Anticyclotomic $p$-adic $L$-functions for Rankin--Selberg product

Abstract

We construct $p$-adic $L$-functions for Rankin--Selberg products of automorphic forms of hermitian type in the anticyclotomic direction for both root numbers. When the root number is $+1$, the construction relies on global Bessel periods on definite unitary groups which, due to the recent advances on the global Gan--Gross--Prasad conjecture, interpolate classical central $L$-values. When the root number is $-1$, we construct an element in the Iwasawa Selmer group using the diagonal cycle on the product of unitary Shimura varieties, and conjecture that its $p$-adic height interpolates derivatives of cyclotomic $p$-adic $L$-functions. We also propose the nonvanishing conjecture and the main conjecture in both cases.