TY - JOUR

T1 - Elliptic curves with full 2-torsion and maximal adelic galois representations

AU - Corwin, David

AU - Feng, Tony

AU - Li, Zane Kun

AU - Trebat-Leder, Sarah

N1 - Publisher Copyright:
© 2014 American Mathematical Society.

PY - 2014/1/1

Y1 - 2014/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84916887027&partnerID=8YFLogxK

U2 - 10.1090/S0025-5718-2014-02804-4

DO - 10.1090/S0025-5718-2014-02804-4

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AN - SCOPUS:84916887027

VL - 83

SP - 2925

EP - 2951

JO - Mathematics of Computation

JF - Mathematics of Computation

SN - 0025-5718

IS - 290

ER -