## 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.

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## Acknowledgements

We thank Masato Kurihara and Robert Pollack for an interesting correspondence and conversations on *p*-adic versions of the Birch and Swinnerton-Dyer conjectures, and Christian Wuthrich for a helpful comment on regulators. We also thank the anonymous referee for pointing out some inaccuracies and for a suggestion that improved the exposition of this article.

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Sprung, F. A formulation of *p*-adic versions of the Birch and Swinnerton-Dyer conjectures in the supersingular case.
*Res. number theory* **1**, 17 (2015). https://doi.org/10.1007/s40993-015-0018-2

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

### Keywords

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