Elliptic curves with maximal Galois action on their torsion points

Details Elliptic curves with maximal Galois action on their torsion points Elliptic curves with maximal Galois action on their torsion points

2008-09-20

arxiv.org/abs/0809.3482v1

Mathematics

Number Theory

14 pages

Scientific paper

Given an elliptic curve E over a number field k, the Galois action on the torsion points of E induces a Galois representation, \rho_E : Gal(\bar{k}/k) \to GL_2(\hat{Z}). For a fixed number field k, we describe the image of \rho_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 \rho_E will be surjective for "most" elliptic curves over k.

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