Prym covers, theta functions and Kobayashi curves in Hilbert modular surfaces (Q2879439)

By definition, Kobayashi curves on Hilbert modular surfaces are totally geodesic for the Kobayashi metric. Up to now, they are mainly constructed by diagonal embeddings of the upper half plane into the Hilbert half space or by twists of such embeddings. In the present paper, the author considers new kinds of Kobayashi curves, and on the way he determines invariants of certain Teichmüller curves such as Euler characteristics and Lyapunov exponents. The method of construction is interesting in itself; it uses ideas of \textit{C. T. McMullen} [Duke Math. J. 133, No. 3, 569--590 (2006; Zbl 1099.14018)] and \textit{E. Lanneau} and \textit{D.-M. Nguyen} [J. Topol. 7, No. 2, 475--522 (2014; Zbl 1408.32014)] and offers the following two different descriptions.NEWLINENEWLINE As curves on Hilbert modular surfaces, they can be described as vanishing loci of determinants of theta functions and their derivatives. On the other hand, they are constructed via flat surfaces generating Teichmüler curves in the respective moduli spaces \({\mathcal M}_3\) and \({\mathcal M}_4\), then passing to appropriate Prym varieties for families of certain abelian varieties (not in all cases principally polarized). These Prym varieties are of dimension 2, have real multiplication and define curves on Hilbert surfaces. For these Prym covers a kind of Torelli theorem turns out to be true. For more information about the use of theta functions describing Teichmüller curves on Hilbert surfaces, see Part III of the author's recent joint paper with \textit{D. Zagier} [``Modular embeddings of Teichmüller curves'' [\url{arxiv:1503.05690}, to appear in Compositio Math.].