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 Nekovář’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 Nekovář’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 $s=0$.