ON THE AUTOMORPHY OF l-ADIC GALOIS REPRESENTATIONS WITH SMALL RESIDUAL IMAGE

Automorphy for some l-adic lifts of automorphic modl Galois representations. II

AUTOMORPHY FOR SOME l-ADIC LIFTS OF AUTOMORPHIC

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 spi...

On Symmetric Power L-invariants of Iwahori Level Hilbert Modular Forms

We compute the arithmetic L-invariants (of Greenberg–Benois) of twists of symmetric powers of p-adic Galois representations attached to Iwahori level Hilbert modular forms (under some technical conditions). Our method uses the automorphy of symmetric powers and the study of analytic Galois representations on p-adic families of automorphic forms over symplectic and unitary groups. Combining thes...

On the Modularity of Wildly Ramified Galois Representations

where GQ = Gal ( Q/Q ) is the absolute Galois group of Q and ` is a fixed rational prime. For example, ρ = ρE,` may be the `-adic representation of an elliptic curve E over Q, or ρ = ρf may be the `-adic representation associated to a modular form. The continuity of such Galois representations implies the image lies in GL2(O) for some ring of integers O with maximal ideal λ in a finite extensio...

Proving modularity for a given elliptic curve over an imaginary quadratic field

We present an algorithm to determine if the L-series associated to an automorphic representation and the one associated to an elliptic curve over an imaginary quadratic field agree. By the work of Harris-Soudry-Taylor, Taylor, and Berger-Harcos, we can associate to an automorphic representation a family of compatible -adic representations. Our algorithm is based on Faltings-Serre’s method to pr...