Let $X$ be a modular curve and $\mathcal{L}$ a line bundle on $X$. I found some literature regarding modular forms as global sections on $\mathcal{L}$. Under this context, my question is:

For any line bundle $\mathcal{L}$ on $X$ (given modular curve), can any global section of $\mathcal{L}$ be considered as a modular form? If then, could I see the way of interpretation?


Write your curve as $\Bbb{H}/\Gamma $, with $\Gamma \subset PGL(2,\mathbb{R})$ acting freely on $\Bbb{H}$. The pull back of $\mathcal{L}$ to $\Bbb{H}$ is the trivial line bundle $\Bbb{H}\times \mathbb{C}$; this implies that $\mathcal{L}$ is the quotient of $\Bbb{H}\times \mathbb{C})$ by $\Gamma $ acting by $\gamma \cdot (\tau , z)=(\gamma \tau , e_\gamma (\tau )z)$, where $\gamma \mapsto e_{\gamma }$ is a 1-cocycle of $\Gamma $ with values in $\mathcal{O}(\Bbb{H})^*$ (see the beginning of Mumford's Abelian varieties for a completely analogous analysis). It follows that the sections of $\mathcal{L}$ correspond to functions $f:\Bbb{H}\rightarrow \mathbb{C}$ satisfying $f(\gamma \tau )= e_{\gamma }(\tau )f(\tau )$.

These can be considered as modular forms in a generalized sense. If you take $\mathcal{L}=K_X^m$, the $m$-th power of the canonical bundle, we have $e_{\gamma }(\tau )=(c\tau +d)^{2m}$ for $\gamma =\begin{pmatrix} a & b\\ c & d \end{pmatrix}$, and $\ f$ becomes a genuine modular form of weight $2m$ for $\Gamma $.

  • $\begingroup$ so $e_\gamma(\tau)$ is a factor of automorphy : for $\gamma,\gamma'\in \Gamma, \tau \in \mathbb{H}$, $e_{\gamma \gamma'}(\tau) =e_{\gamma }(\gamma'\tau) e_{\gamma'}(\tau)$ $\endgroup$ – reuns Aug 15 '17 at 17:28

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