[Paper Review] An extraordinary origami curve

Frank Herrlich, Gabriela Schmithuesen|ArXiv.org|Sep 8, 2005

Geometric and Algebraic Topology4 citations

TL;DR

This paper studies the origami curve $W$ associated with the quaternion group of order 8, proving it has Veech group $\mathrm{SL}_2(\mathbb{Z})$ and providing an explicit equation for its associated family of curves in genus 3. The key contribution is the discovery of infinitely many other origami curves that intersect $C(W)$ in the moduli space $M_3$, offering the first known example of such intersections and a combinatorial construction via $n \times n$ square gluings.

ABSTRACT

We study a special Teichmueller curve in the moduli space of curves of genus 3 that is intersected by infinitely many other Teichmueller curves. The Veech group of the underlying translation surface is SL_2(Z). All occurring Teichmueller curves are induced by origamis, i.e. unramified coverings of the once punctured torus.

Motivation & Objective

• To study the Teichmüller curve $C(W)$ associated with the exceptional origami $W$ defined by the quaternion group of order 8.
• To determine the Veech group of $W$ and establish its full $\mathrm{SL}_2(\mathbb{Z})$ structure.
• To derive an explicit algebraic equation for the family of curves over $C(W)$.
• To investigate the intersection properties of $C(W)$ with other origami curves in the moduli space $M_3$.
• To provide a combinatorial description of the infinitely many origamis that intersect $C(W)$.

Proposed method

• Construct $W$ as a square-tiled surface via unramified covering of a punctured torus, using the quaternion group's action.
• Use the Veech group definition via affine diffeomorphisms with linear part in $\mathrm{SL}_2(\mathbb{R})$, and show it equals $\mathrm{SL}_2(\mathbb{Z})$ via explicit computation.
• Derive the family of curves over $C(W)$ using algebraic geometry and known results on elliptic fibrations.
• Analyze the Jacobian of the family, showing a two-dimensional fixed part, which implies $C(W)$ is a Shimura curve.
• Construct new origamis by gluing two $n \times n$ squares under specific edge-pairing rules, ensuring the resulting surface admits a $\mathbb{Z}/2\mathbb{Z}$-action with required fixed points.
• Verify that the induced map to an elliptic curve is ramified only over torsion points, allowing the construction to be lifted to an origami.

Experimental results

Research questions

• RQ1 Does the origami $W$ associated with the quaternion group of order 8 have Veech group $\mathrm{SL}_2(\mathbb{Z})$?
• RQ2 Can an explicit algebraic equation be derived for the family of curves over the Teichmüller curve $C(W)$?
• RQ3 Is the Teichmüller curve $C(W)$ intersected by infinitely many other origami curves in $M_3$?
• RQ4 What combinatorial structure characterizes the origamis that intersect $C(W)$?
• RQ5 Under what conditions does a map from a curve to an elliptic curve with torsion ramification yield a new origami?

Key findings

• The origami $W$ has Veech group $\mathrm{SL}_2(\mathbb{Z})$, making it the smallest nontrivial normal origami with this property.
• The Jacobian of the family over $C(W)$ contains a two-dimensional fixed part, implying $C(W)$ is a Shimura curve.
• The curve $C(W)$ is the only known algebraic curve in $M_g$ for $g \geq 2$ that is both a Teichmüller curve and a Shimura curve.
• An explicit equation for the family of curves over $C(W)$ is derived, confirming known results with a more elementary approach via origami structure.
• Infinitely many other origami curves intersect $C(W)$ in $M_3$, a phenomenon not previously observed.
• The intersecting origamis are combinatorially constructed by gluing two $n \times n$ squares for varying $n$, with edge-pairings determined by specific monodromy conditions.