Rankin-Selberg L-functions in cyclotomic towers, I

TL;DR

This paper proves a conjecture related to the nonvanishing of central values of Rankin-Selberg L-functions for elliptic curves in abelian extensions of imaginary quadratic fields, extending previous results to more general settings.

Contribution

It formulates and proves a conjecture on nonvanishing of L-functions in cyclotomic towers, generalizing prior theorems to abelian extensions of imaginary quadratic fields.

Findings

Generalizes Rohrlich's theorem to abelian extensions of imaginary quadratic fields.

Proves a conjecture in the style of Mazur-Greenberg for nonvanishing L-values.

Extends results to families of degree-four L-functions from GL(2)×GL(2).

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.