This paper deals with group actions of one-dimensional formal groups defined over the ring of integers in a finite extension of the p-adic field, where the space acted upon is the maximal ideal in the ring of integers of an algebraic closure of the p-adic field. Given a formal group F as above, a formal flow is a series Φ(t,x) satisfying the conditions Φ(0,x)=x and Φ(F(s,t),x)=Φ(s,Φ(t,x)). With this definition, any formal group will act on the disk by left translation, but this paper constructs flows Φ with any specified divisor of fixed points, where a point ξ of the open unit disk is a fixed point of order [les ]n if (x−ξ)n|(Φ(t,x)−x). Furthermore, if γ is an analytic automorphism of the open unit disk with only finitely many periodic points, then there is a flow Φ, an element α of the maximal ideal of the ring of constants, and an integer m such that the m-fold iteration of γ(x) is equal to Φ(α,x). All the formal flows constructed here are actions of the additive formal group on the unit disk. Indeed, if the divisor of fixed points of a formal flow is of degree at least two, then the formal group involved must become isomorphic to the additive group when the base is extended to the residue field of the constant ring.

In this paper we study the group K2n(n+1)(F) where F is the function field of a complete, smooth, geometrically irreducible curve C over a number field, assuming the Beilinson–Soulé conjecture on weights. In particular, we compute the Beilinson regulator on a subgroup of K2n(n+1)(F), using the complexes constructed in Compositio Math. 96 (1995), pp. 197–247. We study the boundary map in the localization sequence for n=2 and n=3. We combine our results with results of Goncharov in order to obtain a complete description of the image of the regulator map on K4(3)(C) and K6(4)(C) (which have the same images as K4(C)$⊗ _{\Bbb Z}\, {\Bbb Q}$, and K6(C)$⊗ _{\Bbb Z}\, {\Bbb Q}$, respectively), independent of any conjectures.

We prove a certain Riemann–Roch-type formula for symmetric powers of Galois modules on Dedekind schemes which, in the number field or function field case, specializes to a formula of Burns and Chinburg for Cassou–Noguès–Taylor operations.