Iwasawa main conjecture for Rankin–Selberg $p$-adic $L$-functions

Abstract

In this paper we prove that the $p$-adic $L$-function that interpolates the Rankin–Selberg product of a general modular form and a CM form of higher weight divides the characteristic ideal of the corresponding Selmer group. This is one divisibility of the Iwasawa main conjecture for this $p$-adic $L$-function. We prove this conjecture using congruences between Klingen–Eisenstein series and cusp forms on the group $GU\left(3,1\right)$, following the strategy of recent work by C. Skinner and E. Urban. The actual argument is, however, more complicated due to the need to work with general Fourier–Jacobi expansions. This theorem is used to deduce a converse of the Gross–Zagier–Kolyvagin theorem and the $p$-adic part of the precise BSD formula in the rank one case.