Euler characteristics of Gothic Teichmüller curves

Abstract

We compute the Euler characteristics of the recently discovered series of Gothic Teichmüller curves. The main tool is the construction of “Gothic” Hilbert modular forms vanishing at the images of these Teichmüller 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.