Cambridge Journal of Mathematics
Volume 5 (2017)
Rankin–Eisenstein classes and explicit reciprocity laws
Pages: 1 – 122
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.
2010 Mathematics Subject Classification
11F67, 11F85, 11G40, 14G35