Elliptic curves with full 2-torsion and maximal adelic 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 $\mathrm {GL}_2(\widehat {\mathbb {Z}})$. 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 $\mathbb {Q}(\alpha )$ with maximal image, where $\alpha$ is the real root of $x^3 + x + 1$. 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.