Elliptic curves with surjective adelic Galois representations

TL;DR

This paper establishes criteria for when elliptic curves over number fields have surjective adelic Galois representations and provides an explicit example demonstrating this property.

Contribution

It offers necessary and sufficient conditions for surjectivity and constructs a concrete example of an elliptic curve with this property.

Findings

Derived criteria for surjective adelic Galois representations

Computed an explicit example over a specific number field

Confirmed the existence of elliptic curves with surjective adelic representations

Abstract

We prove useful necessary and sufficient conditions for an elliptic curve over a number field to admit a surjective adelic Galois representation. Using these conditions, we compute an example of a number field K and an elliptic curve E/K whose corresponding adelic representation is surjective.