Bessel periods and Selmer groups over ordinary Rankin--Selberg eigenvariety

TL;DR

This paper develops a framework for ordinary distributions and eigenvarieties for unitary groups, constructs Bessel periods as elements in Selmer groups, and proposes an Iwasawa conjecture relating their divisors.

Contribution

It introduces the notion of ordinary distributions for unitary groups, constructs Bessel periods on eigenvarieties, and formulates an Iwasawa conjecture linking Bessel periods and Selmer groups.

Findings

Constructed Bessel periods as elements in Selmer groups.

Proposed an Iwasawa type conjecture relating divisors.

Proved one divisibility side under certain conditions.

Abstract

We introduce the notion of ordinary distributions for unitary groups and their Rankin--Selberg products, based on which we (re)define the ordinary eigenvarieties. In both definite and indefinite cases, we construct the Bessel period on the Rankin--Selberg eigenvariety as an ordinary distribution and as an element in the Selmer group of ordinary distributions, respectively. We then propose an Iwasawa type conjecture relating the vanishing divisor of the Bessel period and the characteristic divisor of the Selmer group of the associated Rankin--Selberg Galois module over the eigenvariety, and prove one side of the divisibility under certain conditions.