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Integration on $$p$$-adic varieties. (Intégration sur les variétés $$p$$-adiques.) (French) Zbl 0930.14013
Astérisque. 248. Paris: Société Mathématique de France, vii, 155 p. (1998).
Author’s abstract: We show that there is a unique “reasonable” way of integrating closed 1-forms on smooth algebraic varieties defined over a $$p$$-adic field. In contrast with the theory developed by Coleman, this $$p$$-adic integration does not require that the varieties under consideration have good reduction and can be used to extend to the general case several results obtained by Coleman in the case of good reduction; in particular the construction of $$p$$-adic periods of abelian varieties and the reciprocity law for differentials of the third kind. Having a theory which works for all primes it can be used to adelize certain constructions. For example, if $$X$$ is a smooth and proper algebraic curve defined over a number field, we define, in a purely analytic way, a pairing between divisors of degree 0 using adelic Green functions from which one can recover the Néron-Tate height pairing and $$p$$-adic analogues considered by Gross and Coleman in the case of good reduction.

##### MSC:
 14G20 Local ground fields in algebraic geometry 14K20 Analytic theory of abelian varieties; abelian integrals and differentials 14K25 Theta functions and abelian varieties 32G20 Period matrices, variation of Hodge structure; degenerations