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Elliptic curves and class fields of real quadratic fields: Algorithms and evidence. (English) Zbl 1040.11048
Summary: In [{H. Darmon}, “Integration on \({\mathcal H}_p\times {\mathcal H}\) and arithmetic applications”, Ann. Math. (2) 154, 589–639 (2001; Zbl 1035.11027)] the first author proposed a conjectural \(p\)-adic analytic construction of points on (modular) elliptic curves, points which are defined over ring class fields of real quadratic fields. These points are related to classical Heegner points in the same way as Stark units to circular or elliptic units. For this reason they are called “Stark-Heegner points”, following a terminology introduced in [H. Darmon, Contemp. Math. 210, 41–69 (1998; Zbl 0923.11099)].
If \(K\) is a real quadratic field, the Stark-Heegner points attached to \(K\) are conjectured to satisfy an analogue of the Shimura reciprocity law, so that they can in principle be used to find explicit generators for the ring class fields of \(K\). It is also expected that their heights can be expressed in terms of derivatives of the Rankin \(L\)-series attached to \(E\) and \(K\), in analogy with the Cross-Zagier formula.
The main goal of this paper is to describe algorithms for calculating Stark-Heegner points and supply numerical evidence for the Shimura reciprocity and Gross-Zagier conjectures, focussing primarily on elliptic curves of prime conductor.

MSC:
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11G05 Elliptic curves over global fields
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