On pairs of p-adic L-functions for weight two modular forms

TL;DR

This paper constructs explicit p-adic L-functions for weight two modular forms, extending previous work, and uses them to analyze the Birch and Swinnerton-Dyer conjecture in the cyclotomic setting.

Contribution

It provides a new explicit p-adic analytic construction of two Iwasawa functions for weight two modular forms, generalizing prior supersingular cases.

Findings

Bounded the rank of elliptic curves in the cyclotomic direction.

Estimated growth of the Tate-Shafarevich group analytically.

Identified a new phenomenon for small slopes.

Abstract

The point of this paper is to give an explicit p-adic analytic construction of two Iwasawa functions L_p^\sharp(f,T) and L_p^\flat(f,T) for a weight two modular form \sum a_n q^n and a good prime p. This generalizes work of Pollack who worked in the supersingular case and also assumed a_p=0. The Iwasawa functions work in tandem to shed some light on the Birch and Swinnerton-Dyer conjectures in the cyclotomic direction: We bound the rank and estimate the growth of the Tate-Shafarevich group in the cyclotomic direction analytically, encountering a new phenomenon for small slopes.