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arXiv · 1512.09362

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

Florian Sprung
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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^\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.

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Florian Sprung. 2015-12-31. A formulation for p-adic versions of the Birch and Swinnerton-Dyer conjectures in the supersingular case. https://arxiv.org/abs/1512.09362

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