A signed count of 2-torsion points on real abelian varieties

TL;DR

This paper establishes a universal signed count of 2-torsion points on real abelian varieties, linking algebraic geometry with geometric interpretations and proposing a conjecture for broader fields.

Contribution

It proves a formula for the signed count of 2-torsion points on real abelian varieties and connects it to geometric properties of real curves and theta characteristics.

Findings

Signed count equals 2^g for real principally polarized abelian varieties

Derived signed counts of real odd theta characteristics for Jacobians

Proposed a conjecture for general fields using $\\mathbb{A}^1$-enumerative geometry

Abstract

We prove that a natural signed count of the $2$-torsion points on a real principally polarized abelian variety $A$ always equals to $2^{g}$ where $g$ is the dimension of $A$. When $A$ is the Jacobian of a real curve we derive signed counts of real odd theta characteristics. These can be interpreted in terms of the extrinsic geometry of contact hyperplanes to the canonical embedding of the curve. We also formulate a conjectural generalization to arbitrary fields in terms of $\mathbb{A}^1$-enumerative geometry.