Algebra & Number Theory
- Algebra Number Theory
- Volume 11, Number 4 (2017), 885-928.
On pairs of $p$-adic $L$-functions for weight-two modular forms
The point of this paper is to give an explicit -adic analytic construction of two Iwasawa functions, and , for a weight-two modular form and a good prime . This generalizes work of Pollack who worked in the supersingular case and also assumed . 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.
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
Permanent link to this document
Digital Object Identifier
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
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. https://projecteuclid.org/euclid.ant/1510842781