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.
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. https://doi.org/10.1215/S0012-7094-04-12216-X