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This chapter is devoted to complex abelian surfaces whose endomorphism ring contains an order from a real quadratic field. The moduli spaces of such abelian surfaces are Hilbert modular surfaces. Since the moduli spaces of polarized complex abelian varieties are Siegel modular varieties we find natural maps of Hilbert modular surfaces to Siegel modular threefolds. The images are called Humbert surfaces, after G. Humbert, who studied abelian surfaces with real multiplication, i.e. with an endomorphism ring that contains an order from a real quadratic field.
KeywordsModulus Space Modular Form Boundary Component Abelian Variety Endomorphism Ring
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