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On local constants associated to arithmetical automorphic functions. (English) Zbl 1193.11037

This article concerns with the characterization of the \(\overline Q\)-rational functions of the canonical models of the Shimura curves associated to the indefinite quaternion Q-algebra of discriminant 6, through their expansions at CM-points. The authors derive an interesting formula which expresses a particular quaternionic automorphic function through quotients of Siegel automorphic forms. They introduce some local parameters ad obtain explicit expansions at the elliptic points and SCM-points for the functions defining the canonical models. The authors give the form of these local parameters depending on a constant \(k_p\) of the same transcendence class as certain products of values of Euler’s gamma function at rational arguments. They prove that the constants \(k_p\) agree, up to algebraic elements, with some specific constants \(\pi_d\) which only depend on the discriminant \(d\) of the field of complex multiplication. Finally, the authors stress the arithmetical meaning of the parameters \(\pi_d\) that is, that they allow the characterization of the canonical models through series expansions.

MSC:

11F03 Modular and automorphic functions
11G18 Arithmetic aspects of modular and Shimura varieties
14G35 Modular and Shimura varieties
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
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