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In [[mathematics]], a '''modular form''' is a (complex) [[analyticholomorphic function]] on the [[Upper half-plane#Complex plane|complex upper half-plane]], <math>\,\mathcal{H}\,</math>, that roughly satisfies: a [[functional equation]] with respect to the [[Group action (mathematics)|group action]] of the [[modular group]] and a growth condition. The theory of modular forms has origins in [[complex analysis]], with important connections with [[number theory]]. Modular forms also appear in other areas, such as [[algebraic topology]], [[sphere packing]], and [[string theory]].

EachModular form theory is a special case of the more general theory of [[automorphic form]]s, which are functions defined on [[Lie group]]s that 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>. Every modular form is attached to a [[Galois representation]].<ref name=":0">{{Cite news |last=Van Wyk |first=Gerhard |date=July 2023 |title=Elliptic Curves Yield Their Secrets in a New Number System |work=Quanta |url=https://www.quantamagazine.org/elliptic-curves-yield-their-secrets-in-a-new-number-system-20230706/?mc_cid=e612def96e&mc_eid=506130a407}}</ref>

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

* and a growth condition.

The term "modular form", as a systematic description, is usually attributed to [[Erich Hecke]]. The importance of modular forms across multiple field of mathematics has been humorously represented in a possibly apocryphal quote attributed to [[Martin Eichler]] describing modular forms as being the fifth fundamental operation in mathematics, after addition, subtraction, multiplication and division.<ref>{{Cite web |last=Cepelewicz |first=Jordana |date=2023-09-21 |title=Behold Modular Forms, the 'Fifth Fundamental Operation' of Math |url=https://www.quantamagazine.org/behold-modular-forms-the-fifth-fundamental-operation-of-math-20230921/ |access-date=2025-02-25 |website=Quanta Magazine |language=en}}</ref>

The theory of modular forms therefore belongs to [[complex analysis]]. The main importance of the theory is its connections with [[number theory]]. Modular forms appear in other areas, such as [[algebraic topology]], [[sphere packing]], and [[string theory]].

== Definition ==

Modular form theory is a special case of the more general theory of [[automorphic form]]s, which are functions defined on [[Lie group]]s that 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>.

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]] suchsatisfying thatthe following two conditions are satisfied:

* ''Automorphy condition'': Forfor any <math>\gamma \in \Gamma</math>, therewe ishave<math>f(\gamma(z)) the= equality(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><math>f(\gamma(z)) = (cz + d)^k f(z)</math>and

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

* ''Growth condition'': Forfor 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>.

Each modular form is attached to a [[Galois representation]].<ref name=":0">{{Cite news |last=Van Wyk |first=Gerhard |date=July 2023 |title=Elliptic Curves Yield Their Secrets in a New Number System |work=Quanta |url=https://www.quantamagazine.org/elliptic-curves-yield-their-secrets-in-a-new-number-system-20230706/?mc_cid=e612def96e&mc_eid=506130a407}}</ref>

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

== 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 \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 two conditions are satisfied:

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

* 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. {{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><math>f(\gamma(z)) = (cz + d)^k f(z)</math>

* 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>

Note that <math>\gamma</math> is a matrix

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}

:<math display="inline">\gamma = \begin{pmatrix}

a & b \\

c & d

\end{pmatrix} \in \text{SL}_2(\mathbb{Z}),</math>

\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:

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.

* Cuspidal 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>

=== 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 function ==

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

A modular form of weight {{mvar|<math>k}}</math> for the [[modular group]]

:<math>\text{SL}(2, \mathbf Z) = \left \{ \left. \begin{pmatrix}a & b \\ c & d \end{pmatrix} \right | a, b, c, d \in \mathbf Z,\ ad-bc = 1 \right \}</math>

is a [[complex numbers|complex-valued]] function {{<math|&thinsp;''>f''&thinsp;}}</math> on the [[upper half-plane]] {{<math|'''>\mathcal{H''' {{}=}} \{''z'' ∈ '''\in\C''',\mid [[imaginary part|\operatorname{Im]]}(''z'') > 0\},}}</math> satisfying the following three conditions:

# {{<math|&thinsp;''>f''&thinsp;}}</math> is a [[holomorphic function|holomorphic]] on <math>\mathcal{{math|'''H'''}}</math>.

# For any {{<math|''>z'' ∈ '''\in\mathcal{H'''}}</math> and any matrix in <math>\text{{math|SL}(2, '''\Z''')}} as above</math>, we have:

#:<math> f\left(\frac{az+b}{cz+d}\right) = (cz+d)^k f(z)</math>.

# <math>f</math> is bounded as <math>\operatorname{Im}(z)\to\infty</math>.

# {{math|&thinsp;''f''&thinsp;}} is required to be bounded as {{math|''z'' → [[Imaginary unit|''i'']]∞}}.

Remarks:

* The weight {{mvar|<math>k}}</math> is typically a positive integer.

* For odd {{mvar|<math>k}}</math>, only the zero function can satisfy the second condition.

* The third condition is also phrased by saying that {{<math|&thinsp;''>f''&thinsp;}}</math> is "holomorphic at the cusp", a terminology that is explained below. Explicitly, the condition means that there exist some <math> M, D > 0 </math> such that <math> \operatorname{Im}(z) > M \implies |f(z)| < D </math>, meaning <math>f</math> is bounded above some horizontal line.

* The second condition for

::<math>S = \begin{pmatrix}0 & -1 \\ 1 & 0 \end{pmatrix}, \qquad T = \begin{pmatrix}1 & 1 \\ 0 & 1 \end{pmatrix}</math>

:reads

::<math>f\left(-\frac{1}{z}\right) = z^k f(z), \qquad f(z + 1) = f(z)</math>

:respectively. Since {{mvar|<math>S}}</math> and {{mvar|<math>T}}</math> [[generating set of a group|generate]] the modular group <math>\text{{math|SL}(2, '''\Z''')}}</math>, the second condition above is equivalent to these two equations.

* Since {{<math|&thinsp;''>f''&thinsp;(''z'' + 1) {{=}} &thinsp;''f''&thinsp;(''z'')}}</math>, modular forms are [[periodic function]]s, with period {{math|1}}, and thus have a [[Fourier series]].

===Definition in terms of lattices or elliptic curves===

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===The Riemann surface ''G''\H^&lowast;===

Let {{mvar|G}} be a subgroup of {{math|SL(2, '''Z''')}} that is of finite [[Index of a subgroup|index]]. Such a group {{mvar|G}} [[Group action (mathematics)|acts]] on '''H''' in the same way as {{math|SL(2, '''Z''')}}. The [[quotient topological space]] ''G''\'''H''' can be shown to be a [[Hausdorff space]]. Typically it is not compact, but can be [[compactification (mathematics)|compactified]] by adding a finite number of points called ''cusps''. These are points at the boundary of '''H''', i.e. in '''[[Rational numbers|Q]]'''∪{∞},<ref group="note">Here, a matrix <math>\begin{pmatrix} a & b \\ c & d \end{pmatrix}</math> sends ∞ to ''a''/''c''.</ref> such that there is a parabolic element of {{mvar|G}} (a matrix with [[trace of a matrix|trace]] ±2) fixing the point. This yields a compact topological space ''G''\'''H'''^∗. What is more, it can be endowed with the structure of a [[Riemann surface]], which allows one to speak of holo- and meromorphic functions.

Important examples are, for any positive integer ''N'', either one of the [[congruence subgroup]]s

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==Rings of modular forms==

{{Main|Ring of modular forms}}

For a subgroup {{math|Γ}} of the {{math|SL(2, '''Z''')}}, the ring of modular forms is the [[graded ring]] generated by the modular forms of {{math|Γ}}. In other words, if {{math|M_k(Γ)}} beis the ringvector space of modular forms of weight {{mvar|k}}, then the ring of modular forms of {{math|Γ}} is the graded ring <math>M(\Gamma) = \bigoplus_{k > 0} M_k(\Gamma)</math>.

Rings of modular forms of congruence subgroups of {{math|SL(2, '''Z''')}} are finitely generated due to a result of [[Pierre Deligne]] and [[Michael Rapoport]]. Such rings of modular forms are generated in weight at most 6 and the relations are generated in weight at most 12 when the congruence subgroup has nonzero odd weight modular forms, and the corresponding bounds are 5 and 10 when there are no nonzero odd weight modular forms.

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==Types==

===Entire forms===

If ''f'' is [[Holomorphic function|holomorphic]] at the cusp (has no pole at ''q''&nbsp;=&nbsp;0), it is called an '''entire modular form'''.

If ''f'' is meromorphic but not holomorphic at the cusp, it is called a '''non-entire modular form'''. For example, the [[j-invariant]] is a non-entire modular form of weight 0, and has a simple pole at&nbsp;i∞.

===New forms===

{{Main|Atkin–Lehner theory}}

[[Atkin–Lehner theory|New forms]] are a subspace of modular forms<ref>{{Cite web|last=Mocanu|first=Andreea|title=Atkin-Lehner Theory of <math>\Gamma_1(N)</math>-Modular Forms|url=https://andreeamocanu.github.io/atkin-lehner-theory.pdf|url-status=live|archive-url=https://web.archive.org/web/20200731204425/https://andreeamocanu.github.io/atkin-lehner-theory.pdf|archive-date=31 July 2020}}</ref> of a fixed weightlevel <math>N</math> which cannot be constructed from modular forms of lower weightslevels <math>M</math> dividing <math>N</math>. The other forms are called '''old forms'''. These old forms can be constructed using the following observations: if <math>M \mid N</math> then <math>\Gamma_1(N) \subseteq \Gamma_1(M)</math> giving a reverse inclusion of modular forms <math>M_k(\Gamma_1(M)) \subseteq M_k(\Gamma_1(N))</math>.

===Cusp forms===

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Taniyama and Shimura identified a 1-to-1 matching between certain modular forms and elliptic curves. [[Robert Langlands]] built on this idea in the construction of his expansive [[Langlands program]], which has become one of the most far-reaching and consequential research programs in math.

In 1994 [[Andrew Wiles]] used modular forms to prove [[Fermat’s Last Theorem]]. In 2001 all elliptic curves were proven to be modular over the rational numbers. In 2013 elliptic curves were proven to be modular over real [[quadratic fields]]. In 2023 elliptic curves were proven to be modular over about half of imaginary quadratic fields, including fields formed by combining the [[rational numbers]] with the [[square root]] of integers down to −5. <ref name=":0" />

== See also ==

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*{{citation |author-link=Erich Hecke |first=Erich |last=Hecke |title=Mathematische Werke |___location=Göttingen |publisher=[[Vandenhoeck & Ruprecht]] |year=1970 }}

*{{citation |first=Robert A. |last=Rankin |title=Modular forms and functions |year=1977 |publisher=[[Cambridge University Press]] |___location=Cambridge |isbn=0-521-21212-X }}

*{{citation |firstfirst1=K. |lastlast1=Ribet |first2=W. |last2=Stein |url=https://wstein.org/books/ribet-stein/ |title=Lectures on Modular Forms and Hecke Operators}}

*{{citation |author-link=Jean-Pierre Serre |first=Jean-Pierre |last=Serre |title=A Course in Arithmetic |series=Graduate Texts in Mathematics |volume=7 |publisher=[[Springer-Verlag]] |___location=New York |year=1973 }}. ''Chapter VII provides an elementary introduction to the theory of modular forms''.

*{{citation |firstfirst1=N. P. |lastlast1=Skoruppa |author-link2=Don Zagier |first2=D. |last2=Zagier |title=Jacobi forms and a certain space of modular forms |journal=[[Inventiones Mathematicae]] |year=1988 |volume=94 |page=113 |publisher=[[Springer Publishing|Springer]] |doi=10.1007/BF01394347 |bibcode=1988InMat..94..113S }}

*[https://www.quantamagazine.org/behold-modular-forms-the-fifth-fundamental-operation-of-math-20230921/ Behold Modular Forms, the ‘Fifth Fundamental Operation’ of Math]

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