The Weight Part of Serre’s Conjecture over CM Fields

The coefficient space is a kind of resolution of singularities of the universal flat deformation space for a given Galois representation of some local field. It parametrizes (in some sense) the finite flat models for the Galois representation. The aim of this note is to determine the image of the coefficient space in the universal deformation space.

We prove the modularity of minimally ramified ordinary residually reducible p-adic Galois representations of an imaginary quadratic field F under certain assumptions. We first exhibit conditions under which the residual representation is unique up to isomorphism. Then we prove the existence of deformations arising from cuspforms on GL2(AF) via the Galois representations constructed by Taylor et al. We establish a sufficient condition (in terms of the non-existence of certain field extensions which in many cases can be reduced to a condition on an L-value) for the universal deformation ring to be a discrete valuation ring and in that case we prove an R=T theorem. We also study reducible deformations and show that no minimal characteristic 0 reducible deformation exists.

In [10] (C R Acad Sci Paris Ser I Math 323(2) 117–120, 1996), [11] (Math Res Lett 10(1):71–83 2003), [12] (Can J Math 57(6):1215–1223 2005), Khare showed that any strictly compatible systems of semisimple abelian mod p Galois representations of a number field arises from a unique finite set of algebraic Hecke characters. In this article, we consider a similar problem for arbitrary global fields. We give a definition of Hecke character which in the function field setting is more general than previous definitions by Goss and Gross and define a corresponding notion of compatible system of mod p Galois representations. In this context we present a unified proof of the analog of Khare’s result for arbitrary global fields. In a sequel we shall apply this result to strictly compatible systems arising from Drinfeld modular forms, and thereby attach Hecke characters to cuspidal Drinfeld Hecke eigenforms.

For a fixed mod $p$ automorphic Galois representation, $p$-adic automorphic Galois representations lifting it determine points in universal deformation space. In the case of modular forms and under some technical conditions, Bockle showed that every component of deformation space contains a smooth modular point, which then implies their Zariski density when coupled with the infinite fern of Gouvea-Mazur. We generalize Bockle's result to the context of polarized Galois representations for CM fields, and to two dimensional Galois representations for totally real fields. More specifically, under assumptions necessary to apply a small $R = \mathbb{T}$ theorem and an assumption on the local mod $p$ representation, we prove that every irreducible component of the universal polarized deformation space contains an automorphic point. When combined with work of Chenevier, this implies new results on the Zariski density of automorphic points in polarized deformation space in dimension three.

We obtain a criteria for a pure sheaf to be semisimple. As a corollary, we prove the following: Let X 0 X_0 and S 0 S_0 be two schemes over a finite field F q \mathbf {F}_q , and let f 0 : X 0 → S 0 f_0: X_0\rightarrow S_0 be a proper smooth morphism. Assume S 0 S_0 is normal and geometrically connected, and assume there exists a closed point s s in S 0 S_0 such that the Frobenius automorphism F s F_s acts semisimply on H i ( X s ¯ , Q l ¯ ) H^i(X_{\bar s}, {\overline {\mathbf {Q}_l}}) , where X s ¯ X_{\bar s} is the geometric fiber of f 0 f_0 at s s (this last assumption is unnecessary if the semisimplicity conjecture is true). Then R i f 0 ∗ Q l ¯ R^if_{0\ast } {\overline {\mathbf {Q}_l}} is a semisimple sheaf on S 0 S_0 . This verifies a conjecture of Grothendieck and Serre provided the semisimplicity conjecture holds. As an application, we prove that the galois representations of function fields associated to the l l -adic cohomologies of K 3 K3 surfaces are semisimple. We also get a theorem of Zarhin about the semisimplicity of the Galois representations of function fields arising from abelian varieties. The proof relies heavily on Deligne’s work on Weil conjectures.

This article deals with universal deformations of dihedral representations with a particular focus on the question when the universal deformation is dihedral. Results are obtained in three settings: (1) representation theory, (2) algebraic number theory, (3) modularity. As to (1), we prove that the universal deformation is dihedral if all infinitesimal deformations are dihedral. Concerning (2) in the setting of Galois representations of number fields, we give sufficient conditions to ensure that the universal deformation relatively unramified outside a finite set of primes is dihedral, and discuss in how far these conditions are necessary. As side-results, we obtain cases of the unramified Fontaine–Mazur conjecture, and in many cases positively answer a question of Greenberg and Coleman on the splitting behaviour at p of p-adic Galois representations attached to newforms. As to (3), we prove a modularity theorem of the form ‘ $$R=\mathbb {T}$$ ’ for parallel weight one Hilbert modular forms for cases when the minimal universal deformation is dihedral.

In this article, we prove the non-existence of certain semistable Galois representations of a number field and apply our results to some geometric problems. For example, we prove a special case of a conjecture of Rasmussen and Tamagawa, related with the finiteness of the set of isomorphism classes of abelian varieties with constrained prime power torsion.

We describe the generic blocks in the category of smooth locally admissible mod $2$ representations of $\mathrm{GL}_2(\mathbb{Q}_2)$. As an application we obtain new cases of Breuil--M\'ezard and Fontaine--Mazur conjectures for $2$-dimensional $2$-adic Galois representations.

The irreducible orthogonal and symplectic Galois representations of a p-adic field (the tame case)

We construct infinitely ramified Galois representations ρ such that the al(ρ)'s have distributions in contrast to the statements of Sato–Tate, Lang–Trotter and others. Using similar methods we deform a residual Galois representation for number fields and obtain an infinitely ramified representation with very large image, generalizing a result of Ramakrishna.

The semisimplicity conjecture says that for any smooth projective scheme $X_0$ over a finite field $\mathbf {F}_q$, the Frobenius correspondence acts semisimply on $H^i(X\otimes _{\mathbf { F}_q} \mathbf { F}, \overline {\mathbf { Q}}_l)$, where $\mathbf { F}$ is an algebraic closure of $\mathbf { F}_q$. Based on the works of Deligne and Laumon, we reduce this conjecture to a problem about the Galois representations of function fields. This reduction was also achieved by Laumon a few years ago (unpublished).

We establish new cases of quadratic number fields [Formula: see text] unramified away from a prime [Formula: see text] and [Formula: see text] whose absolute Galois group has no irreducible two-dimensional continuous Galois representations in [Formula: see text]. Our work builds on methods of Moon–Taguchi and Şengün and the usual analytic techniques of Odlyzko and Poitou where we note one of the new conditional cases arises via a correction of Poitou’s estimate. The results here seem optimal in that it seems these methods alone will yield no further cases either due to prohibitive computational issues or a failure of the analytic obstructions.

Let F be a totally real field and p be an odd prime which splits completely in F. We show that a generic p-ordinary non-CM primitive Hilbert modular cuspidal eigenform over F of parallel weight two or more must have a locally non-split p-adic Galois representation, at at least one of the primes of F lying above p. This is proved under some technical assumptions on the global residual Galois representation. We also indicate how to extend our results to nearly ordinary families and forms of non-parallel weight.

We study deformation theory of mod p Galois representations of p-adic fields with values in generalised tori, such as L-groups of (possibly non-split) tori. We show that the corresponding deformation rings are formally smooth over a group algebra of a finite abelian p-group. We compute their dimension and the set of irreducible components.

We give a criterion for two \ell -adic Galois representations of an algebraic number field to be isomorphic when restricted to a decomposition group, in terms of the global representations mod \ell . This is applied to prove a generalization of a conjecture of Rasmussen-Tamagawa under a semistablity condition, extending some results of one of the authors. It is also applied to prove a congruence result on the Fourier coefficients of modular forms.