Cropping Euler factors of modular L-functions
Gonzalez, J. · Jimenez, J. · Lario, J. -C.
Original · EN
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 Af/Q associated to weight 2 newforms for the modular group Gamma₁(N). Specifically, our goal is to relate the order of L(Af/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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