Icosahedral invariants and Shimura curves

Nagano, Atsuhira

Journal de théorie des nombres de Bordeaux, Tome 29 (2017) no. 2, pp. 603-635.

Une courbe de Shimura est un espace de modules de surfaces abéliennes avec multiplication par une algèbre de quaternions. En utilisant les périodes pour une famille des surfaces $K3$ paramétrées par les invariants icosaédriques qui ont été étudiés par Klein, nous obtenons des modèles explicites de certaines courbes de Shimura.

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 $𝔄,𝔅$ and $ℭ$ 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 $\mathrm{Proj}\left(ℂ\right[𝔄,𝔅,ℭ\left]\right)$.

Classification : 11F46, 14J28, 14G35, 11R52
Mots clés : $K3$ surfaces, Abelian surfaces, Shimura curves, Hilbert modular functions, quaternion algebra

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