Cohomological Weight Shiftings for Automorphic Forms on Definite Quaternion Algebras

TL;DR

This paper develops methods to shift weights of mod p automorphic forms on definite quaternion algebras over totally real fields, using cohomological maps and intertwining operators, extending previous techniques to new weight configurations.

Contribution

It introduces new weight shifting techniques for automorphic forms on quaternion algebras, including cases with (2,...,2)-blocks, using cohomological and intertwining operator methods.

Findings

Established weight shiftings without (2,...,2)-blocks via cohomological maps.

Extended weight shifting methods to include (2,...,2)-blocks using existing techniques.

Provided explicit constructions of weight shiftings in the automorphic forms setting.

Abstract

Let F/Q be a totally real field extension of degree g and let D be a definite quaternion algebra with center F. Fix an odd prime p which is unramified in F and D. We produce weight shiftings between (mod p) automorphic forms on the multiplicative group of D having fixed level. When the starting weight does not contain any (2,...,2)-block, we obtain these shiftings via maps induced in cohomology by intertwining operators acting on F_{p}-representations of GL(2,O_{F}/pO_{F}). Shiftings by (p-1,...,p-1) for weights containing (2,...,2)-blocks are obtained following the methods of Edixhoven-Khare, as half of our intertwining operators becomes trivial in this case.