Cropping Euler factors of modular L-functions
Abstract
According to the Birch and Swinnerton-Dyer conjectures, if A/Q is an abelian variety then its L-function must capture substantial part of the arithmetic properties of A. The smallest number field L where A has all its endomorphisms defined must also have a role. This article deals with the relationship between these two objects in the specific case of modular abelian varieties A_f/Q associated to weight 2 newforms for the modular group Gamma_1(N). Specifically, our goal is to relate the order of L(A_f/Q,s) at s = 1 with Euler products cropped by the set of primes that split completely in L. The results we obtain for the case when f has complex multiplication are complete, while in the absence of CM, our results depend on the rate of convergence in Sato-Tate distributions.
Cite
@article{arxiv.1002.4373,
title = {Cropping Euler factors of modular L-functions},
author = {J. Gonzalez and J. Jimenez and J. -C. Lario},
journal= {arXiv preprint arXiv:1002.4373},
year = {2010}
}
Comments
24 pages