Classification of real Riemann surfaces and their Jacobians in the critical case

TL;DR

This paper develops new invariants and criteria to classify real Riemann surfaces and their Jacobians based on their period matrices, enabling the determination of real points using minimal data.

Contribution

It introduces novel invariants for real principally polarized abelian varieties and provides an exhaustive criterion for the existence of real points on real Riemann surfaces.

Findings

Distinguished real period matrices of different topological types.

Established new invariants for real abelian varieties.

Provided a criterion using a single theta constant to detect real points.

Abstract

For every $g\geq 2$ we distinguish real period matrices of real Riemann surfaces of topological type $(g,0,0)$ from the ones of topological type $(g,k,1)$, with $k$ equal to one or two for $g$ even or odd respectively (Theorem B). To that purpose, we exhibit new invariants of real principally polarized abelian varieties of orthosymmetric type (Theorem A.1). As a direct application, we obtain an exhaustive criterion to decide about the existence of real points on a real Riemann surface, requiring only a real period matrix of its and the evaluation of the sign of at most one (real) theta constant (Theorem C). A part of our real, algebro-geometric instruments first appeared in the framework of nonlinear integrable partial differential equations.