Lattices in the cohomology of $U(3)$ arithmetic manifolds

TL;DR

This paper proves that, under certain hypotheses, the lattice structure in the cohomology of compact $U(3)$ arithmetic manifolds is a local invariant of the attached Galois representation, combining integral Hecke theory with the Taylor--Wiles method.

Contribution

It introduces a novel approach combining integral Hecke theory with the Taylor--Wiles method to establish local invariance of lattice structures in cohomology for $U(3)$ forms.

Findings

Lattice structures are local invariants of Galois representations under specified conditions.

Established mod p multiplicity one results for these cohomology spaces.

Combined integral Hecke theory with Taylor--Wiles method effectively.

Abstract

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 $\lambda$-isotypic part of completed cohomology is a local invariant of the Galois representation attached to $\lambda$ 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.