Height pairings, exceptional zeros and Rubin’s formula: the multiplicative group

Abstract

In this paper we prove a formula, much in the spirit of one due to Rubin, which expresses the leading coefficients of various p-adic L-functions in the presence of an exceptional zero in terms of Nekováˇr’s p-adic height pairings on his extended Selmer groups. In a particular case, the Rubin-style formula we prove recovers a p-adic Kronecker limit formula. In a disjoint case, we observe that our computations with Nekováˇr’s heights agree with the Ferrero- Greenberg formula (more generally, Gross’ conjectural formula) for the leading coefficient of the Kubota-Leopoldt p-adic L-function (resp., the Deligne-Ribet p-adic L-function) at s = 0.