Jack A. Thorne

Level-raising and symmetric power functoriality, III

We apply automorphy lifting techniques to establish new cases of symmetric power functoriality for Hilbert modular forms of regular algebraic weight. The proof is based on a novel application of an automorphy lifting theorem for residually reducible Galois representations.

Automorphy lifting for residually reducible l-adic Galois representations

We prove automorphy lifting theorems for residually reducible Galois representations in the setting of unitary groups over CM fields. Our methods are inspired by those of Skinner–Wiles in the setting of GL2.

Level-raising and symmetric power functoriality, I

As the simplest case of Langlands functoriality, one expects the existence of the symmetric power

Potential Automorphy and the Leopoldt conjecture

S^n(\pi )

On the arithmetic of simple singularities of type E

, where

On the GL

\pi

_{n}

is an automorphic representation of

-eigenvariety and a conjecture of Venkatesh

{\rm GL}(2,{\mathbb{A}})

On the automorphy of

and

l

{\mathbb{A}}