2023/07/19 by Pietro Giavedoni, Giavedoni, Pietro
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2307.10486
openalex publication_date 2023/07/19 · openalex created_date 2023/07/22 · openalex updated_date 2026/07/28
For every g≥ 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.