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

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.