In this paper we prove a formula, much in the spirit of one due to Rubin, which expresses the leading coefficients of various -adic -functions in the presence of an exceptional zero in terms of Nekovář’s -adic height pairings on his extended Selmer groups. In a particular case, the Rubin-style formula we prove recovers a -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 -adic -function (resp., the Deligne–Ribet -adic -function) at .
Cite this article
Kâzim Büyükboduk, Height pairings, exceptional zeros and Rubin’s formula: the multiplicative group. Comment. Math. Helv. 87 (2012), no. 1, pp. 71–111DOI 10.4171/CMH/249