Rankin-Selberg L-functions in cyclotomic towers, II

TL;DR

This paper generalizes Mazur's conjecture for elliptic curve L-functions in abelian extensions of imaginary quadratic fields, using p-adic L-functions to relate Mordell-Weil ranks to nonvanishing criteria.

Contribution

It introduces a novel approach leveraging p-adic L-functions to prove a generalized Mazur's conjecture for elliptic curves in specific number field extensions.

Findings

Proves a generalization of Mazur's conjecture for elliptic curves in abelian extensions.

Shows that Mordell-Weil rank in the Z_p^2-extension is finitely generated modulo Heegner points.

Introduces a method using p-adic L-functions to reduce the problem to a nonvanishing criterion.

Abstract

Following the prequel work \cite{VO3}, we prove a generalization of "Mazur's conjecture" for $L$-functions of elliptic curves in abelian extensions of imaginary quadratic fields, including the assertion that the Mordell-Weil rank of an elliptic curve in the ${\bf{Z}}_p^2$-extension is finitely generated modulo Heegner points. The novelty of the approach here is to use the existence of a suitable $p$-adic $L$-function to reduce the problem to a minimal nonvanishing criterion, which should be applicable to a broader class of problems than considered here.