Vol. 11, No. 4, 2017

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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
Abstract

The point of this paper is to give an explicit p-adic analytic construction of two Iwasawa functions, Lp(f,T) and Lp(f,T), for a weight-two modular form anqn and a good prime p. This generalizes work of Pollack who worked in the supersingular case and also assumed ap = 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