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

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.

This paper replaces and corrects most of the arxiv submission arXiv:1211.1352, which has been split into two articles. The major corrections are the non-uniqueness of the sharp/flat p-adic L-functions in the ordinary case, and the correct bounds on the p-adic analytic cyclotomic rank

References in corpus (1)

• Factorisation of two-variable p-adic L-functions

Cited by in corpus (2)

• Mordell-Weil ranks and Tate-Shafarevich groups of elliptic curves with mixed-reduction type over cyclotomic extensions
• On fine Selmer groups and the greatest common divisor of signed and chromatic $p$-adic $L$-functions