Twisted Teichmller Curves [Paperback]

These notes introduce a new class of algebraic curves on Hilbert modular surfaces. These curves are called twisted Teichm?ller curves, because their construction is very reminiscent of Hirzebruch-Zagier cycles. These new objects are analyzed in detail and their main properties are described. In particular, the volume of twisted Teichm?ller curves is calculated and their components are partially classified. The study of algebraic curves on Hilbert modular surfaces has been widely covered in the literature due to their arithmetic importance. Among these, twisted diagonals (Hirzebruch-Zagier cycles) are some of the most important examples.

Introduction.- Background.- Teichm?ller Curves.- Twisted Teichm?ller Curves.- Stabilizer and Maximality.- Calculations for Twisted Teichm?ller Curves.- Prym Varieties and Teichm?ller Curves.- Lyapunov Exponents.- Kobayashi Curves Revisited.- Appendix.- Tables.- List of Symbols.- Index.- Bibliography.

These notes contain detailed background material and a comprehensive survey of recent advances in the field. They can be used as a handbook for mathematicians who are interested in Teichm?ller curves and Hilbert modular surfaces. (Dawei Chen, Mathematical Reviews, February, 2016)

These notes introduce a new class of algebraic curves on Hilbert modular surfaces. These curves are called twisted Teichm?ller curves, because their construction is very reminiscent of Hirzebruch-Zagier cycles. These new objects are analyzed in detail and their main properties are described. In particular, the volume of twisted Teichm?ller curves is calculated and their components are partially classified. The study of algebraic curves on Hilbert modular surfaces has been widely covered in the literature due to their arithmetic importance. Among these, twisted diagonals (Hirzebruch-Zagier cycles) are some of the most important examples.

Contains a comprehensive introduction to the theory of Teichm?ller curves