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Computation of Heegner points. (English) Zbl 0559.14010
Modular forms, Symp. Durham/Engl. 1983, 13-41 (1984).
The authors have been, over several years recently, carrying out an enormous amount of explicit calculations of Heegner points on some elliptic curves \(E\) arising from modular curves \(X_ 0(N)\); on the basis of these computations, they were led to formulate interesting conjectures (on the Tate height of the canonical rational points given by Heegner’s remarkable construction and) involving the values (at 1) of \(L\)-series and derivatives, associated with \(E\) and its ’twists’, which (conjectures) are indeed described in §4. (A nice consequence of two of these conjectures, for example, is that the rank of \(E(\mathbb Q(\sqrt{8d}))\) is 1 for \(E=X_ 0(11)\), whenever \(\pm d\) is a product of distinct primes which are quadratic residues modulo 11 and are, further, not of the form \(x^ 2+11 y^ 2\) with \(x,y\) in \(\mathbb Z\).) The beautiful tables of the authors’ computations for most \(E\) with conductor \(\leq 50\) and many others with larger conductor are reproduced in §3, although most of these conjectures seem to have been proved now by B. Gross and D. Zagier [cf. C. R. Acad. Sci., Paris, Ser. I 297, 85–87 (1983; Zbl 0538.14023); see also D. Goldfeld’s remark: ”Unhappily, it is a consequence of this conjecture (of Birch-Stephens) that the Heegner points turn out to be trivial whenever the rank of \(E(K)\) is more than one”, on page 34 of his excellent article in Bull. Am. Math. Soc., New Ser. 13, 23–37 (1985; Zbl 0572.12004)]. The tables provide a welcome illustration for some earlier work of Gross on the non-triviality of Heegner points of Eisenstein curves.
[For the entire collection see Zbl 0546.00010.]
Reviewer: S.Raghavan

14G05 Rational points
11G05 Elliptic curves over global fields
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
14-04 Software, source code, etc. for problems pertaining to algebraic geometry
14H45 Special algebraic curves and curves of low genus
11D25 Cubic and quartic Diophantine equations