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By Herbert Lange, Wolfgang Barth, Klaus Hulek

ISBN-10: 3110144115

ISBN-13: 9783110144116

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

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In the complement of C\ U C% U C3) is only fixed by the respective involution. The curve C3 is isomorphic to the modular curve X (3) ; branch points lying on the curve C3 have the symmetric group of degree 3 as their isotropy group. The case η — 2 will be the hardest part to deal with. Throughout this paragraph we do not regard —14 G Γι,2 as an involution. We say an element of Sp(4, Z) normalizes an involution if the homomorphism induced by conjugacy maps this involution to its negative. 2. The conjugate matrix ±MI0M~l ments Μ e Sp(4, Z ) .

Then we obtain [Fi] = [A] • h\ = a2zh\h22hz + aZ2h\h23h2. Furthermore, we have [Fi] = d\ · [Fi], hence a2Z = di • a > 6 · 2 = 12 and also 032 > 1 2 . Arguing in the same way with the projections φ2 and φζ we obtain a,ij > 12 for i,j = 1,2,3, i Φ j. 1 then yields α - 1 8 3 < a(a - 27) = ϋ ( 9 ~ a kj) ^ _216 ' (i,j,fc)es3 a contradiction. We conclude that not all of the projections ψι, φ 2 and surjective. can be • 3. 1) T h e o r e m . Let A be an abelian threefold over C, embedded in P2 x P2 χ UV Then A is a product E\ χ E2 x F3, where E\, E2 and are smooth plane cubics.

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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