Hauptmodul - crowlogic/arb4j GitHub Wiki

"Hauptmodul" is a German word which translates to "main module" or "principal module" in English. In mathematics, it refers to a special type of modular function defined on the complex upper half-plane that takes the value 0 at a specified cusp, which is a point on the boundary of the upper half-plane where the group of modular transformations has a nontrivial stabilizer.

Hauptmoduls play an important role in the theory of modular forms and modular curves. They are used to construct explicit examples of modular forms of given weight and level, and to parametrize the space of modular forms of a given weight and level. The theory of hauptmoduls is also closely related to the theory of elliptic curves and abelian varieties.

More specifically, a hauptmodul for a congruence subgroup $\Gamma$ of $SL(2,\mathbb{Z})$ is a meromorphic function $f$ on the upper half-plane such that:

1. $f$ is invariant under the action of $\Gamma$.
2. $f$ has a simple pole at each cusp of $\Gamma$.
3. $f$ has a zero of order 1 at a specified cusp of $\Gamma$.
4. $f$ is holomorphic at all other points in the upper half-plane.

Hauptmoduls are not unique, but they are unique up to a constant multiple and a Möbius transformation. That is, if $f$ and $g$ are hauptmoduls for the same congruence subgroup $\Gamma$, then there exists a constant $c$ and a Möbius transformation $T$ such that $f(z) = c g(T(z))$ for all $z$ in the upper half-plane.