By Herbert Lange, Wolfgang Barth, Klaus Hulek

ISBN-10: 3110144115

ISBN-13: 9783110144116

Booklet by way of Barth, Wolf, Hulek, Klaus

**Read Online or Download Abelian Varieties: Proceedings of the International Conference Held in Egloffstein, Germany, October 3-8, 1993 PDF**

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

**Sample text**

4] • N o t a t i o n . The following three matrices are elements of finite order in Γι ιΤΙ : / Ο R := 0 1 0 \ 0 1 0 0 - 1 0 - 1 0 \ 0 0 0 1 J equal to Ueno III(2)a = J 0 (X 4 , H2) S := f 0 0 -1 0 \ 0 1 0 0 1 0 1 0 \ 0 0 0 1 ) equal to Ueno III(4)a = / 0 (X, 1 2 ) Τ / 0 0 -1 0 1 0 1 0 0 \ 0 0 0 equal to Ueno III(5)a = I 0 (Y, H2). := 0 \ 0 0 1 ) We determine the relations T 2 = S3 = Iq , S 4 = R. Their fixed varieties are given as follows ( ρ := exp (^f 1 )): C\ = {ρ} χ Η is fixed by R,S C2 = {i} x Η is fixed by T.

Every involution in ΓιιΤΙ is conjugate with respect to Γι >η to exactly one of the following matrices in dependence on η mod 4. 1) η ξ ± 1 mod 4 ; /ο,^3· 2) η Ξ 0 mod 4 : Ιο,Ιχ,Ι^,Ι^. 3) η = 2 mod 4 : J 0 , Ji, J2, J3, J4· Proof. Every involution in Γι )ΤΙ is conjugate to Jo or J3 in Sp(4, Z ) by the Uenoclassification. Therefore it suffices to investigate how the conjugacy classes of I0J3 split in Γι ι Τ Ι . 3 we get the involutions Jo, (Ji), (J2) in dependence on n. 3. • 3. The case η > 4 Throughout the following sections we collect elements of Γ ι ) η having a fixed point Ζ G KI2 with isotropy group I s o Z = { j e Γ ι ι η | g • Ζ = Ζ } ; it is clear how to generalize this concept to an arbitrary submanifold of H2 .

35 Branch points in moduli spaces of certain abelian surfaces vi P r o o f . L e t ω υ2 be a fixed point of I3 with gcd (ui, "3 υ4 J c, d , A , B , which satisfy the properties: — mine integers a, b, t>i3 : = gcd(t;i,ü3) = αν ι + bv 3, gcd v24 ( ν 2 , ν 4 ) = c v v + 2 4, dv = V3,V4) , 2 A v 13 + 1. Deter- B v 2 1. = 4 The equation for the fixed point ω of I3 yields the existence of an integer Λ with Λυχ = — υι — V2 mod η λν 2 = v 2 mod η = — U3 mod η Xvs Aui = — vz v^ mod n . Combining the above congruences we get in particular a v 13 (A + 1) mod —ν = 2 η , - dv3 (1 — λ ) mod n.

### 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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