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arXiv · 1207.1672

Rankin-Selberg L-functions in cyclotomic towers, I

Jeanine Van Order
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Abstract

We formulate and for the most part prove a conjecture in the style of Mazur-Greenberg for the nonvanishing of central values of Rankin-Selberg $L$-functions attached to elliptic curves in abelian extensions of imaginary quadratic fields. This in particular generalizes the theorem of Rohrlich on $L$-functions of elliptic curves in cyclotomic towers to the setting of abelian extensions of imaginary quadratic fields, corresponding to families of degree-four $L$-functions given by $\operatorname{GL}(2)\times\operatorname{GL}(2)$ Rankin-Selberg $L$-functions. It also generalizes the theorems of Rohrlich, Greenberg, Vatsal, and Cornut for $L$-functions of elliptic curves in $\operatorname{Z}_p^2$-extensions of imaginary quadratic fields.

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Jeanine Van Order. 2012-07-06. Rankin-Selberg L-functions in cyclotomic towers, I. https://arxiv.org/abs/1207.1672

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