A formulation of p-adic versions of the Birch and Swinnerton-Dyer conjectures in the supersingular case

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Abstract

Given an elliptic curve E and a prime p of (good) supersingular reduction, we formulate p-adic analogues of the Birch and Swinnerton-Dyer conjecture using a pair of Iwasawa functions L(E,T) and L(E,T). They are equivalent to the conjectures of Perrin-Riou and Bernardi. We also generalize work of Kurihara and Pollack to give a criterion for positive rank in terms of the value of the quotient between these functions, and derive a result towards a non-vanishing conjecture. We also generalize a conjecture of Kurihara and Pollack concerning the greatest common divisor of the two functions to the general supersingular case.

Original languageEnglish (US)
Article number17
JournalResearch in Number Theory
Volume1
Issue number1
DOIs
StatePublished - Dec 1 2015
Externally publishedYes

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P-adic
Formulation
Highest common factor
Generalise
L-function
Elliptic Curves
Quotient
Analogue

Keywords

  • Birch and Swinnerton-Dyer conjecture
  • Iwasawa theory
  • p-adic l-functions

ASJC Scopus subject areas

  • Algebra and Number Theory

Cite this

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abstract = "Given an elliptic curve E and a prime p of (good) supersingular reduction, we formulate p-adic analogues of the Birch and Swinnerton-Dyer conjecture using a pair of Iwasawa functions L♯(E,T) and L♭(E,T). They are equivalent to the conjectures of Perrin-Riou and Bernardi. We also generalize work of Kurihara and Pollack to give a criterion for positive rank in terms of the value of the quotient between these functions, and derive a result towards a non-vanishing conjecture. We also generalize a conjecture of Kurihara and Pollack concerning the greatest common divisor of the two functions to the general supersingular case.",
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AB - Given an elliptic curve E and a prime p of (good) supersingular reduction, we formulate p-adic analogues of the Birch and Swinnerton-Dyer conjecture using a pair of Iwasawa functions L♯(E,T) and L♭(E,T). They are equivalent to the conjectures of Perrin-Riou and Bernardi. We also generalize work of Kurihara and Pollack to give a criterion for positive rank in terms of the value of the quotient between these functions, and derive a result towards a non-vanishing conjecture. We also generalize a conjecture of Kurihara and Pollack concerning the greatest common divisor of the two functions to the general supersingular case.

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