Elliptic curves with full 2-torsion and maximal adelic Galois representations

In 1972, Serre showed that the adelic Galois representation associated to a non-CM elliptic curve over a number field has open image in G L 2 ( Z ^ ) \mathrm {GL}_2(\widehat {\mathbb {Z}}) . In (2010), Greicius developed 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’s 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 ( α ) \mathbb {Q}(\alpha ) with maximal image, where α \alpha is the real root of x 3 + x + 1 x^3 + x + 1 . Next, we extend Greicius’s 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.