Selected Topics in Geometry - The modular curve (V5D34), Sommersemester 2020

Dozent: Dr. Matteo Costantini

Sprechstunde: Nach Vereinbarung per email.

Lecture:
Mittwoch 14:00 c.t. - 16:00 Uhr, MATH / N 0.003 - Neubau

The modular curve is a very interesting object studied in many areas of mathematics like number theory, complex algebraic geometry, differential geometry and homogenous dynamics. The modular curve is the quotient of the complex upper-half space by the modular group. It is the moduli space of elliptic curves, or equivalently of pointed genus one compact Riemann surfaces.

In this lecture we will focus on the geometric part of the story, in particular we will focus on the study of the geodesic flow on the modular curve. We will see for example how it is related to continued fractions and show its ergodic properties. In order to do this, we will also provide a basic introduction to ergodic theory. Another result we will show is Ratner's theorem in the easiest case, so for the horocycle flow on the modular curve. We will finally briefly present other aspects of the modular curve, considering it as a moduli space, and show how the previous results can be used in this area.

The prerequisite for this lecture is a basic knowledge of differential geometry. The definition of Riemann surface and some basic notions about complex 1-dimensional geometry is also recommended.

Literatur: The following references are more advanced than what is treated in the lecture, so only a small portion of them will be used. More references will be added later. Handwritten notes of the course will be available here.

• M. Bekka and M. Mayer, Ergodic theory and topological dynamics of group actions on homogeneous spaces., London Mathematical Society Lecture Note Series, 269. Cambridge University Press, 2000.
• M. Einsiedler and T. Ward,Ergodic theory with a view towards number theory,Graduate Texts in Mathematics, 259. Springer-Verlag London, 2011.
• S. Katok and I. Ugarcovici, Symbolic dynamics for the modular surface and beyond, Bull. Amer. Math. Soc. 44 (2007), no. 1, 87–132.
• R. Hain, Lectures on moduli spaces of elliptic curves, Adv. Lect. Math. (ALM), 16, Int. Press, Somerville, MA, 2011.