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arXiv · 0809.3482

Elliptic curves with maximal Galois action on their torsion points

David Zywina
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Abstract

Given an elliptic curve E over a number field k, the Galois action on the torsion points of E induces a Galois representation, ρ_E : Gal(\bar{k}/k) \to GL_2(\hat{Z}). For a fixed number field k, we describe the image of ρ_E for a "random" elliptic curve E over k. In particular, if k\neq Q is linearly disjoint from the cyclotomic extension of Q, then ρ_E will be surjective for "most" elliptic curves over k.

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David Zywina. 2008-09-20. Elliptic curves with maximal Galois action on their torsion points. https://doi.org/10.1112/blms%2Fbdq039

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