arXiv subjects

Yoshihiko Shinomiya

Publications and source records attributed to Yoshihiko Shinomiya.

5 recordsLinked to original sources

arXiv2023

Simple closed geodesics on hyperelliptic translation surfaces and classification theorem for translation surfaces in $\mathcal{H}^{\mathrm{hyp}}(4)$

In this paper, we give the maximum of the numbers $n$ such that we can take $n$ simple closed geodesics without singularities that are disjoint to each other for translation surfaces in the hyperelliptic components $\mathcal{H}^{\rm {hyp}}(2g-2)$ and $\mathcal{H}^{\rm{hyp}}(g-1, g-1)$. The maximum is different from that of the case of hyperbolic surfaces. We also give a classification theorem for translation surfaces in $\mathcal{H}^{\mathrm{hyp}}(4)$ with respect to their Euclidean structures.

math.GT↗

arXiv2021

Period matrices of some hyperelliptic Riemann surfaces

In this paper, we calculate period matrices of algebraic curves defined by $$w^2=z(z^2-1)(z^2-a_1^2)(z^2-a_2^2)\cdots (z^2-a_{g-1}^2)$$ for any $g\geq 2$ and $a_1, a_2, \dots, a_{g-1}\in \mathbb{R}$ with $1<a_1<a_2<\cdots <a_{g-1}$. We construct these algebraic curves from Euclidean polygons. A symplectic basis of these curves are given from the polygons.

math.AG↗

arXiv2012

On holomorphic sections of Veech holomorphic families of Riemann surfaces

We give upper bounds of the numbers of holomorphic sections of Veech holomorphic families of Riemann surfaces. The numbers depend only on the topological types of base Riemann surfaces and fibers. We also show a relation between types of Veech groups and moduli of cylinder decompositions of flat surfaces.

math.CV↗

arXiv2012

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

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.

math.CV↗

arXiv2011

Veech groups of flat structures on Riemann surfaces

In this paper, we construct new examples of Veech groups by extending Schmithusen's method for calculating Veech groups of origamis to Veech groups of unramified finite coverings of regular 2n-gons. We calculate the Veech groups of certain Abelian coverings of regular 2n-gons by using an algebraic method.

math.GT↗