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

Let E be an elliptic curve over Q attached to a newform f of weight 2 on Γ0(N), and let K be a real quadratic field in which all the primes dividing N are split. This paper relates the canonical R/Z-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.