Quaternionic loci in Siegel's modular threefold


Yifan Yang

14:00:00 - 15:00:00

103 , Mathematics Research Center Building (ori. New Math. Bldg.)

Let B be the indefinite quaternion algebra of discriminant D over Q, O be a maximal order in B, and X be the Shimura curve associated to O. An abelian surface over C is said to have QM by O if its endomorphism ring contains O. It is known that the set of moduli points of principally polarized abelian surfaces with QM by O in Siegel's modular threefold is the union of several curves, each of which is the image of X under the forgetful map. In this talk, we will first give an exact formula for the number of such curves in this quaternionic locus. Then for each curve in the locus, we describe a general method to determine the parametrization of the Igusa invariants along the curve in terms of modular functions on X. This provides a  simple criterion for a genus 2 curve to have QM by O.

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