Recall that for a subgroup $\Gamma \subset SL_2(\mathbb{Z})$ a modular form $f$ of weight $k$ is a holomorphic function from the upper-half plane into the complex numbers such that for any

$\begin{pmatrix} a & b \\ c& d \end{pmatrix}\in \Gamma$

and any $z$ in the upper half-plane, we have $f(\frac{az+b}{cz+d}) = (cz+d)^k f(z)$ and $f$ is meromorphic (or holomorphic, depending on taste) at the cusps, which can be formulated as a growth condition, which I won't make precise here.

For many subgroups $\Gamma$, we have a *modular interpretation* of this. For example, for $\Gamma = SL_2(\mathbb{Z})$, a modular form corresponds to a section of $\omega^{\otimes k}$ on the (compactified, if we want holomorphic at the cusps) moduli stack of elliptic curves over $\mathbb{C}$. For $\Gamma = \Gamma_0(N) = \{\begin{pmatrix}
a & b \\
c& d
\end{pmatrix}\in SL_2(\mathbb{Z}) | c \equiv 0 \mod N\}$, we consider instead elliptic curves with a fixed subgroup of order $N$. As this all makes sense over $\mathbb{Z}$ (or, at least, $\mathbb{Z}[\frac{1}{N}]$) instead of $\mathbb{C}$, we also get integral versions of the rings of modular forms.

But in some sense, this seems to be just the tip of the iceberg. Many functions which show some "modular behaviour" are not modular forms in the sense above. In particular, this is true for many kinds of $\Theta$-functions, where both the aspects of a nebentypus and a half-integral weight come in.

Let $\chi: (\mathbb{Z}/N)^\times \to \mathbb{C}$ be a character. Then a modular form of weight $k$, level $N$ (or level group $\Gamma_0(N)$) with nebentypus $\chi$ is a holomorphic function $f$ on the upper half-plane such that for any

$\begin{pmatrix} a & b \\ c& d \end{pmatrix}\in \Gamma_0(n)$

and any $z$ in the upper half-plane, we have

$f(\frac{az+b}{cz+d}) = \chi(d)(cz+d)^k f(z)$

and $f$ is meromorphic (or holomorphic) at the cusps.

Is there any modular interpretation for modular forms with nebentypus? Are there integral versions of rings of modular forms with nebentypus? And how about the story when we have only half-integral weight?