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.
"On pairs of $p$-adic $L$-functions for weight-two modular forms." Algebra Number Theory 11 (4) 885 - 928, 2017. https://doi.org/10.2140/ant.2017.11.885