Duke Mathematical Journal
- Duke Math. J.
- Volume 122, Number 1 (2004), 181-204.
A Birch and Swinnerton-Dyer conjecture for the Mazur-Tate circle pairing
Let be an elliptic curve over attached to a newform of weight 2 on , and let be a real quadratic field in which all the primes dividing are split. This paper relates the canonical -valued "circle pairing" on defined by Mazur and Tate [MT1] to a period integral defined in terms of and . The resulting conjecture can be viewed as an analogue of the classical Birch and Swinnerton-Dyer conjecture, in which replaces the derivative of the complex -series 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.
Duke Math. J., Volume 122, Number 1 (2004), 181-204.
First available in Project Euclid: 24 March 2004
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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]
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