Galois representations and composite moduli

TL;DR

This paper explores generalizations and applications of the known surjectivity properties of Galois representations attached to elliptic curves over ield, focusing on composite moduli and their prime components.

Contribution

It extends the understanding of Galois representation surjectivity for elliptic curves to more general settings and discusses related applications and variants.

Findings

Surjectivity of ield Galois representations for composite moduli depends on prime components.

Generalizations of the known prime-based surjectivity criterion are provided.

Applications to the study of elliptic curves and Galois groups are discussed.

Abstract

It is known that for any elliptic curve $E/\mathbb{Q}$ and any integer $m$ co-prime to $30,$ the induced Galois representation $\rho_{E,m}: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \longrightarrow \text{GL}_{2}(\mathbb{Z}/m\mathbb{Z})$ is surjective if and only if $\rho_{E,\ell}$ is surjective for any prime $\ell|m.$ In this article, we shall discuss some generalizations, applications, and variants of this phenomenon.