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

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^\sharp(E,T) and L^\flat(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.

arXiv commentTo appear in 'Research in Number Theory,' volume 1, issue 1. The generalization of the theorems of Kurihara and Pollack is new, while the p-adic BSD conjectures in terms of the integral p-adic L-functions supersedes part of an earlier arxiv submission [arXiv:1211.1352], which is being withdrawn