Kyoto Journal of Mathematics

A theta expression of the Hilbert modular functions for 5 via the periods of K3 surfaces

Atsuhira Nagano

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In this paper, we give an extension of the classical story of the elliptic modular function to the Hilbert modular case for Q(5). We construct the period mapping for a family F={S(X,Y)} of K3 surfaces with 2 complex parameters X and Y. The inverse correspondence of the period mapping gives a system of generators of Hilbert modular functions for Q(5). Moreover, we show an explicit expression of this inverse correspondence by theta constants.

Article information

Kyoto J. Math., Volume 53, Number 4 (2013), 815-843.

First available in Project Euclid: 21 November 2013

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Zentralblatt MATH identifier

Primary: 14J28: $K3$ surfaces and Enriques surfaces
Secondary: 11F46: Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms 33C05: Classical hypergeometric functions, $_2F_1$


Nagano, Atsuhira. A theta expression of the Hilbert modular functions for $\sqrt{5}$ via the periods of $K3$ surfaces. Kyoto J. Math. 53 (2013), no. 4, 815--843. doi:10.1215/21562261-2366102.

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