Iwasawa-Greenberg main conjecture for non-ordinary modular forms and Eisenstein congruences on $\mathrm{GU}(3,1)$

TL;DR

This paper proves one divisibility of the Iwasawa-Greenberg main conjecture for certain Rankin-Selberg products involving non-ordinary modular forms, using Eisenstein congruences on a unitary group, advancing the understanding of p-adic L-functions.

Contribution

It extends previous results to non-ordinary cusp forms by developing semi-ordinary Hida theory and Eisenstein families on $ ext{GU}(3,1)$, providing key input for supersingular elliptic curve conjectures.

Findings

Established one-sided divisibility in the Iwasawa-Greenberg main conjecture.

Developed semi-ordinary Hida theory along a smaller weight space.

Analyzed semi-ordinary Eisenstein families on $ ext{GU}(3,1)$.

Abstract

In this paper we prove one side divisibility of the Iwasawa-Greenberg main conjecture for Rankin-Selberg product of a weight two cusp form and an ordinary CM form of higher weight, using congruences between Klingen Eisenstein series and cusp forms on $\mathrm{GU}(3,1)$, generalizing earlier result of the third-named author to allow non-ordinary cusp forms. The main result is a key input in the third author's proof for Kobayashi's $\pm$-main conjecture for supersingular elliptic curves. The new ingredient here is developing a semi-ordinary Hida theory along an appropriate smaller weight space, and a study of the semi-ordinary Eisenstein family.