An explicit open image theorem for products of elliptic curves

TL;DR

This paper provides an explicit bound on the index of the adelic Galois representation image associated with a product of non-isogenous, non-CM elliptic curves over a number field, extending understanding of their Galois actions.

Contribution

It establishes an explicit bound for the index of the adelic Galois image in a specific subgroup of the product of general linear groups, for a product of elliptic curves with certain properties.

Findings

Explicit bound for the Galois image index

Applicable to non-isogenous, non-CM elliptic curves

Enhances understanding of Galois representations in elliptic curve products

Abstract

Let $K$ be a number field and $E_1, \ldots, E_n$ be elliptic curves over $K$, pairwise non-isogenous over $\overline{K}$ and without complex multiplication over $\overline{K}$. We study the image of the adelic representation of the absolute Galois group of $K$ naturally attached to $E_1 \times \cdots \times E_n$. The main result is an explicit bound for the index of this image in $\left\{ (x_1,\ldots,x_n) \in \operatorname{GL}_2(\hat{\mathbb{Z}})^n \bigm\vert \det x_i = \det x_j \;\; \forall i,j \right\}$.