Hauptmodi

Hauptmodul is a term used in the theory of modular forms to denote a principal generator of the function field of modular functions for a given discrete group Γ acting on the upper half-plane. The word is German for “main modulus.” If the quotient XΓ = Γ\H* (the compactified modular curve) has genus zero, then there exists a Hauptmodul tΓ such that the field of meromorphic modular functions for Γ is the rational function field C(tΓ). In this setting every modular function for Γ can be expressed as a rational function in tΓ.

A Hauptmodul is typically a modular function of weight zero that is meromorphic on the upper half-plane and at cusps, holomorphic elsewhere. In genus-zero cases it can be chosen so that it has poles only at the cusps, often a simple pole at a distinguished cusp, and it provides a convenient parameter for the modular curve XΓ. The Fourier (q-)expansion of a Hauptmodul is used to study its arithmetic properties, and the theory implies that all modular functions for Γ are rational functions in tΓ.

Prominent examples include the j-function, which is a Hauptmodul for the full modular group SL2(Z); its field of modular functions is C(j). The modular lambda function λ(τ) serves as a Hauptmodul for Γ(2). For congruence subgroups Γ0(N) with genus zero—N in {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18, 25}—appropriate Hauptmoduls can be constructed, often as eta-quotients. Hauptmoduls play a central role in areas such as monstrous moonshine, where genus-zero groups correspond to specific modular functions with remarkable symmetry.