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\(p\)-adic abelian integrals and commutative Lie groups. (English) Zbl 0884.14020

The note gives a self-contained, elementary and very elegant approach to \(p\)-adic abelian integrals introduced by R. Coleman [Ann. Math. 121, 111-168 (1985; Zbl 0578.14038)]. The author first defines in a direct way the logarithm map log: \(G(K) \rightarrow \text{Lie}(G)\) for any commutative algebraic group over a complete local field \(K \supset \mathbb{Q}_p\). Let \(\omega\) be a differential form on a projective variety \(V\) over \(K\) having only logarithmic singularities. Then there exist an algebraic group \(G\) and a rational map \(f: V \rightarrow G\) such that \(\omega = f^*(\tau)\) for some \(\tau \in \text{Lie}(G)\). Then for any two points \(P,Q \in V(K)\) we can put the integral of \(\omega\) from \(P\) to \(Q\) equal to \(\tau (\log f(P) - \log f(Q)) \in K\). This definition can be also worked out for closed regular forms.

MSC:

14K20 Analytic theory of abelian varieties; abelian integrals and differentials
14G20 Local ground fields in algebraic geometry
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)

Citations:

Zbl 0578.14038
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References:

[1] N. Bourbaki,Groupes et Algébres de Lie, Hermann, Paris (1972). · Zbl 0244.22007
[2] R. F. Coleman, ”Torsion points on curves andp-adic abelian integrals,”Ann. Math.,121, 111–168 (1985). · Zbl 0578.14038 · doi:10.2307/1971194
[3] R. F. Coleman, ”Reciprocity laws on curves,”Compos. Math.,72, 205–235 (1989). · Zbl 0706.14013
[4] R. F. Coleman and B. Gross, ”p-Adic heights on curves,” In:Adv. Stud. Pure Math., Vol. 17 (1989), pp. 73–81. · Zbl 0758.14009
[5] G. Faltings and G. Wüstholz, ”Einbettungen commutativer algebraischer Gruppen und einige ihrer Eigenschaften,”J. Reine Angew. Math.,354, 175–205 (1984). · Zbl 0543.14029 · doi:10.1515/crll.1984.354.175
[6] M. Raynaud, ”Varietes abéliennes et geometrie rigide,” In:Actes Congres Intern. Math., Vol. 1 (1970), pp. 473–477.
[7] J.-P. Serre, ”Morphismes universels et varieté d’Albanese,” In:Séminaire C. Chevalley (1958/59), E.N.S. Paris.
[8] J.-P. Serre, ”Morphismes universels et differentielles de troisieme espéce,” In:Séminaire C. Chevalley (1958/59), E.N.S., Paris.
[9] J.-P. Serre,Groupes Algèbriques et Corps de Classes, Hermann, Paris (1959). · Zbl 0097.35604
[10] I. R. Shafarevich,Basic Algebraic Geometry, Springer-Verlag, Berlin-Heidelberg-New York (1974). · Zbl 0284.14001
[11] A. Weil,Foundations of Algebraic Geometry, AMS Colloquium Publications29, Revised edition, American Mathematical Society, Providence, Rhode Island (1962). · Zbl 0168.18701
[12] Yu. G. Zarhin, ”p-adic heights on abelian varieties,” In:Séminaire de Théorie des Nombres, Paris 1987–88 (C. Goldstein, ed.).Progr. Math., Vol. 81, Birkhäuser, Basel (1990), pp. 317–341.
[13] Yu. G. Zarhin, ”Néron pairing and quasicharacters,”Izv. Akad. Nauk SSSR. Ser. Mat.,36, 497–509 (1972); English translation inMath. USSR Izv.,6, 491–503 (1972).
[14] Yu. G. Zarhin, ”Local heights and Néron pairings,” In:Tr. Mat. Inst. Rossiisk. Akad. Nauk, Vol. 208 (1995), pp. 111–127. · Zbl 0879.14008
[15] Yu. G. Zarhin, ”Local heights and abelian integrals,” Exposé 9 dans ”Problemes Diophantiens 88–89” (D. Bertrand, M. Waldschmidt, eds.),Publ. Math. Univ. Paris VI90 (1990).
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