Jacobians with a vanishing theta-null in genus 4

Abstract

Abstract. In this paper we prove a conjecture of Hershel Farkas [8] that if a 4-dimensional principally polarized abelian variety has a vanishing theta-null, and the hessian of the theta function at the corresponding point of order two is degenerate, the abelian variety is a Jacobian.We also discuss possible generalizations to higher genera, and an interpretation of this condition as an infinitesimal version of Andreotti and Mayer's local characterization of Jacobians by the dimension of the singular locus of the theta divisor.

On the Schottky problem for genus five Jacobians with a vanishing theta null

“…In [GM08], it was proved that this is an equality in genus four, and it was asked whether J g ∩ θ null is an irreducible component of θ 3 null in higher genus. In the rest of this paper, we will discuss this problem and present a proof for genus five.…”

The topology of the zero locus of a genus 2 theta function

“…A theta function of characteristic δ is a holomorphic function ϑ δ : h g × C g → C defined by the following series: [14]). The vector δ corresponds to a 2-torsion point of A Ω and ϑ δ is called a theta function of characteristic δ.…”

“…If ϑ δ is even (odd), then the characteristic δ is also said to be even (odd). Equivalently, δ is even (odd) if and only if the number δ (δ ) T ∈ F 2 is equal to 0 (or 1) (see [14]). Direct calculation shows that the theta function ϑ δ (Ω, z) is equal to the product of ϑ 0 (Ω, z + δ + δ Ω) with a non-vanishing holomorphic function on h 2 × C 2 .…”

“…The thetanull ϑ δ (−, 0) is called even or odd, respectively, if the corresponding theta function is. The odd thetanulls vanish identically [14]. Theta functions enjoy special transformation properties with respect to the Sp g (Z)-action on h g ×C g given by the formula M ·(Ω, z) = M · Ω, z · (CΩ + D) −1 .…”

The loci of abelian varieties with points of high multiplicity on the theta divisor

“…[12]. There is also another method to produce scalar valued modular forms vanishing exactly on the component of (∂ 2 θ) null .…”