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

Sprung, Florian

doi:10.2140/ant.2017.11.885

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

Florian Sprung

Vol. 11 (2017), No. 4, 885–928

DOI: 10.2140/ant.2017.11.885

Abstract

The point of this paper is to give an explicit $p$ -adic analytic construction of two Iwasawa functions, $L_{p}^{♯} (f, T)$ and $L_{p}^{♭} (f, T)$ , 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 $a_{p} = 0$ . These 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 Šafarevič–Tate group in the cyclotomic direction analytically, encountering a new phenomenon for small slopes.

Dedicated to Barry, Joe, and Rob

Keywords

Birch and Swinnerton-Dyer, $p$-adic L-function, elliptic curve, modular form, Šafarevič–Tate group, Iwasawa Theory

Mathematical Subject Classification 2010

Primary: 11G40

Secondary: 11F67, 11R23

Milestones

Received: 10 April 2016

Revised: 16 December 2016

Accepted: 13 January 2017

Published: 18 June 2017

Authors

Florian Sprung
School of Mathematics Institute for Advanced Study & Princeton University 1 Einstein Dr Princeton, NJ 08540 United States

Vol. 11, No. 4, 2017