By Herbert Lange, Wolfgang Barth, Klaus Hulek

Booklet by way of Barth, Wolf, Hulek, Klaus

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Additional resources for Abelian Varieties: Proceedings of the International Conference Held in Egloffstein, Germany, October 3-8, 1993

Example text

In particular at least one of the three integers i>i, V3, υ4 is a unit in the ring Z / n Z . CASE 1: V4 = 1 Here Λ ξ 1 mod η and «1 ξ « 3 ξ 0 mod ^ must be satisfied, hence ω = with 3v Ξ 0 mod η. ( Let 1 ν Ν = 0 \ 0 0 1 0 0 1 —v 1 0 0 \ 0 G Sp(4,Z) with Ν ·ω = ω0. -υ 1 / This shows N R N € οΓι, η and the (2,4)-entry computes in a straightforward way to 3v2. Hence, we have the implication NRN~X Ε Γι ιΠ => ν = 0 mod η => ω = ωό . CASE 2: V\ = 1 o r V3 — 1 Making use of the action of i ? - 1 on the set of fixed points of R we reduce the investigation to the case U3 = 1.

By this identification the surfaces correspond to the coordinate planes, the curves to the coordinate axes and the respective (reduced) isotropy groups of these fixed varieties are identified with the subgroup lattice of the reflection group. e. away from the union of curves) is only fixed by the respective involution. The first section gives a solution to the 'conjugacy problem', which arises when we try to decide under which conditions an element of Γι >η is conjugate to some element of ΓιιΤΙ or Sp(4, Z ) .

C\ represents products Ερ χ Ε of elliptic curves, which carry product polarizations trivial on the first factor and of degree η on the second factor. Similarly C2 parametrizes products EiX Ε . Every singularity on C\ resp. C2 is locally isomorphic to a product of the modular curve X (n) and the rational triple point ^31 resp. the rational double point ^2,1 · The branch locus of Ai}n consists of f (n) disjoint surfaces where f (n) = 2, 4, 5 for η odd, 4|n and η = 2 mod 4 respectively. The generic branch point (i.

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