Euler characteristics of Gothic Teichm\"{u}ller curves

TL;DR

This paper calculates the Euler characteristics of Gothic Teichmüller curves using novel Gothic Hilbert modular forms, revealing unique properties that influence Lyapunov exponents, unlike previously known cases.

Contribution

It introduces Gothic Hilbert modular forms and computes Euler characteristics for Gothic Teichmüller curves, showing non-proportionality with ambient surfaces.

Findings

Euler characteristics are computed for Gothic Teichmüller curves.

The Euler characteristic is not proportional to that of the ambient Hilbert modular surfaces.

This leads to varying phenomena in Lyapunov exponents.

Abstract

We compute the Euler characteristics of the recently discovered series of Gothic Teichm\"{u}ller curves. The main tool is the construction of 'Gothic' Hilbert modular forms vanishing at the images of these Teichm\"{u}ller curves. Contrary to all previously known examples, the Euler characteristic is not proportional to the Euler characteristic of the ambient Hilbert modular surfaces. This results in interesting 'varying' phenomena for Lyapunov exponents.