Editing Modular form (section)

== Definition ==
In general,<ref>{{Cite web|last=Lan|first=Kai-Wen|title=Cohomology of Automorphic Bundles|url=http://www-users.math.umn.edu/~kwlan/articles/iccm-2016.pdf|url-status=live|archive-url=https://web.archive.org/web/20200801235440/http://www-users.math.umn.edu/~kwlan/articles/iccm-2016.pdf|archive-date=1 August 2020}}</ref> given a subgroup <math>\Gamma < \text{SL}_2(\mathbb{Z})</math> of [[finite index]] (called an [[arithmetic group]]), a '''modular form''' of level <math>\Gamma</math> and weight <math>k</math> is a [[holomorphic function]] <math>f:\mathcal{H} \to \mathbb{C}</math> from the [[upper half-plane]] satisfying the following two conditions:

* ''Automorphy condition'': for any <math>\gamma \in \Gamma</math>, we have<math>f(\gamma(z)) = (cz + d)^k f(z)</math> ,<ref group="note">Some authors use different conventions, allowing an additional constant depending only on <math>\gamma</math>, see e.g. {{Cite web |title=DLMF: §23.15 Definitions ‣ Modular Functions ‣ Chapter 23 Weierstrass Elliptic and Modular Functions |url=https://dlmf.nist.gov/23.15#E5 |access-date=2023-07-07 |website=dlmf.nist.gov}}</ref> and

* ''Growth condition'': for any <math>\gamma \in \text{SL}_2(\mathbb{Z})</math>, the function <math>(cz + d)^{-k} f(\gamma(z))</math> is bounded for <math>\text{im}(z) \to \infty</math>.

In addition, a modular form is called a '''cusp form''' if it satisfies the following growth condition:

* ''Cuspidal condition'': For any <math>\gamma \in \text{SL}_2(\mathbb{Z})</math>, we have <math>(cz + d)^{-k}f(\gamma(z)) \to 0</math> as <math>\text{im}(z) \to \infty</math>.

Note that <math>\gamma</math> is a matrix
:<math display="inline">\gamma = \begin{pmatrix}
a & b \\
c & d
\end{pmatrix} \in \text{SL}_2(\mathbb{Z}),</math> 

identified with the function <math display="inline"> \gamma(z) = (az+b)/(cz+d) </math>. The identification of functions with matrices makes function composition equivalent to matrix multiplication. 

=== 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]].