On the classification of geometric families of 4-dimensional Galois representations

TL;DR

This paper classifies certain four-dimensional geometric Galois representations over Q, showing that their images are large under specific conditions, with implications for automorphic forms.

Contribution

It provides a new classification theorem for four-dimensional geometric Galois representations attached to motives, detailing conditions for large image and simplifying cases for semistable families.

Findings

Large image results for almost all mbda in irreducible families

Simplified classification for semistable families

Extension of results to automorphic form attached families

Abstract

We give a classification theorem for certain four-dimensional families of geometric $\lambda$-adic Galois representations attached to a pure motive. More precisely, we consider families attached to the cohomology of a smooth projective variety defined over $\Q$ with coefficients in a quadratic imaginary field, non-selfdual and with four different Hodge-Tate weights. We prove that the image is as large as possible for almost every $\lambda$ provided that the family is irreducible and not induced from a family of smaller dimension. If we restrict to semistable families an even simpler classification is given. A version of the main result is given for the case where the family is attached to an automorphic form.