Number Theory Seminar
It is well-known that closed geodesics on the modular surface (and also other geometric invariants) can be related to automorphic forms and ultimately to central values of certain L-functions via spectral theory. I plan to briefly describe the process for the concrete case at hand, and introduce some classical results on the statistical distribution of such geodesics. Afterwards, we will focus on the variance of their distribution in small random balls or annuli, which is a natural quantity to consider and turns out to have somewhat unexpected properties. I will introduce a random model which confirms that such properties are consistent with "generic" behavior and leads to a conjecture for the asymptotics of the variance. I will then discuss how to leverage new moment bounds for the relevant L-functions, coming from the work of Humphries-Radziwill and of Young, to prove the conjecture for sufficiently small balls and annuli.