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arXiv · 1002.4373

Cropping Euler factors of modular L-functions

J. Gonzalez·J. Jimenez·J. -C. Lario
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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.

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J. Gonzalez, J. Jimenez, J. -C. Lario. 2010-03-29. Cropping Euler factors of modular L-functions. https://arxiv.org/abs/1002.4373

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