Abstract
Let E/Q be an elliptic curve with complex multiplication by the ring of integers of an imaginary quadratic field K. In 1991, by studying a certain special value of the Katz two-variable p-adic L-function lying outside the range of p-adic interpolation, K. Rubin formulated a p-adic variant of the Birch and Swinnerton–Dyer conjecture when E(K) is infinite, and he proved that his conjecture is true for E(K) of rank one. When E(K) is finite, however, the statement of Rubin’s original conjecture no longer applies, and the relevant special value of the appropriate p-adic L-function is equal to zero. In this paper we extend our earlier work and give an unconditional proof of an analogue of Rubin’s conjecture when E(K) is finite.
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Agboola A.: On Rubin’s variant of the p-adic Birch and Swinnerton–Dyer conjecture. Comp. Math. 143, 1374–1398 (2007)
Agboola, A.: On certain special values of the Katz two-variable p-adic L-function (2010, in preparation)
Bernardi D., Goldstein C., Stephens N.: Notes p-adiques sur les courbes elliptiques. Crelle 351, 129–170 (1985)
Brumer A.: On the units of algebraic number fields. Mathematika 14, 121–124 (1967)
de Shalit E.: Iwasawa Theory of Elliptic Curves with Complex Multiplication. Academic Press, London (1987)
Greenberg, R.: Trivial zeros of p-adic L-functions. In: p-Adic Monodromy and the Birch and Swinnerton–Dyer Conjecture (Boston, MA, 1991). Contemporary Mathematics, vol. 165, pp. 149–174. American Mathematical Society, Providence (1994)
Mazur B., Tate J., Teitelbaum J.: On p-adic analogues of the conjectures of Birch and Swinnerton–Dyer. Invent. Math. 84, 1–48 (1986)
Perrin-Riou B.: Déscent infinie et hauteurs p-adiques sur les courbes elliptiques à multiplication complexe. Invent. Math. 70, 369–398 (1983)
Perrin-Riou B.: Théorie d’Iwasawa et hauteurs p-adiques. Invent. Math. 109, 137–185 (1992)
Perrin-Riou, B.: Arithmétique des courbes elliptiques et théorie d’Iwasawa. Mem. Soc. Math. 17, (1984)
Rubin K.: Tate–Shafarevich groups and L-functions of elliptic curves with complex multiplication. Invent. Math. 89, 527–560 (1987)
Rubin K.: p-adic L-functions and rational points on elliptic curves with complex multiplication. Invent. Math. 107, 323–350 (1992)
Rubin, K.: p-adic variants of the Birch and Swinnerton–Dyer conjecture for elliptic curves with complex multiplication. In: p-Adic Monodromy and the Birch and Swinnerton–Dyer Conjecture (Boston, MA, 1991). Contemporary Mathematics, vol. 165, pp. 71–80. American Mathematical Society, Providence (1994)
Rubin K.: The “main conjectures” of Iwasawa theory for imaginary quadratic fields. Invent. Math. 109, 25–68 (1992)
Rubin K.: Euler Systems. Princeton University Press, New Jersey (2000)
Rubin, K.: Euler systems and modular elliptic curves. In: Scholl, A.J., Taylor, R.L. (eds.) Galois representations in arithmetic geometry. CUP 1998
Serre J.-P.: Local Fields. Springer, Berlin (1979)
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This paper was partially supported by NSA Grant no. H98230-08-1-0077. I am very grateful to the anonymous referee for a number of very helpful comments. This paper was completed while I was visiting the Humboldt University, Berlin. I would like to thank the members of the Mathematics Department there for their generous hospitality.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Agboola, A. On Rubin’s variant of the p-adic Birch and Swinnerton–Dyer conjecture II. Math. Ann. 349, 807–837 (2011). https://doi.org/10.1007/s00208-010-0533-3
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DOI: https://doi.org/10.1007/s00208-010-0533-3