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Abelian Surfaces with an Automorphism and Quaternionic Multiplication

Published online by Cambridge University Press:  20 November 2018

Matteo Alfonso Bonfanti  and
Bert van Geemen
Matteo Alfonso Bonfanti
Affiliation:
Dipartimento di Matematica, Università di Milano, 20133 Milano, Italia e-mail: matteoalfonso.bonfanti@gmail.com e-mail: lambertus.vangeemen@unimi.it
Bert van Geemen
Affiliation:
Dipartimento di Matematica, Università di Milano, 20133 Milano, Italia e-mail: matteoalfonso.bonfanti@gmail.com e-mail: lambertus.vangeemen@unimi.it
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Abstract

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We construct one-dimensional families of Abelian surfaces with quaternionic multiplication, which also have an automorphism of order three or four. Using Barth's description of the moduli space of (2,4)-polarized Abelian surfaces, we find the Shimura curve parametrizing these Abelian surfaces in a specific case. We explicitly relate these surfaces to the Jacobians of genus two curves studied by Hashimoto and Murabayashi. We also describe a (Humbert) surface in Barth's moduli space that parametrizes Abelian surfaces with real multiplication by $\mathbf{Z}\left[ \sqrt{2} \right]$ .

Keywords

14K1011G1014K20abelian surfacesmodulishimura curves
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 68 , Issue 1 , 01 February 2016 , pp. 24 - 43
Copyright
Copyright © Canadian Mathematical Society 2016

References

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