An elementary proof of the Eichler–Selberg trace formula

Abstract

Abstract We give a purely algebraic proof of the trace formula for Hecke operators on modular forms for the full modular group SL 2 ⁢ ( ℤ ) {\mathrm{SL}_{2}(\mathbb{Z})} , using the action of Hecke operators on the space of period polynomials. This approach, which can also be applied for congruence subgroups, is more elementary than the classical ones using kernel functions, and avoids the analytic difficulties inherent in the latter (especially in weight two). Our main result is an algebraic property of a special Hecke element that involves neither period polynomials nor modular forms, yet immediately implies both the trace formula and the classical Kronecker–Hurwitz class number relation. This key property can be seen as providing a bridge between the conjugacy classes and the right cosets contained in a given double coset of the modular group.