On the images of the Galois representations attached to generic automorphic representations of GSp(4)

TL;DR

This paper demonstrates that Galois representations linked to generic automorphic representations of GSp(4) have large images for most primes, using Langlands functoriality and (n,p)-groups to construct examples with large images at all primes.

Contribution

It establishes the largeness of Galois representation images for generic GSp(4) automorphic forms and constructs explicit examples with large images at every prime.

Findings

Galois representations have large images for almost all primes.

Construction of automorphic representations with large images at all primes.

Application of Langlands functoriality and (n,p)-groups in the analysis.

Abstract

By making use of Langlands functoriality between GSp(4) and GL(4), we show that the images of the Galois representations attached to "genuine" globally generic automorphic representations of GSp(4) are "large" for almost every prime. Moreover, by using the notion of (n,p)-groups (introduced by Khare, Larsen and Savin) and generic Langlands functoriality from SO(5) to GL(4) we construct automorphic representations of GSp(4) such that the compatible system attached to them has large image for all primes.