Lecture 8 was the final lecture of the course. We have derived Selberg's trace formula for the modular group. In order to make use of the learned material, you have to apply it. A few simple applications can be found in Iwaniec's book on pages 152–183. Much more applications are in Hejhal's books: D. A. Hejhal, The Selberg Trace Formula for PSL(2,R). Springer Lecture Notes in Mathematics 548 (1976) and 1001 (1983). One might hesitate reading such two big volumes, however, Hejhal's books are really worth a read. They give a concise and comprehensive treatment of the trace formula and of many of its applications.
Would I have more time, I would continue the lectures with the Selberg zeta function, its analytic properties, its connection with analytic number theory, and its connection with closed geodesics. Next, I would explore Weyl's law, and finally extend the subject to congruence groups.
Further applications of automorphic forms arise from their associated L-functions, see Exercise 27. According to the multiplicativity of Hecke operators, these L-functions have an Euler product. Analytic continuation of the L-functions allows to explore their complex zeros, some of which are on the negative real axis, the others are conjectured to be on the critical line Re(s)=1/2 (Generalized Riemann Hypothesis).
Also in physics, there are a number of applications of automorphic forms. In quantum mechanics, the Laplace operator plays the role of the Hamilton operator of a point particle sliding freely on a surface. If for some reason the system lives on a hyperbolic surface Γ\H then the Schrödinger equation is solved by the spectral expansion into automorphic forms.
In cosmology, it is natural that space can be curved. From astronomical observations we know that our Universe is almost isotropic. In addition, we expect it to be nearly homogeneous. Hence the curvature of space is constant (if averaged over large scales). In consequence, the Universe is expected to be elliptic, flat, hyperbolic — or any quotient thereof, e.g. Γ\H. In these almost homogeneous and isotropic spaces, the Einstein field equations of general relativity reduce in leading order to the Friedmann equations. These are two ordinary differential equations whose solutions are known analytically and describe the expansion history of our Universe. In next order, the Einstein field equations reduce to a set of partial differential equations. From this set of partial differential equations, we can isolate the eigenvalue equation of the Laplace operator. Expanding the initial conditions of the Universe into eigenfunctions of the Laplace operator allows to separate variables and we are left with a simpler set of differential equations. Hence, the spectral expansion into automorphic forms simplifies the task of solving the Einstein field equations enormeously.
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