\(p\)-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.


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