Kings, Guido; Loeffler, David; Zerbes, Sarah Livia Rankin-Eisenstein classes and explicit reciprocity laws. (English) Zbl 1428.11103 Camb. J. Math. 5, No. 1, 1-122 (2017). 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. Cited in 5 ReviewsCited in 52 Documents 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 Keywords:critical values; \(L\)-functions; Rankin-Eisenstein classes PDFBibTeX XMLCite \textit{G. Kings} et al., Camb. J. Math. 5, No. 1, 1--122 (2017; Zbl 1428.11103) Full Text: DOI arXiv