## $$p$$-adic $$L$$-functions and rational points on elliptic curves with complex multiplication.(English)Zbl 0770.11033

The main result is the construction of rational points in $$E(\mathbb{Q})$$, $$E$$ an elliptic $$CM$$-curve over $$\mathbb{Q}$$. More precisely the author uses a method of Perrin-Riou to construct elements in the Selmer group, which come from rational points if the Tate-Shafarevich group is finite (which is true in many cases by results of Kolyvagin and the author, which ultimately depend on the work of Gross-Zagier about Heegner points). The $$p$$-adic height of these points is then related to special values of $$p$$- adic $$L$$-functions.

### MSC:

 11G18 Arithmetic aspects of modular and Shimura varieties 14G20 Local ground fields in algebraic geometry 14H52 Elliptic curves
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