Introduction

The aim of this paper is to rework the material in Chapter III of Gross and Zagier's “Heegner points and derivatives of L-series” —see [GZ] in the list of references—based on more systematic deformation-theoretic methods, so as to treat all imaginary quadratic fields, all residue characteristics, and all j-invariants on an equal footing. This leads to more conceptual arguments in several places and interpretations for some quantities which appear to otherwise arise out of thin air in [GZ, Ch. III]. For example, the sum in [GZ, Ch. III, Lemma 8.2] arises for us in (9−6), where it is given a deformation-theoretic meaning. Provided the analytic results in [GZ] are proven for even discriminants, the main results in [GZ] would be valid without parity restriction on the discriminant of the imaginary quadratic field. Our order of development of the basic results follows [GZ, Ch. III], but the methods of proof are usually quite different, making much less use of the “numerology” of modular curves.

Here is a summary of the contents. In Section 2 we consider some background issues related to maps among elliptic curves over various bases and horizontal divisors on relative curves over a discrete valuation ring. In Section 3 we provide a brief survey of the Serre–Tate theorem and the Grothendieck existence theorem, since these form the backbone of the deformation-theoretic methods which underlie all subsequent arguments.