A modification of the Kontsevich-Zagier conjecture on periods. (Une version relative de la conjecture des périodes de Kontsevich-Zagier.)(French. English summary)Zbl 1408.32016

The class of algebraic numbers is too narrow to include many common mathematical constants, while the set of transcendental numbers is not countable, and its members are in general not computable. Periods, which are complex numbers constructed from algebraic objects, form a suitable enlargement, which remains computable, and at the same time contains many interesting numbers. Among them are, at least conjecturally [P. Deligne, Proc. Symp. Pure Math. 33, 313–346 (1979; Zbl 0449.10022)], the special $$L$$-values of motivic $$L$$-functions.
Theorem 1.8, the main result of this article, is equivalent to a relative version of the period conjecture of M. Kontsevich and D. B. Zagier as stated in [“Periods”, in: Mathematics unlimited – 2001 and beyond. Berlin: Springer. 771–808 (2001; doi:10.1007/978-3-642-56478-9_39), §4.1].
More precisely, let $$\overline{\mathbb D}{}^n=\{ z=(z_1,\dots ,z_n) \in \mathbb C^n:|z_i| \leq 1\}$$ be the closed polydisc of radius one. Let $$f_r=f_r(z)$$ be a holomorphic function defined on an open neighbourhood of $$\overline{\mathbb D}{}^n$$. Let $$F=F_z(\varpi) = \sum_{r \gg -\infty} f_r(z) \varpi^r$$ be a formal series in the variable $$\varpi$$. Integrating the series on the real cube $$[0,1]^n$$ yields a Laurent series $$\int_{[0,1]^n} F = \int_{[0,1]^n} F_z(\varpi) \,\, dz$$. When $$F$$ is algebraic then $$\int_{[0,1]^n} F$$ is a series of periods. The main goal of this article is to determine the algebraic series $$F$$ such that $$\int_{[0,1]^n} F=0$$ which provides information about the transcendence properties of the series of periods.
Theorem 1.8 confirms that a linear or even algebraic relationship between series of periods is always due to an ‘almost-algebraic’ relationship. In this sense it delivers a transcendence result for series of periods. Though the announcement of the theorem is relatively elementary, the proof relies on a number of heavy machineries from motivic cohomology theory and neighbouring branches. It is unknown whether a more elementary proof of Theorem 1.8 can be obtained.

MSC:

 32G20 Period matrices, variation of Hodge structure; degenerations 11J81 Transcendence (general theory) 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols 14F42 Motivic cohomology; motivic homotopy theory

Zbl 0449.10022
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