Algebra & Number Theory

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

Florian Sprung

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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.

Article information

Algebra Number Theory, Volume 11, Number 4 (2017), 885-928.

Received: 10 April 2016
Revised: 16 December 2016
Accepted: 13 January 2017
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture [See also 14G10]
Secondary: 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 11R23: Iwasawa theory

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


Sprung, Florian. On pairs of $p$-adic $L$-functions for weight-two modular forms. Algebra Number Theory 11 (2017), no. 4, 885--928. doi:10.2140/ant.2017.11.885.

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