Editing Modular form (section)

=== As sections of a line bundle ===
Modular forms can also be interpreted as sections of a specific [[line bundle]] on [[Modular curve|modular varieties]]. For <math>\Gamma < \text{SL}_2(\mathbb{Z})</math> a modular form of level <math>\Gamma</math> and weight <math>k</math> can be defined as an element of
:<math>f \in H^0(X_\Gamma,\omega^{\otimes k}) = M_k(\Gamma),</math>
where <math>\omega</math> is a canonical line bundle on the [[modular curve]]
:<math>X_\Gamma = \Gamma \backslash (\mathcal{H} \cup \mathbb{P}^1(\mathbb{Q})).</math>
The dimensions of these spaces of modular forms can be computed using the [[Riemann–Roch theorem]].<ref>{{Cite web|last=Milne|title=Modular Functions and Modular Forms|url=https://www.jmilne.org/math/CourseNotes/mf.html|page=51}}</ref> The classical modular forms for <math>\Gamma = \text{SL}_2(\mathbb{Z})</math> are sections of a line bundle on the [[moduli stack of elliptic curves]].