On pairs of P-Adic L-Functions for weight-two modular forms

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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. (f, T) and L p. (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.

Original languageEnglish (US)
Pages (from-to)885-928
Number of pages44
JournalAlgebra and Number Theory
Volume11
Issue number4
DOIs
StatePublished - Jan 1 2017
Externally publishedYes

Keywords

  • Birch and Swinnerton-Dyer
  • Elliptic curve
  • Iwasawa Theory
  • Modular form
  • P-adic L-function
  • Šafarevič-Tate group

ASJC Scopus subject areas

  • Algebra and Number Theory

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