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

We prove the two-variable anticyclotomic Iwasawa main conjecture for Hida families and discuss its arithmetic application to a definite version of the horizontal non-vanishing conjecture, which is formulated in [LV11]. Our approach is based on the two-variable anticyclotomic control theorem for Selmer groups 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 [CL16] and [CKL17] and the self-dual twist of the specialisation to the anticyclotomic line of the three-variable p-adic L-function of Skinner–Urban [SU14].