Joint IAS/Princeton University Number Theory Seminar

It is classical that an element of the class group of a real quadratic field corresponds to a closed geodesic on the modular surface, but not every closed geodesic arises this way; we call those that do "fundamental." Given a fixed compact subset W of (the unit tangent bundle of) the modular surface, we say a closed geodesic is "low-lying" if it is contained in W; in particular, it does not enter "high" into the cusp. In joint work with Bourgain, we exhibit a region W which contains infinitely many fundamental geodesics, answering a question of Einsiedler-Lindenstrauss-Michel-Venkatesh.