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Zbl 1015.11028
Breuil, Christophe
Integration on $p$-adic varieties (after Coleman, Colmez). (Intégration sur les variétés $p$-adiques (d'après Coleman, Colmez).)
(French)
[A] Séminaire Bourbaki. Volume 1998/99. Exposés 850-864. Paris: Société Mathématique de France, Astérisque. 266, 319-350, Exp. No. 860 (2000).

The purpose of this article is to introduce the reader to the theory of integration on $p$-adic varieties due to {\it P. Colmez} [Astérisque 248 (1998; Zbl 0930.14013)] and {\it R. F. Coleman} [Ann. Math. (2) 121, 111--168 (1985; Zbl 0578.14038)]. Colmez's construction applies to smooth varieties over a local field $K$ and gives us the following: There exists a unique way to associate to a smooth variety $X$ over $K$ and a rational closed 1-form $\omega$ on $X$, a function $f_\omega$, which is well defined up to addition of a constant and locally analytic on $X(K)- \{\text{poles of }\omega\}$, such that (i) $df_\omega=\omega$, (ii) $f_{\lambda\omega+ \lambda'\omega'}= \lambda f_\omega+ \lambda'f_{\omega'}$ $(\lambda,\lambda'\in K)$, (iii) $f_\omega= f$ if $\omega=df$ with $f\in K(X) [\log(h)$; $h\in K(X)^\times]$, (iv) $f_{g^*\omega}= g^*f_\omega$ if $g:X'\to X$ is a morphism of smooth varieties over $K$. \par The proof by Colmez reduces the problem to the case of Abelian varieties by using the Albanese morphisms, and it constructs effectively this integral on the Abelian varieties by using the theorem of square. Coleman's construction relies on rigid geometry, and the Frobenius morphism plays a role analogous to the theorem of square in Colmez's integral. \par The author also explains two applications of the theory: $p$-adic periods of Abelian varieties and $p$-adic polylogarithms.
[Fumio Hazama (Hatoyama)]
MSC 2000:
*11G25 Varieties over finite and local fields
11G10 Abelian varieties of dimension $>1$
14F40 De Rham cohomology
14G22 Rigid analytic geometry
11G20 Curves over finite and local fields
14K15 Arithmetic ground fields (abelian varieties)

Keywords: rational 1-form; locally analytic function; abelian variety; $p$-adic period; Frobenius operator; $p$-adic polylogarithm; Coleman integration; Colmez integration; integration on $p$-adic varieties; Albanese morphisms; rigid geometry; Colmez's integral

Citations: Zbl 0930.14013; Zbl 0578.14038

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