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Atsuhira Nagano
Icosahedral invariants and Shimura curves
Journal de théorie des nombres de Bordeaux, 29 no. 2 (2017), p. 603-635, doi: 10.5802/jtnb.993
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Class. Math.: 11F46, 14J28, 14G35, 11R52
Keywords: $K3$ surfaces, Abelian surfaces, Shimura curves, Hilbert modular functions, quaternion algebra

Résumé - Abstract

Shimura curves are moduli spaces of abelian surfaces with quaternion multiplication. Models of Shimura curves are very important in number theory. Klein’s icosahedral invariants $\mathfrak{A},\mathfrak{B}$ and $\mathfrak{C}$ give the Hilbert modular forms for $\sqrt{5}$ via the period mapping for a family of $K3$ surfaces. Using the period mappings for several families of $K3$ surfaces, we obtain explicit models of Shimura curves with small discriminant in the weighted projective space $\mathrm{Proj}(\mathbb{C}[\mathfrak{A},\mathfrak{B},\mathfrak{C}])$.

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