The conjecture of Birch and Swinnerton-Dyer for elliptic curves with complex multiplication by a nonmaximal order

The Conjecture of Birch and Swinnerton-Dyer relates an analytic invariant of an elliptic curve---the value of the L-function, to an algebraic invariant of the curve---the order of the Tate-Safarevic group. Gross has refined the Birch-Swinnerton-Dyer Conjecture in the case of an elliptic curve with complex multiplication by the full ring of integers in a quadratic imaginary field. It is this version which interests us here. Gross' Conjecture has been reformulated, by Fontaine and Perrin-Riou, in the language of derived categories and determinants of perfect complexes. Burns and Flach then realized that this immediately leads to a refined conjecture for elliptic curves with complex multiplication by a nonmaximal order. The conjecture is now expressed as a statement concerning a generator of the image of a map of 1-dimensional modules. We prove this conjecture of Burns and Flach.