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

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

Journal:
Math. Comp. **83** (2014), 2925-2951

MSC (2010):
Primary 11F80, 11G05

Published electronically:
January 30, 2014

MathSciNet review:
3246816

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Abstract | References | Similar Articles | Additional Information

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 . 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 with maximal image, where is the real root of . 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.

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Additional Information

**David Corwin**

Affiliation:
Department of Mathematics, Princeton University, Princeton, New Jersey 08544

Address at time of publication:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Email:
corwind@mit.edu

**Tony Feng**

Affiliation:
479 Quincy Mail Center, 58 Plympton Street, Cambridge, Massachusetts 02138

Email:
tfeng@college.harvard.edu

**Zane Kun Li**

Affiliation:
Department of Mathematics, Princeton University, Princeton, New Jersey 08544

Address at time of publication:
Department of Mathematics, UCLA, Los Angeles, California 90095

Email:
zkli@math.ucla.edu

**Sarah Trebat-Leder**

Affiliation:
Department of Mathematics, Princeton University, Princeton, New Jersey 08544

Address at time of publication:
Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322

Email:
strebat@emory.edu

DOI:
http://dx.doi.org/10.1090/S0025-5718-2014-02804-4

Received by editor(s):
July 20, 2012

Received by editor(s) in revised form:
February 4, 2013

Published electronically:
January 30, 2014

Article copyright:
© Copyright 2014
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.