Teichmüller curves in genus two: Torsion divisors and ratios of sines

Curtis T Mcmullen

doi:10.1007/s00222-006-0511-2

Abstract

Teichmüller curves in genus two: Torsion divisors and ratios of sines

Highlights

Let Mg denote the moduli space of Riemann surfaces of genus g, and ΩMg → Mg the bundle of pairs (X, ω), where ω = 0 is a holomorphic 1-form on X ∈ Mg.There is a natural action of GL+2 (R) on ΩMg, whose orbits project to complex geodesics in Mg
A Teichmuller curve is primitive if it does not arise from a curve in Mh, h < g, via a covering construction
Our main result completes the classification of primitive Teichmuller curves in genus two

Summary

Introduction

If ω has a pair of simple zeros and the Teichmuller curve it generates is primitive, P is equivalent to the 18-72-90 triangle by Theorem 1.1. The proof uses an elementary fact about cyclotomic fields (§2): Theorem 1.5 There are only 15 pairs of rational numbers 0 < α < β < 1/2 such that sin(πα)/ sin(πβ) is a quadratic irrational To see how these results are related to Theorem 1.1, let f : V → M2 be a primitive Teichmuller curve generated by a form (X, ω) ∈ ΩM2 with simple zeros. This result follows from: Theorem 1.7 The primitive Teichmuller curves generated by forms of genus two coincide with the 1-dimensional irreducible components of WD[n].

Ratios of sines

Torsion divisors

Teichmuller curves in genus two

The regular decagon

Uniqueness