On the syntomic regulator for K 1 of a surface

Abstract

We consider elements of K 1(S), where S is a proper surface over a p-adic field with good reduction, which are given by a formal sum Σ(Z i , f i ) with Z i curves in S and f i rational functions on the Z i in such a way that the sum of the divisors of the f i is 0 on S. Assuming compatibility of pushforwards in syntomic and motivic cohomologies, our result computes the syntomic regulator of such an element, interpreted as a functional on H 2dR (S), when evaluated on the cup product ω∪[η] of a holomorphic form ω by the first cohomology class of a form of the second kind η. The result is Σ i 〈F η , log(f i ); F ω 〉gl,Z i , where F ω and F η are Coleman integrals of ω and η, respectively, and the symbol in brackets is the global triple index, as defined in our previous work.