Matrix-valued Hilbert modular forms
Enrico Da Ronche
TL;DR
The paper extends Knopp–Mason’s unrestricted vector-valued modular form framework to matrix-valued Hilbert modular forms over totally real fields, defining matrix-valued functions with a weight- and representation-driven transformation law and proving that each component admits a unique polynomial Fourier expansion. It develops the polynomial Fourier expansion theory via linear-algebraic block-triangularizability, establishes a robust space of matrix-valued Hilbert modular forms with a graded-module structure over the classical Hilbert modular forms ring, and constructs explicit Poincaré series under unitary-type assumptions that yield bona fide matrix-valued Hilbert modular forms. Key contributions include existence and uniqueness of polynomial Fourier expansions, the construction of spaces $M^F_{\uuline{k}}(\rho)$ with conjectured Cohen–Macaulay properties, and explicit Poincaré-series-based examples. This framework lays groundwork for deeper structural and algebraic investigation of matrix-valued Hilbert modular forms across totally real fields, with potential implications for automorphic representation theory and arithmetic geometry.
Abstract
In this paper we generalize the notion of logarithmic vector-valued modular form in order to give a general definition of matrix-valued Hilbert modular forms. We prove that they admit unique polynomial Fourier expansions and we build examples in some particular cases.