Deformations of Galois representations and exceptional monodromy, II: raising the level

TL;DR

This paper generalizes level-raising techniques for Galois representations from SL(2) to broader groups, enabling the construction of geometric Galois representations with exceptional monodromy groups for more primes and with more flexible Hodge structures.

Contribution

It extends level-raising methods to general (semi-)simple groups and enhances the construction of geometric Galois representations with exceptional monodromy groups.

Findings

Achieved level-raising results for almost all primes l.

Constructed geometric Galois representations with exceptional monodromy groups for broader primes.

Improved flexibility in Hodge numbers of the lifts.

Abstract

Building on lifting results of Ramakrishna, Khare and Ramakrishna proved a purely Galois-theoretic level-raising theorem for two-dimensional odd representations of the Galois group of Q. In this paper, we generalize these techniques from type A1 to general (semi-)simple groups. We then strengthen our previous results on constructing geometric Galois representations with exceptional monodromy groups, achieving such constructions for almost all l, rather than a density-one set, and achieving greater flexibility in the Hodge numbers of the lifts; the latter improvement requires the new level-raising result.