Inventiones mathematicae

, Volume 133, Issue 2, pp 417-447

First online:

Syntomic regulators and special values of p-adic L-functions

  • Manfred KolsterAffiliated with McMaster University, Department of Mathematics, 1250 Main Street West, Hamilton, Ontario L8S 4K1, Canada
  • , Thong Nguyen Quang DoAffiliated with Université de Franche-Comté, Laboratoire de Mathématiques, 16, Route de Gray, F-25030 Besancon Cedex, France

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In this paper p-adic analogs of the Lichtenbaum Conjectures are proven for abelian number fields F and odd prime numbers p, which generalize Leopoldt's p-adic class number formula, and express special values of p-adic L-functions in terms of orders of K-groups and higher p-adic regulators. The approach uses syntomic regulator maps, which are the p-adic equivalent of the Beilinson regulator maps. They can be compared with étale regulators via the Fontaine-Messing map, and computations of Bloch-Kato in the case that p is unramified in F lead to results about generalized Coates-Wiles homomorphisms and cyclotomic characters.