Elliptic Curves with Full 2-Torsion and Maximal Adelic Galois Representations

TL;DR

This paper extends criteria and computational methods to identify elliptic curves with full 2-torsion that have maximal adelic Galois representations, including new infinite families over cubic fields.

Contribution

It develops new necessary and sufficient conditions and effective tests for maximal adelic Galois representations in elliptic curves with full 2-torsion, expanding previous work to broader settings.

Findings

Constructed an infinite family of curves over Q(α) with maximal Galois image.

Extended tests to non-semistable elliptic curves over cubic fields.

Provided a general discussion on torsion subgroup-related Galois representations.

Abstract

In 1972, Serre showed that the adelic Galois representation associated to a non-CM elliptic curve over a number field has open image in GL_2(\hat{Z}). In Greicius' thesis, he develops necessary and sufficient criteria for determining when this representation is actually surjective and exhibits such an example. However, verifying these criteria turns out to be difficult in practice; Greicius describes tests for them that apply only to semistable elliptic curves over a specific class of cubic number fields. In this paper, we extend Greicius' methods in several directions. First, we consider the analogous problem for elliptic curves with full 2-torsion. Following Greicius, we obtain necessary and sufficient conditions for the associated adelic representation to be maximal and also develop a battery of computationally effective tests that can be used to verify these conditions. We are able to use our tests to construct an infinite family of curves over Q(alpha) with maximal image, where alpha is the real root of x^3 + x + 1. Next, we extend Greicius' tests to more general settings, such as non-semistable elliptic curves over arbitrary cubic number fields. Finally, we give a general discussion concerning such problems for arbitrary torsion subgroups.