Modular form: Difference between revisions

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{{Redirect|Modular function|text=A distinct use of this term appears in relation to [[Haar measure#The modular function|Haar measure]]}}

In [[mathematics]], a '''modular form''' is a (complex) [[analytic function]] on the [[upper half-plane]] that satisfies:

In [[mathematics]], a '''modular form''' is a (complex) [[analytic function]] on the [[upper half-plane]] satisfying a certain kind of [[functional equation]] with respect to the [[Group action (mathematics)|group action]] of the [[modular group]], and also satisfying a growth condition. The theory of modular forms therefore belongs to [[complex analysis]] but the main importance of the theory has traditionally been in its connections with [[number theory]]. Modular forms appear in other areas, such as [[algebraic topology]], [[sphere packing]], and [[string theory]].

* a kind of [[functional equation]] with respect to the [[Group action (mathematics)|group action]] of the [[modular group]],

A '''modular function''' is a function that is invariant with respect to the modular group, but without the condition that {{math|''f'' (''z'')}} be [[Holomorphic function|holomorphic]] in the upper half-plane (among other requirements). Instead, modular functions are [[Meromorphic function|meromorphic]] (that is, they are holomorphic on the complement of a set of isolated points, which are poles of the function).

* and a growth condition.

In [[mathematics]], a '''modular form''' is a (complex) [[analytic function]] on the [[upper half-plane]] satisfying a certain kind of [[functional equation]] with respect to the [[Group action (mathematics)|group action]] of the [[modular group]], and also satisfying a growth condition. The theory of modular forms therefore belongs to [[complex analysis]]. but theThe main importance of the theory has traditionally been inis its connections with [[number theory]]. Modular forms appear in other areas, such as [[algebraic topology]], [[sphere packing]], and [[string theory]].

Modular form theory is a special case of the more general theory of [[automorphic form]]s which are functions defined on [[Lie group]]s which transform nicely with respect to the action of certain [[discrete subgroup]]s, generalizing the example of the modular group <math>\mathrm{SL}_2(\mathbb Z) \subset \mathrm{SL}_2(\mathbb R)</math>.

Modular form theory is a special case of the more general theory of [[automorphic form]]s, which are functions defined on [[Lie group]]s whichthat transform nicely with respect to the action of certain [[discrete subgroup]]s, generalizing the example of the modular group <math>\mathrm{SL}_2(\mathbb Z) \subset \mathrm{SL}_2(\mathbb R)</math>.

== General definition of modular forms ==

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 \subset \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]] such that the following two conditions are satisfied:<blockquote>

== Definition ==

1. ('''automorphy condition''') For any <math>\gamma \in \Gamma</math> there is the equality<ref group="note">Some authors use different conventions, allowing an additional constant depending only on <math>\gamma</math>, see e.g. https://dlmf.nist.gov/23.15#E5</ref> <math>f(\gamma(z)) = (cz + d)^k f(z)</math>

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 \subset \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]] such that the following two conditions are satisfied:<blockquote>

2.* ('''growthAutomorphy condition'''): For any <math>\gamma \in \text{SL}_2(\mathbb{Z})Gamma</math> there is the functionequality<ref <mathgroup="note">(czSome +authors d)^{-k}use f(\gamma(z))</math>different isconventions, boundedallowing foran additional constant depending only on <math>\text{im}(z) \to \inftygamma</math>, see e.g. https://dlmf.nist.gov/23.15#E5</blockquoteref> where <math display=inline> f(\gamma(z)) = \frac{az+b}{(cz + d})^k f(z)</math> and the function <math display=inline> \gamma </math> is identified with the matrix <math display=inline>\gamma = \begin{pmatrix}

* 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>

where <math display="inline"> \gamma(z) = \frac{az+b}{cz+d} </math> and the function <math display="inline"> \gamma </math> is identified with the matrix <math display="inline">\gamma = \begin{pmatrix}

a & b \\

c & d

\end{pmatrix} \in \text{SL}_2(\mathbb{Z}).\,</math> (The identification of such functions with such matrices causes composition of such functions to correspond to matrix multiplication.) In addition, it is called a '''cusp form''' if it satisfies the following growth condition:<blockquote>3.

* ('''cuspidalCuspidal condition''') : For any <math>\gamma \in \text{SL}_2(\mathbb{Z})</math> the function <math>(cz + d)^{-k}f(\gamma(z)) \to 0</math> as <math>\text{im}(z) \to \infty</math></blockquote>

=== 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 \subset \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<blockquote><math>f \in H^0(X_\Gamma,\omega^{\otimes k}) = M_k(\Gamma)</math></blockquote>where <math>\omega</math> is a canonical line bundle on the [[modular curve]]<blockquote><math>X_\Gamma = \Gamma \backslash (\mathcal{H} \cup \mathbb{P}^1(\mathbb{Q}))</math></blockquote>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]].

== Modular forms for SL(2,function Z)==

A '''modular function''' is a function that is invariant with respect to the modular group, but without the condition that {{math|''f''&thinsp;(''z'')}} be [[Holomorphic function|holomorphic]] in the upper half-plane (among other requirements). Instead, modular functions are [[Meromorphic function|meromorphic]] (that is,: they are holomorphic on the complement of a set of isolated points, which are poles of the function).

== Modular forms for SL(2, Z) ==

=== Standard definition ===

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==Modular functions==

When the weight ''k'' is zero, it can be shown using [[Liouville's theorem (complex analysis)|Liouville's theorem]] that the only modular forms are constant functions. However, relaxing the requirement that ''f'' be holomorphic leads to the notion of ''modular functions''. A function ''f'' : '''H''' → '''C''' is called modular [[iff]] it satisfies the following properties:

#* ''f'' is [[meromorphic function|meromorphic]] in the open [[upper half-plane]] ''H''.

#* For every integer [[matrix (mathematics)|matrix]] <math>\begin{pmatrix}a & b \\ c & d \end{pmatrix}</math> in the [[modular group|modular group {{math|Γ}}]], <math> f\left(\frac{az+b}{cz+d}\right) = f(z)</math>.

#* As pointed out above, theThe second condition implies that ''f'' is periodic, and therefore has a [[Fourier series]]. The third condition is that this series is of the form

::<math>f(z) = \sum_{n=-m}^\infty a_n e^{2i\pi nz}.</math>

It is often written in terms of <math>q=\exp(2\pi i z)</math> (the square of the [[nome (mathematics)|nome]]), as:

Line 198 ⟶ 208:

==History==

{{Unreferenced section|date=October 2019}}

The theory of modular forms was developed in four periods:

* first inIn connection with the theory of [[elliptic function]]s, in the first part of theearly nineteenth century; then
* byBy [[Felix Klein]] and others towards the end of the nineteenth century as the automorphic form concept became understood (for one variable);
* then byBy [[Erich Hecke]] from about 1925;
* and then inIn the 1960s, as the needs of number theory and the formulation of the [[modularity theorem]] in particular made it clear that modular forms are deeply implicated.

The term "modular form", as a systematic description, is usually attributed to Hecke.