We prove a ‘minimal’ type automorphy lifting theorem for 2-adic Galois representations of unitary type, over imaginary CM fields. We use this to improve an automorphy lifting theorem of Kisin for GL_2.

Given a smooth plane quartic curve C over a field $\textit{k}$ of characteristic 0, with Jacobian variety $\textit{J}$, and a marked rational point P $\in$ C($\textit{k}$), we construct a reductive group $\textit{G}$ and ...

We establish the automorphy of some families of 2-dimensional representations of the absolute Galois group of a totally real field, which do not satisfy the so-called ‘Taylor–Wiles hypothesis’. We apply this to the problem ...

We show that if $\textit{p}$ is a prime, then all elliptic curves de ned over the cyclotomic $\mathbb{Z}$$_{p}$-extension of Q are modular.

We study the problem of variation of Frobenius eigenvalues on the cohomology of families of local systems of algebraic curves over finite fields.

The simplest case of the Langlands functoriality principle asserts the existence of the symmetric powers Symn of a cuspidal representation of GL.2/ over the adèles of F , where F is a number field. In 1978, Gelbart and ...

We study the variation of the $\phi$-Selmer groups of the elliptic curves y$^2$ = x$^3$ − Dx under quartic twists by square-free integers. We obtain a complete description of the distribution of the size of this group when ...

Let π be a cuspidal, cohomological automorphic representation of GL$_{n}$(A). Venkatesh has suggested that there should exist a natural action of the exterior algebra of a certain motivic cohomology group on the π-part of ...

We construct the compatible system of $\textit{l}$-adic representations associated to a regular algebraic cuspidal automorphic representation of GL$_{n}$ over a CM (or totally real) field and check local-global compatibility ...

We study in this paper Hida's p-adic Hecke algebra for GL_n over a CM field F. Hida has made a conjecture about the dimension of these Hecke algebras, which he calls the non-abelian Leopoldt conjecture, and shown that his ...

We construct algebras of endomorphisms in the derived category of the cohomology of arithmetic manifolds, which are generated by Hecke operators. We construct Galois representations with coefficients in these Hecke algebras ...