## 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 GL_{2}(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 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.

Original language | English |
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Pages (from-to) | 2925-2951 |

Number of pages | 27 |

Journal | Mathematics of Computation |

Volume | 83 |

Issue number | 290 |

DOIs | |

State | Published - 1 Jan 2014 |

Externally published | Yes |

## ASJC Scopus subject areas

- Algebra and Number Theory
- Computational Mathematics
- Applied Mathematics