HECKE OPERATORS ON HILBERT–SIEGEL MODULAR FORMS

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...

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...

Explicit Matrices for Hecke Operators on Siegel Modular Forms

We present an explicit set of matrices giving the action of the Hecke operators T (p), Tj(p ) on Siegel modular forms. Introduction It is well-known that the space of elliptic modular forms of weight k has a basis of simultaneous eigenforms for the Hecke operators, and the Fourier coefficients of an eigenform (and hence the eigenform) are completely determined by its eigenvalues and first Fouri...

HECKE OPERATORS AND THE q-EXPANSION OF MODULAR FORMS

is a weight k ∈ 2Z meromorphic modular form on SL2(Z). It is well known that f(z) is distinguished by its weight k and its “first few coefficients”. The Riemann-Roch Theorem provides the number of coefficients which are sufficient for distinguishing such a form f(z). Here we obtain a combinatorial extension of this classical fact; specifically, we give universal recursion relations which produc...