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Download PDF by Herbert Lange, Wolfgang Barth, Klaus Hulek: Abelian Varieties: Proceedings of the International

By Herbert Lange, Wolfgang Barth, Klaus Hulek

ISBN-10: 3110144115

ISBN-13: 9783110144116

Ebook by means of Barth, Wolf, Hulek, Klaus

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

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

Example text

Proof. Choose arbitrary elements g Ε Iso Τ , Ζ € J- . If h is contained in the invariance group ln\ Τ we compute h~lgh · Ζ = h~lh • Ζ = Ζ showing that h is contained in the normalizer of Iso J*7. Conversely let h be an element of the normalizer. We consider the identity h_1gh • Ζ = Ζ to conclude that the submanifold h · Τ is fixed pointwise under the action of the group Iso Τ. This yields h € Inv Τ by the assumptions. • Before we state the theorem, which describes the branch and singular locus in A\

To be more precise: the matrices Ιο,Ιι, I3 are involutions of Γ 1 η and / 2 is contained in Γι „ iff η ξ 0 mod 2 holds. The involutions are of special interest since they are up to conjugation the involutions contained in Sp(4, Z). Note, that Io = —Ueno II(l)a and J3 is conjugate to Ueno 11(1 )b because we have 33 Branch points in moduli spaces of certain abelian surfaces f N~ I3N = Ueno II(l)b for Ν = 0 1 0 0 1 0 0 0 ^ 0 -1 0 0 0 \ 0 1 0 ) For each r = 0, 1, 2, 3 there is a deeper correspondence between the special matrix Ir and the general element Ir (Δι, Δ2).

Proof. We have to show that π (C 0 r, n (/)) contains a generating set for the group of upper triangular matrices in SL(2, Z / n Z ) . - 1 ) i t s % in the ring Z / n Z . We will treat the several cases. ). Case: I - h We may assume, that Η = 0 mod 2 holds, since otherwise IQ and I\ are conjugate to each other. Now we have to check when a symplectic matrix of type /χ (Δι, Δ2) is contained in οΓχ, η . This gives as a necessary and sufficient condition: Δ ι , Δ2 € SL(2, Ζ), Δ ι - Δ 2 ξ *) mod 2, b = 0 mod 2 , β = 0 mod η.

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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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