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\(p\)-adic \(L\)-functions and Selmer varieties associated to elliptic curves with complex multiplication. (English) Zbl 1223.11080

The principle of Birch and Swinnerton-Dyer states that nonvanishing of \(L\)-values accounts in some appropriate sense for the finiteness of integral points. The author aims at extending these ideas to hyperbolic curves. The present paper discusses the special case of a hyperbolic curve of genus 1 over \(\mathbb{Q}\) obtained from an elliptic curve over \(\mathbb{Q}\) with complex multiplication by removing the origin.

MSC:

11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11G05 Elliptic curves over global fields
11G15 Complex multiplication and moduli of abelian varieties
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