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15 March 2004 A Birch and Swinnerton-Dyer conjecture for the Mazur-Tate circle pairing
Massimo Bertolini, Henri Darmon
Duke Math. J. 122(1): 181-204 (15 March 2004). DOI: 10.1215/S0012-7094-04-12216-X


Let E be an elliptic curve over Q attached to a newform f of weight 2 on Γ 0 (N) , and let K be a real quadratic field in which all the primes dividing N are split. This paper relates the canonical / -valued "circle pairing" on E(K) defined by Mazur and Tate [MT1] to a period integral I (f,k) defined in terms of f and k . The resulting conjecture can be viewed as an analogue of the classical Birch and Swinnerton-Dyer conjecture, in which I (f,k) replaces the derivative of the complex L -series L(f,K,s) and the circle pairing replaces the Néron-Tate height. It emerges naturally as an archimedean fragment of the theory of anticyclotomic p-adic L-functions developed in [BD], and has been tested numerically in a variety of situations. The last section formulates a conjectural variant of a formula of Gross, Kohnen, and Zagier [GKZ] for the Mazur-Tate circle pairing.


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Massimo Bertolini. Henri Darmon. "A Birch and Swinnerton-Dyer conjecture for the Mazur-Tate circle pairing." Duke Math. J. 122 (1) 181 - 204, 15 March 2004.


Published: 15 March 2004
First available in Project Euclid: 24 March 2004

zbMATH: 1072.11036
MathSciNet: MR2046811
Digital Object Identifier: 10.1215/S0012-7094-04-12216-X

Primary: 11G40
Secondary: 11F04 , 11G05 , 11G50

Rights: Copyright © 2004 Duke University Press


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Vol.122 • No. 1 • 15 March 2004
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