Images de repr\'esentations galoisiennes associ\'ees \`a certaines formes modulaires de Siegel de genre $2$

TL;DR

This paper investigates the images of $ ext{l}$-adic Galois representations linked to certain vector-valued Siegel modular forms of genus 2, identifying conditions for irreducibility and the size of their images.

Contribution

It determines the primes for which these Galois representations are absolutely irreducible and shows their images are 'full' in most cases, extending understanding of these representations.

Findings

Identifies primes where representations are irreducible

Shows images are 'full' for most primes

Provides exceptions in two specific cases

Abstract

We study the image of the $\ell$-adic Galois representations associated to the four vector valued Siegel modular forms appearing in the work of Chenevier and Lannes. These representations are symplectic of dimension $4$. Following a method of Dieulefait, we determine the primes $\ell$ for which these representations are absolutely irreducible. In addition, we show that their image is "full" for all primes $\ell$ such that the associated residual representation is absolutely irreducible, except in two cases.