Maass wave forms, quantum modular forms and Hecke operators

Harmonic Maass Forms, Mock Modular Forms, and Quantum Modular Forms

This short course is an introduction to the theory of harmonic Maass forms, mock modular forms, and quantum modular forms. These objects have many applications: black holes, Donaldson invariants, partitions and q-series, modular forms, probability theory, singular moduli, Borcherds products, central values and derivatives of modular L-functions, generalized Gross-Zagier formulae, to name a few....

Lectures on Modular Forms and Hecke Operators

Hecke Operators and Hilbert Modular Forms

Let F be a real quadratic field with ring of integers Ø and with class number 1. Let Γ be a congruence subgroup of GL2(Ø). We describe a technique to compute the action of the Hecke operators on the cohomology H(Γ ;C). For F real quadratic this cohomology group contains the cuspidal cohomology corresponding to cuspidal Hilbert modular forms of parallel weight 2. Hence this technique gives a way...

Hecke Operators on Hilbert–siegel Modular Forms

We define Hilbert–Siegel modular forms and Hecke “operators” acting on them. As with Hilbert modular forms (i.e. with Siegel degree 1), these linear transformations are not linear operators until we consider a direct product of spaces of modular forms (with varying groups), modulo natural identifications we can make between certain spaces. With Hilbert–Siegel forms (i.e. with arbitrary Siegel d...

New Models for the Action of Hecke Operators in Spaces of Maass Wave Forms

Utilizing the theory of the Poisson transform, we develop some new concrete models for the Hecke theory in a space Mλ(N) of Maass forms with eigenvalue 1/4− λ on a congruence subgroup Γ1(N). We introduce the field Fλ = Q(λ, √ n, nλ/2 | n ∈ N) so that Fλ consists entirely of algebraic numbers if λ = 0. The main result of the paper is the following. For a packet Φ = (νp | p ∤N) of Hecke eigenvalu...