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).