Let $E$
be an elliptic curve over $\mathbb{Q}$
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 $\mathbb{R}/\mathbb{Z}$-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.
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