Hecke operators, Hecke Eigensystems, and Formal Modular Forms over Number Fields
J. E. Cremona
TL;DR
This work formalizes an explicit, algebraic framework for modular forms over arbitrary number fields by modeling them as functions on modular points and equipping these points with a rich Hecke–Atkin–Lehner calculus. Central results show that full Hecke eigensystems can be recovered from their principal components, up to unramified quadratic twists, enabling practical computation via principal operators. The paper provides detailed constructions of lattices, modular points, and explicit matrix realizations of principal Hecke and Atkin–Lehner operators, including level-changing and dual operators, and discusses the grading by the class group. These developments unify and extend prior work for $K={\mathbb Q}$ and imaginary quadratic fields, and supply tools for computing automorphic forms for $ ext{GL}(2,K)$ over general number fields, with applications to Bianchi cusp forms and data in the LMFDB. The explicit machinery is implemented in the imaginary quadratic setting and is poised to support broader explorations of automorphic forms across number fields.
Abstract
We develop an explicit theory of formal modular forms over arbitrary number fields $K$, as functions of modular points. We define modular points for $Γ_0({\mathfrak n})$ and $Γ_1({\mathfrak n})$, where the level ${\mathfrak n}$ is an integral ideal of $K$; Hecke operators and generalized Atkin-Lehner operators as functions of modular points; and associated Hecke eigensystems. We show how complete eigensystems may be recovered, uniquely up to unramified quadratic twist, from their restrictions to principal Hecke operators, and we give explicit formulas for principal operators suitable for machine computation. These have been implemented by the author in the case of imaginary quadratic fields, and used in his systematic computation of Bianchi cusp forms, which are available in the L-functions and modular forms database (LMFDB). While our description incorporates the classical theory for $K={\mathbb Q}$, and also extends work of the author and his students for imaginary quadratic fields, it applies to arbitrary number fields, and may be useful in the computation of spaces of automorphic forms for GL$(2,K)$ over number fields, whether via modular symbols or other methods.