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arXiv · 1601.00010

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

Florian Sprung
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Abstract

The point of this paper is to give an explicit p-adic analytic construction of two Iwasawa functions L_p^\sharp(f,T) and L_p^\flat(f,T) 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. The 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 Tate-Shafarevich group in the cyclotomic direction analytically, encountering a new phenomenon for small slopes.

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Florian Sprung. 2015-12-31. On pairs of p-adic L-functions for weight two modular forms. https://doi.org/10.2140/ant.2017.11.885

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