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Algebraic Geometry

By Herbert Lange, Wolfgang Barth, Klaus Hulek

ISBN-10: 3110144115

ISBN-13: 9783110144116

Publication via Barth, Wolf, Hulek, Klaus

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

Example text

9 to every element /3,574p -1 and L corresponds a unitary (2 χ 2)-matrix of finite order. We may assume that the matrix ( J corresponds -1 to I3 and U = represents ghg under this isomorphism of finite groups. Since U2 = II2 must hold, we get u 2 = Ü3 . Prom this we conclude that _02) U is not an involution, because this would yield the contradiction U2 — —«3, hence U = ± (g1 . For this reason we see L2 φ Ά2. 10 (i) it must be conjugate to some element of the Ueno-class 11(2). Taking all former results into account Ζ must be equivalent to a point on one of the curves C4, C5 or CQ .

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 . Before developing the main results we have to add some simple properties of (2 χ 2)-matrices.

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.

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