arXiv subjects
Publications and source records attributed to Daniel Le.
We prove the weight elimination direction of the Serre weight conjectures as formulated by Herzig for forms of $U(n)$ which are compact at infinity and split at places dividing $p$ in generic situations. That is, we show that all modular weights for a mod $p$ Galois representation are contained in the set predicted by Herzig. Under some additional hypotheses, we also show modularity of all the "obvious" weights.
We prove the weight part of Serre's conjecture in generic situations for forms of $U(3)$ which are compact at infinity and split at places dividing $p$ as conjectured by Herzig. We also prove automorphy lifting theorems in dimension three. The key input is an explicit description of tamely potentially crystalline deformation rings with Hodge-Tate weights $(2,1,0)$ for $K/\mathbb{Q}_p$ unramified combined with patching techniques. Our results show that the (geometric) Breuil-Mézard conjectures hold for these deformation rings.
Under hypotheses required for the Taylor-Wiles method, we prove for forms of $U(3)$ which are compact at infinity that the lattice structure on upper alcove algebraic vectors or on principal series types given by the $λ$-isotypic part of completed cohomology is a local invariant of the Galois representation attached to $λ$ when this Galois representation is residually irreducible locally at places dividing $p$. As a crucial input, we establish corresponding mod $p$ multiplicity one results. Our main innovation is the combination of integral Hecke theory and the Taylor--Wiles method.
Suppose that F/F+ is a CM extension of number fields in which the prime p splits completely and every other prime is unramified. Fix a place w|p of F. Suppose that rbar : Gal(F-bar/F) -> GL_3(Fp-bar) is a continuous irreducible Galois representation such that rbar|_{Gal(F_w-bar/F_w)} is upper-triangular, maximally non-split, and generic. If rbar is automorphic, and some suitable technical conditions hold, we show that rbar|_{\Gal(F_w-bar/F_w)} can be recovered from the GL_3(F_w)-action on a space of mod p automorphic forms on a compact unitary group. On the way we prove results about weights in Serre's conjecture for rbar, show the existence of an ordinary lifting of rbar, and prove the freeness of certain Taylor-Wiles patched modules in this context. We also show the existence of many Galois representations rbar to which our main theorem applies.