Cropping Euler factors of modular L-functions
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According to the Birch and Swinnerton-Dyer conjectures, if A/Q is an abelian variety, then its L-function must capture a substantial part of the properties of A. The smallest number field L where A has all its endomorphisms defined must also play 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 group t1(N). Specifically, our goal is to relate ords=1 L(Af =Q, s), with the order at s D 1 of Euler products restricted to primes that split completely in L. This is attained when a power of Af is isogenous over Q to the Weil restriction of the building block of Af . We give separated formulae for the CM and non-CM cases.
CitationGonzalez, J.; Jimenez, J.; Lario, J. Cropping Euler factors of modular L-functions. "Forum mathematicum", Setembre 2013, vol. 25, núm. 5, p. 1039-1066.