An extraordinary origami curve

Abstract

We study the origami W defined by the quaternion group of order 8 and its Teichmüller curve C(W ) in the moduli space M3. We prove that W has Veech group SL2(Z), determine the equation of the family over C(W ) and find several further properties. As main result we obtain infinitely many origami curves in M3 that intersect C(W ). We present a combinatorial description of these origamis.Origami curves are certain special Teichmüller curves in some moduli space of curves. They are obtained from an unramified covering of a once punctured torus, see Section 1.1 for a precise definition.First examples of Teichmüller curves were already given by Veech in [V]. In recent years they have attracted a lot of attention, partly because of their relation to rational billiards, see e.g. [McM] and references therein; a survey of examples defining primitive Teichmüller curves can be found in [HuSc]. Another topic of interest is the action of the absolute Galois group of the rationals on them as discussed in [Lo] and [M1].In some respects Teichmüller curves arising via origamis are more accessible than general ones. Precisely for them, the Veech group (as defined in Section 1.1) is a subgroup of SL 2 (Z), cf. [GJ]. It can be determined for each origami explicitly, cf. [S]. Nevertheless, it is still difficult to approach the question how their Teichmüller curves are located in the moduli space. There are only a few origami curves for which explicit equations have been found so far, cf. [H] and [M1].In this note we present an extraordinary origami curve in genus 3. It is the smallest nontrivial example of a normal origami having as Veech group the full group SL 2 (Z), see Prop. 2. The Jacobian of the associated family of curves has a two-dimensional fixed part, see Proposition 7. M. Möller has observed that this implies that our origami curve is also a Shimura curve; recently he proved that it is the only algebraic curve in a moduli space of curves of genus at least 2 which is at the same time a Teichmüller curve and a Shimura curve, cf.[M2]. An explicit equation for the associated family of curves is given in Proposition 5. This family of curves has been studied by several authors, and some of the results of Section 1 have been known previously, cf.[Gi], [Gu2], [KK]; nevertheless the use of the "origami" structure allows for new proofs and makes the exposition considerably more elementary.A very remarkable property of this origami curve, and the main new result of this paper, is the fact that it intersects infinitely many other origami curves, see Theorem 1 in Section 3. To our knowledge this is the first example of origami curves that intersect in moduli space.The decomposition of the Jacobian gives a second map onto an elliptic curve; if this map is ramified only over torsion points it can be made into an origami. In Section 2 we develop explicit formulas for these maps to determine when this particular type of ramification occurs.We give a combinatorial description of the infinitely many origamis that intersect our ori...

Cyclic Coverings, Calabi-Yau Manifolds and Complex Multiplication

“…Moreover we will make some remarks aboutḠ 4 and E 4 . We obtain due to [24], Proposition 9 and the notations of [24], Section 2: The involutions of E 4 are given by (αβ) ν , α ζ , β ς and γ κ for ν = 2, ζ 2 = 1 − λ, ς 2 = 1 − 1 λ and κ 2 = λ. The group E 4 has a subgroup isomorphic to the quarternion group given by (αβ) ν , α ζ , β ς and γ κ for ν = 0, 2, ζ 2 = −1 + λ, ς 2 = −1 + 1 λ and κ 2 = −λ.…”

Book Review: Twisted Teichmüller curves

“…Gutkin and Judge [27] showed that every arithmetic surface is a cover of a torus, with possible ramification above one point. These so-called square-tiled surfaces (also known as origami) are dense in ΩM g , and provide key specific examples in the theory [25,29]. Schmithüsen [71] gave an algorithm for computing the Veech group in this setting.…”

“…Möller and Viehweg [63] characterize all Kobayashi curves in A g , showing rigidity. Möller [60] shows that for g = 5 there are exactly two Teichmüller curves (those given in [25,29]), that are simultaneously Shimura curves.…”

On the boundary of Teichmüller disks in Teichmüller and in Schottky space

“…Recall from (18) in Chapter 3 that a base of open neighbourhoods of (X ∞ , f ∞ ) is given by the open sets…”

“…Using the description of origamis by gluing squares it is not difficult to see that there are, for any g ≥ 2, infinitely many Teichmüller curves in M g that come from origamis. In genus 3 there is even an explicit example of an origami curve that is intersected by infinitely many others, see [18].…”