Bence Hevesi: Local-global compatibility at p = ell for CM fields

Given a regular algebraic automorphic representation of $GL(n)$ over a CM field, the celebrated works of Harris--Lan--Taylor--Thorne and Scholze show the existence of a Galois representation whose $L$-function matches that of the automorphic representation. In joint work with Lambert A'Campo, Jack Thorne and Dmitri Whitmore we show that these automorphic Galois representations are de Rham with the expected Hodge—Tate weights and are compatible with the semisimple local Langlands correspondence at $p = \ell$. The novelty of our work is that we impose no conditions on the mod $\ell$ reduction of the Galois representation.