In this paper we prove the p-adic analogue of a result of Hamida [11], namely that the p-adic Borel regulator introduced by Huber and Kings for the K-theory of a p-adic number field equals Karoubi's p-adic regulator up to an explicit rational factor.

In this paper we define a p-adic analogue of the Borel regulator for the K-theory of p-adic fields. The van Est isomorphism in the construction of the classical Borel regulator is replaced by the Lazard isomorphism. The main result relates this p-adic regulator to the Bloch–Kato exponential and the Soulé regulator. On the way we give a new description of the Lazard isomorphism for certain formal groups. We also show that the Soulé regulator is induced by continuous and even analytic classes.

Lazard showed in his seminal work (Groupes analytiques p-adiques, Publ. Math. Inst. Hautes Études Sci. 26 (1965), 389–603) that for rational coefficients, continuous group cohomology of p-adic Lie groups is isomorphic to Lie algebra cohomology. We refine this result in two directions: first, we extend Lazard’s isomorphism to integral coefficients under certain conditions; and second, we show that for algebraic groups over finite extensions K/ℚp, his isomorphism can be generalized to K-analytic cochains andK-Lie algebra cohomology.