Cropping Euler factors of modular L-functions

TL;DR

This paper investigates the relationship between the order of the L-function of modular abelian varieties at s=1 and Euler products truncated by primes splitting in the endomorphism field, with complete results for CM cases.

Contribution

It establishes a connection between the order of L-functions at s=1 and cropped Euler products for modular abelian varieties, especially clarifying the CM case.

Findings

Complete results for CM modular abelian varieties.

Results depend on convergence rates in Sato-Tate distributions for non-CM cases.

Relates L-function behavior to prime splitting in endomorphism fields.

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.