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 and give the Hilbert modular forms for via the period mapping for a family of surfaces. Using the period mappings for several families of surfaces, we obtain explicit models of Shimura curves with small discriminant in the weighted projective space .
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 paramétrées par les invariants icosaédriques qui ont été étudiés par Klein, nous obtenons des modèles explicites de certaines courbes de Shimura.
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Keywords: $K3$ surfaces, Abelian surfaces, Shimura curves, Hilbert modular functions, quaternion algebra

@article{JTNB_2017__29_2_603_0, author = {Atsuhira Nagano}, title = {Icosahedral invariants and {Shimura} curves}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {603--635}, publisher = {Soci\'et\'e Arithm\'etique de Bordeaux}, volume = {29}, number = {2}, year = {2017}, doi = {10.5802/jtnb.993}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.993/} }
TY - JOUR AU - Atsuhira Nagano TI - Icosahedral invariants and Shimura curves JO - Journal de théorie des nombres de Bordeaux PY - 2017 SP - 603 EP - 635 VL - 29 IS - 2 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.993/ DO - 10.5802/jtnb.993 LA - en ID - JTNB_2017__29_2_603_0 ER -
%0 Journal Article %A Atsuhira Nagano %T Icosahedral invariants and Shimura curves %J Journal de théorie des nombres de Bordeaux %D 2017 %P 603-635 %V 29 %N 2 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.993/ %R 10.5802/jtnb.993 %G en %F JTNB_2017__29_2_603_0
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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