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

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.