Serre's uniformity problem in the split Cartan case
Number Theory
2009-03-03 v5 Algebraic Geometry
Abstract
We prove that there exists an integer p_0 such that X_split(p)(Q) is made of cusps and CM-points for any prime p>p_0. Equivalently, for any non-CM elliptic curve E over Q and any prime p>p_0 the image of the Galois representation induced by the Galois action on the p-division points of E is not contained in the normalizer of a split Cartan subgroup. This gives a partial answer to an old question of Serre.
Keywords
Cite
@article{arxiv.0807.4954,
title = {Serre's uniformity problem in the split Cartan case},
author = {Yuri Bilu and Pierre Parent},
journal= {arXiv preprint arXiv:0807.4954},
year = {2009}
}
Comments
11 pages; Version 5; minor revision (a few bugs corrected, some references added)