Iwasawa Main Conjecture for Rankin-Selberg p-adic L-functions

TL;DR

This paper proves a divisibility in the Iwasawa main conjecture for p-adic L-functions associated with Rankin-Selberg products, using congruences of Eisenstein series and cusp forms, with implications for BSD conjecture.

Contribution

It establishes one divisibility of the Iwasawa main conjecture for Rankin-Selberg p-adic L-functions using a novel approach involving congruences on GU(3,1).

Findings

Proves divisibility of p-adic L-function by the characteristic ideal of the Selmer group.

Uses congruences between Klingen Eisenstein series and cusp forms.

Enables deduction of converse BSD and Gross-Zagier-Kolyvagin results.

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 the p-adic L-function. We prove this conjecture using the congruences between Klingen Eisensteinseries and cusp forms on the group GU(3; 1), following the strategy of a recent work of C. Skinner and E. Urban. This theorem can be used to deduce a converse of Gross-Zagier-Kolyvagin theorem and the precise BSD formula in the rank one case.