On the images of the Galois representations attached to genus 2 Siegel modular forms

TL;DR

This paper investigates the images of Galois representations linked to genus 2 Siegel cusp forms, demonstrating that they are maximal for most primes under certain conditions, advancing understanding of their symplectic nature.

Contribution

It proves that the Galois representation images are as large as possible for almost all primes for a broad class of genus 2 Siegel cusp forms, under verifiable conditions.

Findings

Images are maximal for almost every prime.

Results apply to non-Maass spezialforms.

Provides criteria for large Galois images.

Abstract

We address the problem of the determination of the images of the Galois representations attached to genus 2 Siegel cusp forms of level 1 having multiplicity one. These representations are symplectic. We prove that the images are as large as possible for almost every prime, if the Siegel cusp form is not a Maass spezialform and verifies two easy to check conditions.