Vol. 14, No. 2, 2020

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

Xin Wan

Vol. 14 (2020), No. 2, 383–483
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(3,1), 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.

Keywords
Iwasawa main conjecture, Rankin–Selberg
Mathematical Subject Classification 2010
Primary: 11R23
Milestones
Received: 14 January 2018
Revised: 14 August 2019
Accepted: 14 September 2019
Published: 29 May 2020
Authors
Xin Wan
Academy of Mathematics and Systems Science
Chinese Academy of Sciences and University of Chinese Academy of Sciences
Haidian District
Beijing
China