Anticyclotomic p-ordinary Iwasawa Theory of Elliptic Modular Forms

TL;DR

This paper proves the Iwasawa main conjecture for p-ordinary elliptic modular forms over anticyclotomic Zp-towers of imaginary quadratic fields, using Beilinson-Flach elements to analyze the conjecture in both definite and indefinite cases.

Contribution

It establishes the Iwasawa main conjecture for suitable twists of p-ordinary elliptic modular forms in the anticyclotomic setting, a novel result in this area.

Findings

Proves the Iwasawa main conjecture for p-ordinary forms

Analyzes Beilinson-Flach elements in both definite and indefinite cases

Advances understanding of Iwasawa theory for elliptic modular forms

Abstract

This is the first in a series of articles where we will study the Iwasawa theory of an elliptic modular form f along the anticyclotomic Zp-tower of an imaginary quadratic field K where the prime p splits completely. Our goal in this portion is to prove the Iwasawa main conjecture for suitable twists of f assuming that f is p-ordinary, both in the definite and indefinite setups simultaneously, via an analysis of Beilinson-Flach elements.