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Rankin-Eisenstein classes and explicit reciprocity laws. (English) Zbl 1428.11103

Summary: We construct three-variable \(p\)-adic families of Galois cohomology classes attached to Rankin convolutions of modular forms, and prove an explicit reciprocity law relating these classes to critical values of \(L\)-functions. As a consequence, we prove finiteness results for the Selmer group of an elliptic curve twisted by a 2-dimensional odd irreducible Artin representation when the associated \(L\)-value does not vanish.

MSC:

11F85 \(p\)-adic theory, local fields
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
14G35 Modular and Shimura varieties
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