Reciprocity laws on curves. (English) Zbl 0706.14013

Coleman, Robert F.

Compos. Math. 72, No. 2, 205-235 (1989).

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

Keywords:

reciprocity law for Green’s functions; algebraic differentials; de Rham cohomology group; rigid geometry

References:

[1]	S. Bosch , Eine bemerkenswerte Eigenshaft der formellen Fasern affinoider Räume , Math. Ann. 229, (1977) 25-45. · Zbl 0385.32008
[2]	S. Bosch , Guntzer and R. Remmert , Non-Archimedean Analysis , Springer-Verlag, 1984. · Zbl 0539.14017
[3]	S. Bosch , Lutkeböhmert, Stable Reduction and uniformization of Abelian Varieties I , Math. Ann. 270 (1985). · Zbl 0554.14012
[4]	Coleman, R. , Dilogarithms, Regulators and p-adic Abelian Integrals , Invent. math. 69, (1982) 171-208. · Zbl 0516.12017
[5]	-, Torsion Points on Curves and p-adic Abelian Integrals , Ann. of Math., 121 (1985) 111-168. · Zbl 0578.14038
[6]	-, Bi-extensions, Universal Vectorial Extensions and Differentials of the Third Kind , to appear.
[7]	-, E. De Shalit , p-adic Regulators on Curves and Special Values of p-adic L-functions , to appear in Inventiones. · Zbl 0655.14010
[8]	-, B. Gross , p-adic heights and p-adic Green’s Functions , to appear.
[9]	-, W. Mccallum , Stable Reduction of Fermat Curves and Jacobi sum Hecke Characters , to appear in J. Reine und Angew. Math. · Zbl 0654.12003
[10]	Grauert, H. , Affinoide Überdeckungen einedimensionaler affinoider Räume . Publ. Math. IHES 34 (1968). · Zbl 0197.17302
[11]	Grothendieck, A. , Local Cohomology , Lecture Notes in Math. 41, Springer Verlag (1967). · Zbl 0185.49202
[12]	Hartshorne R. , On the de Rham Cohomology of Algebraic Varieties , Publ. Math. 45, (1976). · Zbl 0326.14004
[13]	Kiehl, R. , Der Einlichkeitsatz für eigenliche Abbildungen in der nichtarchimedishen Funktionen theorie , Math. 2, (1967) 191-214. · Zbl 0202.20101
[14]	, Die De Rham-Kohomologie algebraischer Mannigfaltigkeiten über einem bewerten Korper . Publ. Math. IHES 33 (1967). · Zbl 0159.22404
[15]	Lang, S. , Introduction to Diophantine Geometry . · Zbl 0214.06302
[16]	Raynaud M. , Variétiés Abéliennes et géométrie rigide, Actes , Congrès intern. math. Tome 1, (1970) 473-477. · Zbl 0223.14021
[17]	Serre, J-P. , Groupes algèbriques et corps de class . · Zbl 0718.14001