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

Abstract

Let $E$ be an elliptic curve over $Q$ attached to a newform $f$ of weight 2 on ${\Gamma }_0(N)$, and let $K$ be a real quadratic field in which all the primes dividing $N$ are split. This paper relates the canonical $ℝ/\mathbb{Z}$-valued "circle pairing" on $E(K)$ defined by Mazur and Tate [MT1] to a period integral $I^{\prime }(f,k)$ 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 $I^{\prime }(f,k)$ 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.