## Algebra & Number Theory

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

Florian Sprung

#### 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.

#### Article information

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

Dates
Revised: 16 December 2016
Accepted: 13 January 2017
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.ant/1510842781

Digital Object Identifier
doi:10.2140/ant.2017.11.885

Mathematical Reviews number (MathSciNet)
MR3665640

Zentralblatt MATH identifier
06735374

#### Citation

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

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