By Herbert Lange, Wolfgang Barth, Klaus Hulek
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Additional resources for Abelian Varieties: Proceedings of the International Conference Held in Egloffstein, Germany, October 3-8, 1993
Brasch The moduli space A\, n has at worst quotient singularities. One can view the quotient map H2 —• Ai
The singular locus of «4ι,η consists of the two disjoint curves C\ and C2 each isomorphic to the modular curve Χ (η), which are contained in the Humbert surface Τίο . Every singularity on C\ resp. C2 is locally isomorphic to a product of the modular curve X (η) and the rational triple point Ast 1 resp. e. in the complement of Cx U C2 ) is only fixed by the respective involution. -J. Brasch 42 Proof. Singular points or branch points arise as fixed points of elements of finite order in Γ ι ) η . 2 with non-trivial isotropy group Iso Z.
Let χ be an integer prime to n. By eventually substituting χ by χ + η we may assume that Branch points in moduli spaces of certain abelian surfaces 37 x is odd. This implies the existence of integers r, t with xr — Atn2 = 1. Setting we observe that the matrix ( 0 χ χ - ι ) is contained in the image. Combining everything the image is the group of upper triangular matrices for η odd and in the remaining case 'η even' a subgroup of index 2 as claimed. 5. Every involution in ΓιιΤΙ is conjugate with respect to Γι >η to exactly one of the following matrices in dependence on η mod 4.
Abelian Varieties: Proceedings of the International Conference Held in Egloffstein, Germany, October 3-8, 1993 by Herbert Lange, Wolfgang Barth, Klaus Hulek