On Hecke algebras for Hilbert modular forms

Abstract

We use twisting operators defined by characters of order two of Cl + ( F ) to represent the Hecke algebra acting on spaces of Hilbert modular forms as an extension of a canonically defined subalgebra. The extension is generated by quadratic subextensions which are explicitly parametrized by cosets in Cl + ( F ) relative to canonically defined subgroups. Here Cl + ( F ) is the ideal class group of F in the narrow sense. In addition, we give numerical examples of characteristic polynomials of Hecke operators for real quadratic fields F and for totally real non-Galois cubic fields F with nontrivial Cl + ( F ) / Cl + ( F ) 2 .