Icosahedral invariants and Shimura curves
Atsuhira Nagano1
1 Department of Mathematics King’s College London Strand, London, WC2R 2LS, UK
Journal de théorie des nombres de Bordeaux, Volume 29 (2017) no. 2, pp. 603-635.

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 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 Proj(ℂ[𝔄,𝔅,ℭ]).

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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.993
Classification: 11F46, 14J28, 14G35, 11R52
Keywords: $K3$ surfaces, Abelian surfaces, Shimura curves, Hilbert modular functions, quaternion algebra
Atsuhira Nagano 1

1 Department of Mathematics King’s College London Strand, London, WC2R 2LS, UK
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Atsuhira Nagano. Icosahedral invariants and Shimura curves. Journal de théorie des nombres de Bordeaux, Volume 29 (2017) no. 2, pp. 603-635. doi : 10.5802/jtnb.993. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.993/

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