Let \(E\) be an elliptic curve defined over \(\mathbb{Q}\) by the Weierstrass equation \(y^ 2 = 4x^ 3 - g_ 2x - g_ 3\) with integer coefficients \(g_ 2\) and \(g_ 3\). Assume that the conductor of \(E/ \mathbb{Q}\) is an odd integer \(N\) and, in addition, that \(E\) satisfies the Shimura-Taniyama- Weil conjecture for its \(L\)-series.
The present paper provides a survey on the arithmetic of those elliptic curves \(E/ \mathbb{Q}\) (of the above type) which also satisfy the so-called Heegner hypothesis. This means that \(\left({D \over p}\right) = 1\) for all primes \(p\) dividing the conductor \(N\), where \(D\) denotes the discriminant of the Weierstrass equation of \(E\). The Heegner construction associates to such a curve the set of all primitive quadratic forms \(Ax^ 2 + Bxy + Cy^ 2\) of the same discriminant \(D\), whose coefficients satisfy a certain divisibility condition with respect to the conductor \(N\). These forms are called Heegner forms, and the author describes, in the first part of the paper, how each Heegner form leads to a so-called Heegner object which belongs to a certain \(\mathbb{Z}\)-module \(M_ D\) depending on the discriminant \(D\). If \(D < 0\), then this Heegner object associated with a Heegner form is a certain complex point of \(E\) (Heegner point), and if \(D > 0\), then it will be an element of \(H_ 1 (E(\mathbb{C}), \mathbb{Z})\), represented by a cycle in \(E(\mathbb{C})\) (Heegner cycle).
The author then discusses the properties of these Heegner objects with respect to the action of the Hecke operators and the Atkin-Lehner involution, the relation with \(L\)-functions and, in this context, the respective main results of B. Gross and D. Zagier [Invent. Math. 84, 225-320 (1986; Zbl 0608.14019)], B. Gross, W. Kohnen and D. Zagier [Math. Ann. 278, 497-562 (1987; Zbl 0641.14013)] and J.-L. Waldspurger [Compos. Math. 54, 173-242 (1985; Zbl 0567.10021)].
In the second part of the paper, the author explains his refined versions of the conjecture of Birch and Swinnerton-Dyer, which are motivated by the results obtained in his Ph. D. thesis [Refined class number formulas for derivatives of \(L\)-series, Ph. D. thesis, Harvard University (1991)] and in some of his subsequent papers [Euler systems and refined conjectures of Birch-Swinnerton-Dyer type, in: \(p\)-adic monodromy and the Birch-Swinnerton-Dyer conjecture, Boston University 1991, Contemp. Math. 165, 265-276 (1994); A refined conjecture of Mazur-Tate type for Heegner points, Invent. Math. 110, 123-146 (1992; Zbl 0781.11023)].
In the concluding two subsections, he gives some more examples for the theoretical and computational evidence of his refined conjectures.
For the entire collection see [Zbl 0788.00052].