Quadratic forms and Hecke operators

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...

Computing Hecke Operators on Bianchi Forms

Bianchi forms are generalizations of classical modular forms. The arithmetic significance of these forms is revealed through the system eigenvalues determined by a family of linear operators known as Hecke operators.

Quadratic Forms in Unitary Operators