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

By Herbert Lange, Wolfgang Barth, Klaus Hulek

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

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The curves £4,^5 and C& of the configuration K. parametrize certain quotients Ε χ EjZ2 where the product Ε χ Ε of elliptic curves carries a product polarization of type (2,2). To be more precise: For ζ G IHI let E, F be the elliptic curves Ε = C/(z,2) • Z2, F = C/(2z,2) • 1? e. 1 - ζ X X = c 1 - ζ =C2 2 \ ζ —ζ —ζ 2ζ 1 0 0 2 \ \ • Ζ4 1 0 0 2 or Χ = ζ4 / / ζ —ζ —ζ 2z + 2 1 0 \ 0 2 C2 7 or \ • Ζ4 / if Q belongs to the curves C4, C5, Cö respectively. The following table exhibits X then as a quotient of a product of isomorphic elliptic curves by a subgroup of order 2.

11 (ii) we have therefore established a part of the assertion. The corresponding proof for the curve C4 demands a lot of more work. a (IS0C4). We proceed in steps: Step I: We claim that gl3gΕ {±h} , gQzg~l G {±<^2} holds. 8 we conclude that the involution I3Q2 must be conjugate to I4 in Γ12 · Because conjugacy with g induces an automorphism of IS0C4 we get the claim. -J. Brasch S t e p I I I : Define the following matrices in Sp(4, Z) / Τ := for integers / \ 1 0 1 0 - 1 - 1 - 1 - 1 -1 V 0 1 1 0 0 0 0 M / a U b\ d) a b c b a d c —c d a - b d —c -6 a — 62 + c2 :— \ which satisfy the diophantine equation a, b,c,d, \ d a2 / — d 2 1.

This shows _1 that 4|b and diag(l, 2) Δ ι d i a g ( l , 2 ) G Γ (2). The different possibilities Δ ι = ill2 lead to I4 and I\. 4) A lenghty but straightforward computation yields that an element of type I2 ( Δ ι , Δ 2 ) is an element of finite order in Γι, η commuting with I2 iff Δχ, Δ2 are elements of finite order in SL(2, Z) with γΐ γι*ο a = d = 1 mod 2, ο; ξ δ = 1 mod —, β = 0 mod — , b = 0 mod 4. We conclude that diag(l, 2) Δ ι diag(l, 2 - 1 ) resp. diag(l, Δ2 diag(l, is congruent to the unit matrix modulo 2 resp.

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