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

TL;DR

This paper develops p-adic analogues of the Birch and Swinnerton-Dyer conjecture for supersingular elliptic curves using Iwasawa functions, connecting to existing conjectures and providing criteria for positive rank.

Contribution

It introduces new p-adic formulations of BSD conjectures in the supersingular case and extends previous work to more general settings.

Findings

Formulation of p-adic BSD conjectures using L^lat and L^lat functions.

A criterion for positive rank based on the quotient of these functions.

Progress towards a non-vanishing conjecture and generalization of gcd conjecture.

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.