On the syntomic regulator for K1 of a surface

We consider elements of K1(S), where S is a proper surface over a p-adic field with good reduction, which are given by a formal sum Σ(Zi, fi) with Zi curves in S and fi rational functions on the Zi in such a way that the sum of the divisors of the fi 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 HdR2(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 Σi〈Fη, log(fi); Fω〉gl,Zi, where Fω and Fη are Coleman integrals of ω and η, respectively, and the symbol in brackets is the global triple index, as defined in our previous work.

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