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

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).

v3: 103 pages; Sec 5.2 rewritten; add Sec 5.7. arXiv admin note: text overlap with arXiv:2211.06673 by other authors