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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: 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.

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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

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

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

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

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.

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