#### 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, ${L}_{p}^{♯}\left(f,T\right)$ and ${L}_{p}^{♭}\left(f,T\right)$, 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