Bounds for Serre's open image theorem
Number Theory
2011-02-24 v1
Abstract
Let E be an elliptic curve over the rationals without complex multiplication. The absolute Galois group of Q acts on the group of torsion points of E, and this action can be expressed in terms of a Galois representation rho_E:Gal(Qbar/Q) \to GL_2(Zhat). A renowned theorem of Serre says that the image of rho_E is open, and hence has finite index, in GL_2(Zhat). We give the first general bounds of this index in terms of basic invariants of E. For example, the index can be bounded by a polynomial function of the logarithmic height of the j-invariant of E. As an application of our bounds, we settle an open question on the average of constants arising from the Lang-Trotter conjecture.
Cite
@article{arxiv.1102.4656,
title = {Bounds for Serre's open image theorem},
author = {David Zywina},
journal= {arXiv preprint arXiv:1102.4656},
year = {2011}
}