## Reciprocity laws on curves.(English)Zbl 0706.14013

Let C be a compact Riemann surface and f,g harmonic functions on C with (disjoint sets of) logarithmic singularities. Then the reciprocity law for Green’s functions asserts that $$\sum_{p}ord_ p(f)g(p) =\sum_{p}ord_ p(g)f(p)$$ where $$ord_ p(f)$$ is the order of the logarithmic singularity of f at $$P\in C$$. The paper under review contains a new proof of this result, but this is merely a by-product on the way to the proof of the following p-adic analogue:
Let now C be a smooth projective curve over $${\mathbb{C}}_ p$$ with “arboreal” reduction and F, G abelian integrals of the third kind on C (thus dF and dG are algebraic differentials with simple poles and integral residues). The author’s reciprocity law now asserts $\sum_{p}(Res_ pdF)\cdot G(p) -\sum_{p}(Res_ pdG)\cdot F(p) =(\psi (dF),\psi (dG))$ if dF and dG have no common pole. Here $$\psi$$ is a $${\mathbb{C}}_ p$$-linear map from the space of algebraic differentials on C into the first algebraic de Rham cohomology group and the pairing is the cup product. The precise definition of the map $$\psi$$ as well as the proof of the theorem require a careful analysis of the rigid geometry of C and its reduction, based in turn on properties of what the author calls wide open sets (i.e. complements of closed disks).
Reviewer: F.Herrlich

### MSC:

 14G20 Local ground fields in algebraic geometry 14H05 Algebraic functions and function fields in algebraic geometry 30F15 Harmonic functions on Riemann surfaces 14K20 Analytic theory of abelian varieties; abelian integrals and differentials 14H25 Arithmetic ground fields for curves 30G06 Non-Archimedean function theory
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