Geodesic
Abelian Surfaces with an Automorphism and Quaternionic Multiplication
Canadian journal of mathematics, Tome 68 (2016) no. 1, pp. 24-43

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DOI

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 $\mathbf{Z}\left[ \sqrt{2} \right]$ .
DOI : 10.4153/CJM-2014-045-4
Mots-clés : 14K10, 11G10, 14K20, abelian surfaces, moduli, shimura curves 
Bonfanti, Matteo Alfonso; Geemen, Bert van. Abelian Surfaces with an Automorphism and Quaternionic Multiplication. Canadian journal of mathematics, Tome 68 (2016) no. 1, pp. 24-43. doi: 10.4153/CJM-2014-045-4
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     title = {Abelian {Surfaces} with an {Automorphism} and {Quaternionic} {Multiplication}},
     journal = {Canadian journal of mathematics},
     pages = {24--43},
     year = {2016},
     volume = {68},
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     doi = {10.4153/CJM-2014-045-4},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-2014-045-4/}
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