172.

[Dia96] F. Diamond, On deformation rings and Hecke rings, Ann. Math. 144 (1996), 137–166.

[DS74] P. Deligne and J-P. Serre, Formes modulaires de poids 1, Ann. Sci. Ec. Norm. Sup. 7 (1974), 507–530.

[FI93] J.-M. Fontaine and L. Illusie, p-adic periods: a survey, Proceedings of the Indo-French Conference on Geometry, Hindustan Book Agency, 1993, pp. 57–93.

[FM95] J.-M. Fontaine and B. Mazur, Geometric Galois Representations, Elliptic curves, modular forms, & Fermat’s last theorem (J. Coates and S.-T. Yau, eds.), International Press, 1995, pp. 41–78.

[Fon94] J.-M. Fontaine, Representations p-adiques semi-stables, Asterisque 223 (1994), 113–184.

[Lan80] R. P. Langlands, Base Change for GL(2), Annals of Math. Studies 96 (1980).

[Shi71] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton University Press, 1971.

[SW97] C. Skinner and A. Wiles, Ordinary representations and modular forms, Proc. Nat. Adam. Sci. U.S.A. 94 (1997), no. 20, 10520–10527.

[Tay00] R. Taylor, Remarks on a conjecture of Fontaine and Mazur, preprint, 2000.

[Tun81] J. Tunnell, Artin’s Conjecture for representations of octahedral type, Bull. AMS 5 (1981), 173–175.

[TW95] R. Taylor and A. Wiles, Ring-theoretic properties of certain Hecke algebras, Annals of Mathematics 142 (1995), 553–572.

[Wil95] A. Wiles, Modular elliptic curves and Fermat’s last theorem, Annals of Mathematics 142 (1995), 443–551.