Veech holomorphic families of Riemann surfaces, holomorphic sections, and Diophantine problems

2012/02/22 by Yoshihiko Shinomiya, Shinomiya, Yoshihiko
Mathematics · #11D45 #14H05 #30F35 #30F60 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Complex Variables (math.CV) #FOS: Mathematics #General Topology (math.GN) #Mathematical Dynamics and Fractals #math.AG #math.CV #math.GN #msc:11D45 #msc:14H05 #msc:30F35 #msc:30F60

paper · pdf · doi:10.48550/arxiv.1202.5002

28 pages; Exposition improved. New results are added

openalex publication_date 2012/02/22 · arxiv created 2012/04/10 · arxiv updated 2012/04/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we construct holomorphic families of Riemann surfaces from Veech groups and characterize their sections by some points of corresponding flat surfaces. The construction gives us concrete solutions for some Diophantine equations over function fields. Moreover, we give upper bounds of the numbers of holomorphic sections of certain holomorphic families of Riemann surfaces.

Citations

• Veech groups of flat structures on Riemann surfaces

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