Christophe Breuil

On the Modularity of Elliptic Curves Over Q: Wild 3-Adic Exercises

In this paper, building on work of Wiles [Wi] and of Wiles and one of us (R.T.) [TW], we will prove the following two theorems (see §2.2). Theorem A. If E/Q is an elliptic curve, then E is modular. Theorem B. If ρ : Gal(Q/Q) → GL2(F5) is an irreducible continuous representation with cyclotomic determinant, then ρ is modular. We will first remind the reader of the content of these results and then briefly outline the method of proof. If N is a positive integer then we let Γ1(N) denote the subgroup of SL2(Z) consisting of matrices that modulo N are of the form ( 1 ∗ 0 1 ) .

Une application de Corps des Normes

Let k be a perfect field of characteristic p > 0, K0 = Frac(W(k)), π a uniformizer in K0 and πn ∈ K 0 (n∈ N) such that π0 = π and πn+1p = πn. We write K∞ = ∪n∈N K0 (πn), H∞ = Gal (K0/ K∞ and G = Gal(K0/ K0). The main result of this paper is that the functor ‘restriction of the Galois action’ from the category of crystalline representations of G with Hodge–Tate weights in an interval of length ≤ p - 2 to the category of p-adic representations of H∞ is fully faithful and its essential image is stable by sub-object and quotient. The proof uses the comparison between two ways of building mod. p representations of H∞: one thanks to the norm field of K∞, the other thanks to some categories of ‘filtered’ modules with divided powers previously introduced by the author.

Smoothness and Classicality on eigenvarieties

Let p be a prime number and f an overconvergent p-adic automorphic form on a definite unitary group which is split at p. Assume that f is of “classical weight” and that its Galois representation is crystalline at p, then f is conjectured to be a classical automorphic form. We prove new cases of this conjecture in arbitrary dimensions by making crucial use of the patched eigenvariety constructed in Breuil et al. (Math. Ann. 2016).