Maass forms on the full modular surface $M=SL(2,\mathbb{z})\backslash \mathbb{H}$
have two faces. Firstly, they are eigenstates of the Schr{\"o}dinger equation on
the hyperbolic surface $M$. Secondly, they are automorphic forms on $GL(2)/\mathbb{Q}$,
which are building blocks of modern analytic number theory. These two seemingly
unrelated interpretations of these objects allow one to study problems in Quantum
Chaos via arithmetic means, or problems in number theory via analytic/geometric means.
The main purpose of this course is to introduce the basic analytic/geometric theory
on arithmetic hyperbolic surfaces. At the beginning of the semester, I will briefly
review basic materials that will be used throughout the course. For the first half
of the semester, I will thoroughly go over the spectral theory of the Laplacian on
finite volume non-compact arithmetic surfaces. Then I will introduce Selberg's trace
formula, a version for compact symmetric spaces, and a version for finite volume
non-compact arithmetic surfaces. Toward the end of the course, I will talk about some
direct applications of Selberg's trace formula, including spectral gap of the Laplacian
for hyperbolic surfaces those arise from congruence subgroups, hyperbolic lattice point
counting problem, prime geodesic theorem, and asymptotic formula for the partial sum of
class numbers of real quadratic fields over $\mathbb{Q}$.
Prerequisites: Some knowledge in Complex analysis (fractional transformation, contour integral,
analytic continuation, etc.), Differential geometry (definition of symmetric spaces, Lie group,
Riemannian metric, geodesic, volume form, etc.), and/or Number theory (Prime number theorem,
quadratic fields over $\mathbb{Q}$, class number, etc.). These are going to be helpful,
but most of them are going to be covered/reviewed throughout the course.