Relations monomiales entre périodes p-adiques. (Monomial relations between p-adic periods)
type: Item
instance of: scholarly article
title: Relations monomiales entre périodes p-adiques. (Monomial relations between p-adic periods)
zbMATH Open document ID: 0658.14023
DOI: 10.1007/BF01394337
author: Roland Gillard
published in: Inventiones Mathematicae
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143601
review text: Let K be a number field which is an imaginary quadratic extension of a totally real field and consider the family S of complex abelian varieties with complex multiplication by the ring of integers in K. By a theorem of Shimura, S is defined over a number field. One can define a canonical differential m-form \(\Omega\) (\(\tau)\) on S (depending on an embedding \(\tau:\quad K\hookrightarrow {\mathbb{C}}),\) and comparing with a suitable algebraic m-form \(\Omega^{alg}(\tau)\) on S one obtains a complex period P(\(\tau)\) on S which is constant as \(\Omega\) and \(\Omega^{alg}\) are horizontal for the Gauß-Manin connection. On the other hand one can define periods p(\(\tau\),\(\Phi)\) for each CM- type \(\Phi\) by suitable integration. Comparing P(\(\tau)\) and p(\(\tau\),\(\Phi)\) the author obtains a new proof of a theorem of Shimura saying that for two families \((\Phi_ i)\), \((\Phi '_ i)\), \(i=1,...,m\) of CM-types with equal formal sum the products of the periods are equal up to an algebraic number. The main achievement of the paper is an analogue construction of p-adic periods (for a prime p at which the given abelian variety has ordinary good reduction) and to prove the analogue of Shimura's theorem for these p-adic periods, thus confirming a conjecture of de Shalit.
Mathematics Subject Classification ID: 14K22
Mathematics Subject Classification ID: 14G25
Mathematics Subject Classification ID: 32G20
Mathematics Subject Classification ID: 14K20
Mathematics Subject Classification ID: 11R80
zbMATH DE Number: 4075272
zbMATH Keywords: abelian varieties with complex multiplication
zbMATH Keywords: p-adic periods
reviewed by: Frank Herrlich
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4075272