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Relations monomiales entre périodes p-adiques. (Monomial relations between p-adic periods). (French) Zbl 0658.14023
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.
Reviewer: F.Herrlich

MSC:
14K22 Complex multiplication and abelian varieties
14G25 Global ground fields in algebraic geometry
32G20 Period matrices, variation of Hodge structure; degenerations
14K20 Analytic theory of abelian varieties; abelian integrals and differentials
11R80 Totally real fields
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