Speaker: Min Lee (Univ. of Bristol)

Time: 2019.10.22 14:00 -

Place: 208 Math building

Title : The Selberg trace formula and its applications

Abstract : The spectral theory of non-holomorphic automorphic forms began with H. Maass in the 1940s. A Maass form is a function on a hyperbolic surface which is also an eigenfunction of the Laplace-Beltrami operator. Although Maass discovered some examples by using Hecke L-functions, in general, the construction of explicit examples of Maass forms remains mysterious. Even the existence of such functions (except the examples discovered by Maass) was not clear.
In 1956, A. Selberg introduced his famous trace formula, now called the Selberg trace formula, which relates the spectrum of the Laplace operator on a hyperbolic surface to its geometry. By using his trace formula, Selberg obtained Weyl's law, which gives an asymptotic count for the number of Maass forms with Laplacian eigenvalues up to a given bound.
Let $mathbb{H}$ be the Poincar’e upper half plane and $Gamma$ be a congruence subgroup of $SL_2(mathbb{Z})$. The aim of talk is to develop Selberg’s trace formulas for $Gammabackslash mathbb{H}$ and study their applications.