Deformation of rigid Galois representations and cohomology of certain quaternionic unitary Shimura variety

TL;DR

This paper investigates the deformation theory of Galois representations in the symplectic group to establish a freeness result for the cohomology of specific quaternionic unitary Shimura varieties, aiding in arithmetic level raising proofs.

Contribution

It introduces a new deformation-theoretic approach to prove cohomology freeness for quaternionic unitary Shimura varieties, crucial for level raising theorems.

Findings

Freeness of cohomology over universal deformation rings established.

Application to arithmetic level raising for symplectic groups.

New deformation techniques for Galois representations in symplectic groups.

Abstract

In this article, we use deformation theory of Galois representations valued in the symplectic group of degree four to prove a freeness result for the cohomology of certain quaternionic unitary Shimura variety over the universal deformation ring for certain type of residual representation satisfying a property called rigidity. This result plays an important role in the proof of the arithmetic level raising theorem for the symplectic similitude group of degree four over the field of rational numbers by the author.