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

2021/09/17 by Francesc Castella, Zheng Liu, Castella, Francesc +3 · 2 citations
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #math.NT

paper · pdf · doi:10.48550/arxiv.2109.08375

83 pages

arxiv created 2021/09/17 · arxiv updated 2021/09/20

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 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 ±-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.

Cited by

• Test Schwartz functions at p for theta lifts of Hida families
• The refined Tamagawa number conjectures for GL₂
• Iwasawa Main Conjecture for Rankin-Selberg p-adic L-functions: Non-Ordinary Case

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