Singular homologies of non-archimedean analytic spaces and integrals along cycles : a research announcement (Algebraic Number Theory and Related Topics 2012)

Abstract

This article is an announcement of the author' s results in [Mih]. We construct a new homology, which admits a canonical action of the absolute Galois group G_{k} of the base field k, for Berkovich 's non-Archimedean analytic space. We also dene a new integration of an overconvergent differential form along a cycle. The integration takes its value in Fontaine' s p-adic period ring, and it induces a Galois equivariant pairing of the homology and the space of overconvergent differential forms. Moreover, we verify the homology and the integration satisfy many appropriate properties for homology and integration theories. We compute the homology of certain analytic spaces.