Download e-book for iPad: Abelian Varieties: Proceedings of the International by Herbert Lange, Wolfgang Barth, Klaus Hulek

By Herbert Lange, Wolfgang Barth, Klaus Hulek

ISBN-10: 3110144115

ISBN-13: 9783110144116

Booklet by means of Barth, Wolf, Hulek, Klaus

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

Sample text

Proof Let ω e Ω be a fixed point of J 0 . D. reads as Ac ξ —c mod η , Λd = d mod η for some integer A. In particular this gives λ 2 ξ 1 mod η and 2cd = 0 mod η . -J. Brasch Nc,d 0 0 d -b ( a b c d := 0 0 \ 0 0 0 \ 0 -c a J with the property Nc>d Γ0Ί 0 c -d- ' 0" 0 0 1 Now we calculate ( Nc,d Iq = -(ad + bc) —2 cd 0 0 V 2ab be + ad 0 0 0 0 (ad + bc) 2ab \ 0 0 e οΓΐ,η· —2 cd ad + bc J This shows that the intersection of the conjugacy class [ NCtd Iq ]o C οΓι,η with ΓιιΤΙ is not empty only in the case that the congruence ad + be = 1 mod τι is fulfilled.

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. 5 every element of Iso Ζ is conjugate to Jo, I3, 5,±1,T±1 with respect to Sp(4,Z). In particular every element of Iso Ζ contains 1 as an eigenvalue of multiplicity 2.

Since Μ is an element of finite order, the equation shows that (^ is an element of finite order, too. An immediate calculation now gives diag(l,2) (Q diag(l, 2 _ 1 ) e Γ (2). -J. 11. i) Suppose Μ is an element of finite order in Γι,2 and normalizes I5, then Μ is up to conjugacy and sign one of the matrices I3, 14, I5I3, I5I4. ii) The isotropy group of the fixed curve C5 (resp. ) is generated by the involutions Is and I3 (resp. e. the subgroup of quasi-reflections coincides with the full isotropy group for both curves.

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Abelian Varieties: Proceedings of the International Conference Held in Egloffstein, Germany, October 3-8, 1993 by Herbert Lange, Wolfgang Barth, Klaus Hulek


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