Patching and multiplicities of p-adic eigenforms

TL;DR

This paper establishes the existence of non-classical p-adic automorphic eigenforms linked to specific Galois representations on definite unitary groups, using advanced patching and geometric techniques.

Contribution

It introduces a novel approach combining patching methods with geometric sheaf comparisons to construct non-classical p-adic eigenforms with critical crystalline Galois representations.

Findings

Existence of non-classical p-adic automorphic eigenforms.

Connection between eigenforms and crystalline Galois representations.

Application of patching techniques with geometric sheaf comparisons.

Abstract

We prove the existence of non-classical $p$-adic automorphic eigenforms associated to a classical system of eigenvalues on definite unitary groups in $3$ variables. These eigenforms are associated to Galois representations which are crystalline but very critical at $p$. We use patching techniques related to the trianguline variety of local Galois representations and its local model. The new input is a comparison of the coherent sheaves appearing in the patching process with coherent sheaves on the Grothendieck--Springer version of the Steinberg variety given by a functor constructed by Bezrukavnikov.