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