Height pairings, exceptional zeros and Rubin's Formula: The multiplicative group

TL;DR

This paper establishes a formula relating leading coefficients of p-adic L-functions with exceptional zeros to Nekovar's p-adic height pairings, unifying several classical and conjectural formulas in number theory.

Contribution

It proves a Rubin-style formula connecting p-adic L-function coefficients with p-adic heights, extending and unifying previous results and conjectures.

Findings

Recovers a p-adic Kronecker limit formula.

Aligns Nekovar's heights with Ferrero-Greenberg and Gross conjectures.

Provides a new perspective on exceptional zeros in p-adic L-functions.

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 Nekovar'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 Nekovar'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.