Open Access

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

Research in Number Theory20151:17

https://doi.org/10.1007/s40993-015-0018-2

Received: 22 April 2015

Accepted: 4 August 2015

Published: 30 September 2015

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.

Keywords

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