ABELIAN VARIETIES ASSOCIATED TO GAUSSIAN LATTICES

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ABELIAN VARIETIES ASSOCIATED TO GAUSSIAN LATTICES ARNAUD BEAUVILLE ABSTRACT. We associate to a unimodular lattice ? , endowed with an automorphism of square ?1 , a principally polarized abelian variety A? = ?R/? . We show that the configuration of i -invariant theta divisors of A? follows a pattern very similar to the classical theory of theta char- acteristics; as a consequence we find that A? has a high number of vanishing thetanulls. When ? = E8 we recover the 10 vanishing thetanulls of the abelian fourfold discovered by R. Varley. INTRODUCTION A Gaussian lattice is a free, finitely generated Z[i] -module ? with a positive hermitian form ???? Z[i] . Equivalently, we can view ? as a lattice over Z endowed with an automorphism i of square ?1? . This gives a complex structure on the vector space ?R := ??Z R ; we associate to ? the complex torus A? := ?R/? . As a complex torus A? is isomorphic to Eg , where E is the complex elliptic curve C/Z[i] and g = 12 rkZ ? . More interestingly, the hermitian form provides a polarization on A? (see (1.3) below); in particular, if ? is unimodular, A is a principally polarized abelian variety (p.p.a.

• f2 -vector space


• quadratic forms


• then ?


• let ? ?


• let ?


• linear algebra over

Sujets

• Quadratic form
• multiplication

Informations

• Publié par: pefav
• Nombre de lectures: 14
• Langue: English

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ABELIAN VARIETIES ASSOCIATED TO GAUSSIAN LATTICES

ARNAUD BEAUVILLE

A B S T R A C T . We associate to a unimodular latticeΓ, endowed with an automorphism of square −1, a principally polarized abelian varietyA Γ = Γ R /Γ. We show that the configuration of i-invariant theta divisors ofA Γ follows a pattern very similar to the classical theory of theta char- acteristics; as a consequence we find thatA Γ has a high number of vanishing thetanulls. When Γ =E 8 we recover the 10 vanishing thetanulls of the abelian fourfold discovered by R. Varley.

I NTRODUCTION

AGaussian latticeis a free, finitely generatedZ[i]-moduleΓwith a positive hermitian form Γ×Γ→Z[i]. Equivalently, we can viewΓas a lattice overZendowed with an automorphism iof square−1 Γ . This gives a complex structure on the vector spaceΓ R := Γ⊗ Z R; we associate toΓthe complex torusA Γ := Γ R /Γ. g As a complex torusA Γ is isomorphic toE, whereEis the complex elliptic curveC/Z[i] 1 andg= rk Z Γ. More interestingly, the hermitian form provides apolarizationonA Γ (see (1.3) 2 below); in particular, ifΓis unimodular,Ais a principally polarized abelian variety (p.p.a.v. for short), which is indecomposable ifΓis indecomposable. The first non-trivial case isg= 4, withΓthe root lattice of typeE 8 The(Example 1.2.1). resulting p.p.a.v. is the abelian fourfold discovered by Varley [V] with a different (and more geometric) description; it has 10 “vanishing thetanulls” (even theta functions vanishing at 0), the maximum possible for a 4-dimensional indecomposable p.p.a.v. In fact this property char- acterizes the Varley fourfold outside the hyperelliptic Jacobian locus [D]. Our aim is to explain this property from the lattice point of view, and to extend it to all unimodular lattices. It turns out that we can mimic the classical theory of theta characteristics, replacing the automorphism(−1)byi. We will show: •The groupA i ofi-invariant points ofA Γ is aF 2 -vector space of dimensiong; it admits a natural non-degenerate bilinear symmetric formb. •The set ofi-invariant theta divisors ofA Γ is an affine space overA i , isomorphic to the space of quadratic forms onA i associated tob(see (2.1)). •LetΘbe ani-invariant theta divisor, andQThethe corresponding quadratic form. multiplicitym 0 (Θ)ofΘat0satisfies

2m 0 (Θ)≡σ(Q) +g

whereσis theBrown invariantof the formQ(2.1).

Date: January 20, 2012.

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(mod.8),