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Given an elliptic curve E and a prime p of (good) supersingular reduction, weformulate p-adic analogues of the Birch and Swinnerton-Dyer conjecture using apair of Iwasawa functions L^\sharp(E,T) and L^\flat(E,T). They are equivalentto the conjectures of Perrin-Riou and Bernardi. We also generalize work ofKurihara and Pollack to give a criterion for positive rank in terms of thevalue of the quotient between these functions, and derive a result towards anon-vanishing conjecture. We also generalize a conjecture of Kurihara andPollack concerning the greatest common divisor of the two functions to thegeneral supersingular case.

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