Related papers: Modularity of some potentially Barsotti-Tate Galoi…

The Galois representation associated to a p-divisible group over a complete noetherian normal local ring with perfect residue field is described in terms of its Dieudonn\'e display. As a corollary we deduce in arbitrary characteristic…

We prove Breuil's lattice conjecture for higher Hodge-Tate weights in the case of $\mathrm{GL}_2(K)$ where $K$ is an unramified extension of $\mathbb{Q}_p$. More precisely, under some genericity conditions, we show that the lattice inside a…

In this paper, a strong multiplicity one theorem for Katz modular forms is studied. We show that a cuspidal Katz eigenform which admits an irreducible Galois representation is in the level and weight old space of a uniquely associated Katz…

Fontaine-Mazur Conjecture is one of the core statements in modern arithmetic geometry. Several formulations were given since its original statement in 1993, and various angles have been adopted by numerous authors to try to tackle it.…

Let $F$ be a CM number field. We prove modularity lifting theorems for regular $n$-dimensional Galois representations over $F$ without any self-duality condition. We deduce that all elliptic curves $E$ over $F$ are potentially modular, and…

We prove the Breuil-Mezard conjecture for split non-scalar residual representations of Gal(Qp/Qp) by local methods. Combined with the cases previously proved in [18] and [24], this completes the proof of the conjecture (when p>3). As a…

Let $p\geq 5$ be a prime. We construct modular Galois representations for which the $\mathbb{Z}_p$-corank of the $p$-primary Selmer group (i.e., $\lambda$-invariant) over the cyclotomic $\mathbb{Z}_p$-extension is large. More precisely, for…

In this article, I discuss material which is related to the recent proof of Fermat's Last Theorem: elliptic curves, modular forms, Galois representations and their deformations, Frey's construction, and the conjectures of Serre and of…

In this paper we study deformations of mod $p$ Galois representations $\tau$ (over an imaginary quadratic field $F$) of dimension $2$ whose semi-simplification is the direct sum of two characters $\tau_1$ and $\tau_2$. As opposed to our…

We propose in this paper an approach to Breuil's conjecture on a Langlands correspondence between $p$-adic Galois representations and representations of $p$-adic Lie groups in $p$-adic topological vector spaces. We suggest that Berthelot's…

Let L/K be a finite Galois extension of number fields with Galois group G. Let p be a rational prime and let r be a non-positive integer. By examining the structure of the p-adic group ring Z_p[G], we prove many new cases of the p-part of…

We define deformation rings for potentially semi-stable deformations of fixed discrete series inertial type in dimension $2$. In the case of representations of the Galois group of $\mathbf{Q}_p$, we prove an analogue of the Breuil-M\'ezard…

We extend the classical duality results of Poitou and Tate for finite discrete Galois modules over local and global fields (local duality, nine-term exact sequence, etc.) to all affine commutative group schemes of finite type, building on…

Let f be a modular eigenform of even weight k>0 and new at a prime p dividing exactly the level, with respect to an indefinite quaternion algebra. The theory of Fontaine-Mazur allows to attach to f a monodromy module D_FM(f) and an…

Let $F$ be a finite extension of $\mathbb{Q}_p$. We determine the Lubin-Tate $(\varphi,\Gamma)$-modules associated to the absolutely irreducible mod $p$ representations of the absolute Galois group ${\rm Gal}(\bar{F}/F)$.

This paper develops the fundamentals of modular representation theory for finite monoids, introducing the decomposition matrix and exploring its connection to Brauer characters. We define modular characteristic and explain how the…

In this chapter, we want to have an overview of the Taylor--Wiles patching method. For this purpose, at the first, we recall Mazur's theory of deforming Galois representations and study both local and global deformation problems. Then, we…

Let X_d be the p-adic analytic space classifying the d-dimensional (semisimple) p-adic Galois representations of the absolute Galois group of Q_p. We show that the crystalline representations are Zarski-dense in many irreducible components…

Let R be a complete discrete valuation ring of mixed characteristic (0, p) with perfect residue field, K the fraction field of R. Suppose G is a Barsotti-Tate group (p-divisible group) defined over K which acquires good reduction over a…

Associated to an abelian variety over a number field are several interesting and related groups: the motivic Galois group, the Mumford-Tate group, $\ell$-adic monodromy groups, and the Sato-Tate group. Assuming the Mumford-Tate conjecture,…