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 …

Let p be an odd prime, ka finite field of characteristic p, and W (k) the ring of Witt vectors of k.
The study of Galois representations p: GQ Gal (Q/Q)-+ GL2 (k) and their deformations to …

We prove a modularity lifting theorem for two dimensional, 2-adic, potentially Barsotti-Tate
representations. This proves hypothesis (H) of Khare-Wintenberger, and completes the proof …

We prove new modularity lifting theorems for p-adic Galois representations in situations
where the methods of Wiles and Taylor–Wiles do not apply. Previous generalizations of …

We prove the existence in many cases of minimally ramified p-adic lifts of 2-dimensional
continuous, odd, absolutely irreducible, mod p representations p̄ of the absolute Galois …

This book provides a comprehensive account of the theory of moduli spaces of elliptic
curves (over integer rings) and its application to modular forms. The construction of Galois …

Let k be a non-archimedean field: a field that is complete with respect to a specified
nontrivial non-archimedean absolute value|·|. There is a classical theory of k-analytic …

We prove a variety of results on the existence of automorphic Galois representations lifting a
residual automorphic Galois representation. We prove a result on the structure of …

Let $ K/\mathbb {Q} _p $ be a finite extension and $ G_K $ the absolute Galois group of $ K
$. For $(A^{\circ},\mathfrak {m}) $ a complete local ring with finite residue and $ V_ {A^{\circ}} …

We prove that some new infinite families of odd two dimensional icosahedral
representations of the absolute Galois group of [inline-graphic xmlns: xlink=" http://www. w3 …