Duke Mathematical Journal

A Birch and Swinnerton-Dyer conjecture for the Mazur-Tate circle pairing

Massimo Bertolini and Henri Darmon

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Abstract

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.

Article information

Source
Duke Math. J., Volume 122, Number 1 (2004), 181-204.

Dates
First available in Project Euclid: 24 March 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1080137206

Digital Object Identifier
doi:10.1215/S0012-7094-04-12216-X

Mathematical Reviews number (MathSciNet)
MR2046811

Zentralblatt MATH identifier
1072.11036

Subjects
Primary: 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture [See also 14G10]
Secondary: 11F04 11G05: Elliptic curves over global fields [See also 14H52] 11G50: Heights [See also 14G40, 37P30]

Citation

Bertolini, Massimo; Darmon, Henri. A Birch and Swinnerton-Dyer conjecture for the Mazur-Tate circle pairing. Duke Math. J. 122 (2004), no. 1, 181--204. doi:10.1215/S0012-7094-04-12216-X. https://projecteuclid.org/euclid.dmj/1080137206


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