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

David Corwin, Tony Feng, Zane Kun Li, Sarah Trebat-Leder

Research output: Contribution to journalArticlepeer-review

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 GL2(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) with maximal image, where a is the real root of x3 + 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.

Original languageEnglish
Pages (from-to)2925-2951
Number of pages27
JournalMathematics of Computation
Volume83
Issue number290
DOIs
StatePublished - 1 Jan 2014

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