Non-abelian $p$-adic Rankin-Selberg $L$-functions and non-vanishing of central $L$-values

TL;DR

This paper establishes new congruences for special values of Rankin-Selberg $L$-functions over number fields, enabling control over $p$-adic $L$-functions and proving non-vanishing of central $L$-values under certain conditions.

Contribution

It introduces novel congruences between $L$-values and constructs non-abelian $p$-adic $L$-functions for Hida families on $ ext{GL}(n+1) imes ext{GL}(n)$.

Findings

Established congruences between special $L$-values.

Constructed non-abelian $p$-adic $L$-functions.

Proved non-vanishing of twisted central $L$-values.

Abstract

We prove new congruences between special values of Rankin-Selberg $L$-functions for $\mathrm{GL}(n+1)\times\mathrm{GL}(n)$ over arbitrary number fields. This allows us to control the behavior of $p$-adic $L$-functions under Tate twists and to prove the existence of non-abelian $p$-adic $L$-functions for Hida families on $\mathrm{GL}(n+1)\times\mathrm{GL}(n)$. As an application, we prove strong non-vanishing results for central $L$-values: We give sufficient local conditions for twisted central Rankin-Selberg $L$-values to be generically non-zero.