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

Wan, Xin

doi:10.2140/ant.2020.14.383

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

Xin Wan

Vol. 14 (2020), No. 2, 383–483

DOI: 10.2140/ant.2020.14.383

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

Vol. 14, No. 2, 2020