We extend the results of [CHT] by removing the ‘minimal ramification’ condition on the lifts. That is we establish the automorphy of suitable conjugate self-dual, regular (de Rham with distinct Hodge–Tate numbers), l-adic lifts of certain automorphic mod l Galois representations of any dimension. The main innovation is a new approach to the automorphy of non-minimal lifts which is closer in spirit to the methods of [TW] than to those of [W], which relied on Ihara’s lemma.

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Automorphy for some l-adic lifts of automorphic mod l Galois representations

We prove that, under some mild conditions, a two dimensional p-adic Galois representation which is residually modular and potentially Barsotti-Tate at p is modular. This provides a more conceptual way of establishing the Shimura-Taniyama-Weil conjecture, especially for elliptic curves which acquire good reduction over a wildly ramified extension of ℚ 3 . The main ingredient is a new technique for analyzing flat deformation rings. It involves resolving them by spaces which parametrize finite flat group scheme models of Galois representations. Introduction.

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Moduli of finite flat group schemes, and modularity

Let K/Qp be a finite extension and Gk = Gal(K/K) the Galois group of an alge braic closure K. Let F be a finite field of characteristic p, and Vf a finite dimensional F-vector space equipped with a continuous action of GkThe study of the defor mation theory of Galois representations was initiated by Mazur [Ma], who showed that if Vf has no non-trivial endomorphisms, then it admits a universal deformation ring Ryw. After the work of Wiles [Wi] and the conjectures of Fontaine-Mazur [FM] it became clear that for arithmetic applications it was important to understand cer tain quotients of Ry? corresponding to deformations satisfying certain conditions. For example Wiles uses deformations which arise from finite flat group schemes, and the corresponding quotient of Ryw was constructed by Ramakrishna [Ra]. Suppose that L/K is a finite extension and let a ^ b be integers. It seems to be a kind of folklore conjecture that there should be a quotient of Ry? whose points in finite extensions of W(F)[l/p] correspond to deformations of Vf which become semi-stable over L and have Hodge-Tate weights in the interval [a, b]. This is closely related to the final conjecture of [Fo 3]. Special cases of this, when Vf is 2-dimensional, are conjectured in the papers of Fontaine-Mazur [FM, p. 191], Breuil Conrad-Diamond-Taylor [BCDT, Conj. 1.1.1] and Breuil-M?zard [BM, 2.2.2.4]. The purpose of this paper is to prove such a result.

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Potentially semi-stable deformation rings

Introduction 641 1 Breuil-Mezard conjecture and the p-adic local Langlands 644 (11) The Breuil-Mezard conjecture 644 (12) Review of Colmez’s functor 647 (13) Hilbert-Samuel multiplicities 650 (14) Representations and pseudo-representations 653 (15) GL2(Qp)-representations mod p 657 (16) Local patching and multiplicities 661 (17) From pseudo-representations to representations 669 2 Modularity via the Breuil-Mezard conjecture 675 (21) Quaternionic forms 675 (22) Global patching and multiplicities 677 (23) The Breuil-Mezard conjecture 686 Addendum to [Ki 1] 687 References 689

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The Fontaine-Mazur conjecture for {GL}_2

The motivation at the origin of this article is to investigate some ways of constructing moduli for curves, and covers above them, using tools from stack theory. This idea arose from reading Bertin–Mezard [BeM, esp. Sec. 5] and Abramovich– Corti–Vistoli [ACV]. Our approach is in the spirit of most recent works, where one uses the flexibility of the language of algebraic stacks. This language has two (twin) aspects: category-theoretic on one side and geometric on the other. Some of our arguments, especially in Section 8, are formal arguments involving general constructions concerning group actions on algebraic stacks (this is more on the categoric side). They are, intrinsically, natural enough to preserve the “modular” aspect. In trying to isolate these arguments, we were led to write results of independent interest. It seemed therefore more adequate to present them in a separate, self-contained part. Thus the article is split into two parts of comparable size. More specifically, groups are ubiquitous in algebraic geometry (when one focuses on curves and maps between them, examples include the automorphism group, fundamental group, monodromy group, permutation group of the ramification points, . . .). It is natural to ask whether we can handle group actions on stacks in the same fashion as we do on schemes. For example, we expect: that the quotient of the stack of curves with ordered marked points Mg,n by the symmetric group should classify curves with unordered marked points; that if G acts on a scheme X then the fixed points of the stack Pic(X) under G should be related to G-linearized line bundles on X; and that the quotient of the modular stack curve X1(N ) by (Z/NZ)× should be X0(N ) (the notation is, we hope, well known to the reader). Other important examples appear in the literature: action of tori on stacks of stable maps in Gromov–Witten theory [Ko; GrPa], and action of the symmetric group Sd on a stack of multisections in [L-MB, (6.6)]. Our aim is to provide the material necessary to handle the questions raised here and then answer them, as well as to give other applications. Let us now explain in more detail the structure and results of this paper. In Part A we discuss the notion of a group action on a stack. We are mainly interested in giving general conditions under which the fixed points and the quotient of an algebraic stack are algebraic. In Sections 1 and 2 we give definitions and basics on actions. For simplicity let us now consider a flat group scheme G and an algebraic stack M, both of finite presentation (abbreviated fp) over some

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Group Actions on Stacks and Applications

We formulate a conjecture giving a link between the various rings parametrizing the $2$-dimensional potentially semistable $p$-adic representations of ${\rm Gal}(\overline {\mathbf {Q}}\sb p/\mathbf {Q}\sb p)$ with Hodge-Tate weights $(0,k-1)(k\in \mathbf {Z},1<k<p)$ having the same reduction modulo $p$ and the representations of ${\rm GL}\sb 2(\mathbf {Z}\sb p)$ that are used, via compact induction, to build the smooth irreducible representations of ${\rm GL}\sb 2(\mathbf {Q}\sb p)$. We prove this conjecture for semistable representations and $k$ even. In doing this, we obtain precise results on the restriction to ${\rm Gal}(\overline {\mathbf {Q}}\sb p/\mathbf {Q}\sb p)$ of the representations of ${\rm Gal}(\overline {\mathbf {Q}}/\mathbf {Q})$ over $\overline {\mathbf {F}}\sb p$ that are associated to modular forms on $\Gamma\sb 0(pN)(p\nmid N)$ of weight smaller than $p$. In an appendix, G. Henniart determines which smooth irreducible representations $\lambda$ of ${\rm GL}\sb 2(\mathfrak {O}\sb F)$ are typical for ${\rm GL}\sb 2(F)$ (where $F$ is a locally compact non-Archimedean field and $\mathfrak {O}\sb F$ its ring of integers), in the sense that there is a component $s(\lambda)$ in the Bernstein decomposition (for the category of smooth representations of ${\rm GL}\sb 2(F)$) such that $\lambda$ appears only in smooth representations of ${\rm GL}\sb 2(F)$ belonging to $s(\lambda)$. We need Henniart's results, in the case $F=\mathbf {Q}\sb p$, to state the aforementioned conjecture.

Multiplicités modulaires et représentations de ${\rm GL}\sb 2(\mathbf {Z}\sb p)$ et de ${\rm Gal}(\overline {\mathbf {Q}}\sb p/\mathbf {Q}\sb p)$ en $\ell=p$. Appendice par Guy Henniart. Sur l'unicité des types pour ${\rm GL}\sb 2$

In this paper we study formal moduli for wildly ramified Galois covering. We prove a local-global principle. We then focus on the infinitesimal deformations of the Z/pZ-covers. We explicitly compute a deformation of an automorphism of order p which implies a universal obstruction for p>2. By deforming Artin-Schreier equations we obtain a lower bound on the dimension of the local versal deformation ring. At last, by comparing the global versal deformation ring to the complete local ring in a point of a moduli space, we determine the dimensions of the global and local versal deformation rings.

Déformations formelles des revêtements sauvagement ramifiés de courbes algébriques

We classify the filtered modules with coefficients corresponding to two-dimensional potentially semi-stable p-adic representations of the Galois group of Qp under some assumptions (e.g., p is odd). We focus on the new features that arise when the coefficients are not necessarily Qp.

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Filtered modules with coefficients

Multiplicités modulaires raffinées

Let F be an unramified extension of Qp. The first aim of this work is to develop a purely local method to compute the potentially Barsotti-Tate deformations rings with tame Galois type of irreducible two-dimensional representations of the absolute Galois group of F. We then apply our method in the particular case where F has degree 2 over Q_p and determine this way almost all these deformations rings. In this particular case, we observe a close relationship between the structure of these deformations rings and the geometry of the associated Kisin variety. As a corollary and still assuming that F has degree 2 over Q_p, we prove, except in two very particular cases, a conjecture of Kisin which predicts that intrinsic Galois multiplicities are all equal to 0 or 1.

Ariane Mézard

Papers

Multiplicités modulaires et représentations de ${\rm GL}\sb 2(\mathbf {Z}\sb p)$ et de ${\rm Gal}(\overline {\mathbf {Q}}\sb p/\mathbf {Q}\sb p)$ en $\ell=p$. Appendice par Guy Henniart. Sur l'unicité des types pour ${\rm GL}\sb 2$

Déformations formelles des revêtements sauvagement ramifiés de courbes algébriques

Filtered modules with coefficients

Multiplicités modulaires raffinées

Un calcul d'anneaux de d\'eformations potentiellement Barsotti--Tate