Twisted Teichm\"uller curves

Abstract

Let $X_D$ denote the Hilbert modular surface $\HH \times \HH^- / \SL_2(\OD)$. In \cite{HZ76}, F. Hirzebruch and D. Zagier introduced Hirzebruch-Zagier cycles, that could also be called twisted diagonals. These are maps $\HH \to \HH \times \HH^-$ given by $z \mapsto (Mz,-M^\sigma z)$ where $M \in \GL_2^+(K)$ and $\sigma$ denotes the Galois conjugate. The projection of a twisted diagonal to $X_D$ yields a Kobayashi curve, i.e. an algebraic curve which is a geodesic for the Kobayashi metric on $X_D$. Properties of Hirzebruch-Zagier cycles have been abundantly studied in the literature.\\Teichm\"uller curves are algebraic curves in the moduli space of Riemann surfaces $\mathcal{M}_g$, which are geodesic for the Kobayashi metric. Some Teichm\"uller curves in $\mathcal{M}_2$, namely the primitive ones, can also be regarded as Kobayashi curves on $X_D$. This implies that in the universal cover the curve is of the form $z \mapsto (z,\varphi(z))$ for some holomorphic map $\varphi$. A possibility to construct even more Kobayashi curves on $X_D$ is to consider the projection of $(Mz,M^\sigma\varphi(z))$ to $X_D$ where again $M \in \GL_2^+(K)$. These new objects are called twisted Teichm\"uller curves because their construction reminds very much of twisted diagonals. In these notes we analyze twisted Teichm\"uller curves in detail and describe some of their main properties. In particular, we calculate their volume and partially classify components.