Iwasawa's main conjecture for Rankin-Selberg motives in the anticyclotomic case

TL;DR

This paper advances Iwasawa theory for Rankin-Selberg motives over CM fields, establishing divisibility results for p-adic L-functions and Selmer groups in the anticyclotomic setting, linked to the Gan--Gross--Prasad conjecture.

Contribution

It proves one-sided divisibility of the Iwasawa main conjecture for automorphic representations in the anticyclotomic case, relating p-adic L-functions to Selmer groups.

Findings

When the root number is 1, the p-adic L-function is in the characteristic ideal of the Selmer group.

When the root number is -1, the square of a certain Iwasawa module's characteristic ideal is contained in the torsion part of the Selmer group.

The results connect Iwasawa theory with the Gan--Gross--Prasad conjecture for unitary groups.

Abstract

In this article, we study the Iwasawa theory for cuspidal automorphic representations of $\mathrm{GL}(n)\times\mathrm{GL}(n+1)$ over CM fields along anticyclotomic directions, in the framework of the Gan--Gross--Prasad conjecture for unitary groups. We prove one-side divisibility of the corresponding Iwasawa main conjecture: when the global root number is $1$, the $p$-adic $L$-function belongs to the characteristic ideal of the Iwasawa Bloch--Kato Selmer group; when the global root number is $-1$, the square of the characteristic ideal of a certain Iwasawa module is contained in the characteristic ideal of the torsion part of the Iwasawa Bloch--Kato Selmer group (analogous to Perrin-Riou's Heegner point main conjecture).