Anticyclotomic main conjecture and the non-triviality of Rankin-Selberg $L$-values in Hida families

TL;DR

This paper proves the two-variable anticyclotomic Iwasawa main conjecture for Hida families and establishes a non-vanishing result for Rankin-Selberg $L$-values, advancing understanding in $p$-adic number theory and automorphic forms.

Contribution

It introduces a new proof of the anticyclotomic main conjecture for Hida families using control theorems and $p$-adic $L$-functions, connecting several advanced concepts.

Findings

Proved the two-variable anticyclotomic Iwasawa main conjecture for Hida families.

Established a definite version of the horizontal non-vanishing conjecture.

Linked $p$-adic $L$-functions with Selmer groups and automorphic forms.

Abstract

The aim of this paper is to prove the two-variable anticyclotomic Iwasawa main conjecture for Hida families and a definite version of the horizontal non-vanishing conjecture, which are formulated in Longo-Vigni. Our approach is based on the two-variable anticyclotomic control theorem for Selmer groups for Hida families and the relation between the two-variable anticyclotomic $L$-function for Hida families built out of $p$-adic families of Gross points on definite Shimura curves studied in Castella-Longo and Castella-Kim-Longo and the self-dual twist of the specialisation to the anticyclotomic line of the three-variable $p$-adic $L$-function of Skinner-Urban.