Applications
The trace formula has applications to arithmetic geometry and number theory. For instance, using the trace theorem Eichler and Shimura calculated the Hasse-Weil L-functions associated to modular curves; Goro Shimura's methods by-passed the analysis involved in the trace formula. The development of parabolic cohomology (from Eichler cohomology) provided a purely algebraic setting based on group cohomology, taking account of the cusps characteristic of non-compact Riemann surfaces and modular curves.
The trace formula also has purely differential-geometric applications. For instance, by a result of Buser, the length spectrum of a Riemann surface is an isospectral invariant, essentially by the trace formula.
Read more about this topic: Selberg Trace Formula