10 - Hecke operators

Summary

A natural set of mutually commuting linear operators acting on the space of modular forms are the Hecke operators. They map holomorphic functions to holomorphic functions, weight-k modular forms to weight-k modular forms, and weight-k cusp forms to weight-k cusp forms. For the full modular group SL(2,Z), the Hecke operators map the space of holomorphic modular forms into itself and map the subspace of cusp forms into itself. For congruence subgroups, the Hecke operators map weight-k modular forms of one congruence subgroup into those of another congruence subgroup. Hecke operators commute with the Laplace–Beltrami operator on the upper half plane so that Maass forms and cusp forms are simultaneous eigenfunctions of all Hecke operators. Finally, given a modular form with positive integer Fourier coefficients, the Hecke transforms also have positive integer Fourier coefficients. For this reason, Hecke operators are relevant in a number of physical problems, such as two-dimensional conformal field theory, that we shall discuss.