Cambridge Journal of Mathematics

Volume 5 (2017)

Number 1

Rankin–Eisenstein classes and explicit reciprocity laws

Pages: 1 – 122



Guido Kings (Fakultät für Mathematik, Universität Regensburg, Germany)

David Loeffler (Mathematics Institute, University of Warwick, Coventry, United Kingdom)

Sarah Livia Zerbes (Department of Mathematics, University College London, United Kingdom)


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

Published 28 March 2017