Modular form

Inmathematics,amodular formis a (complex)analytic functionon theupper half-plane, $\,{\mathcal {H}}\,$ ,that satisfies:

a kind offunctional equationwith respect to thegroup actionof themodular group,
and a growth condition.

The theory of modular forms therefore belongs tocomplex analysis.The main importance of the theory is its connections withnumber theory.Modular forms appear in other areas, such asalgebraic topology,sphere packing,andstring theory.

Modular form theory is a special case of the more general theory ofautomorphic forms,which are functions defined onLie groupsthat transform nicely with respect to the action of certaindiscrete subgroups,generalizing the example of the modular group $\mathrm {SL} _{2}(\mathbb {Z} )\subset \mathrm {SL} _{2}(\mathbb {R} )$ .

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

Each modular form is attached to aGalois representation.^[1]

Definition[edit]

In general,^[2]given a subgroup $\Gamma \subset {\text{SL}}_{2}(\mathbb {Z} )$ offinite index,called anarithmetic group,amodular form of level $\Gamma$ and weight $k$ is aholomorphic function $f:{\mathcal {H}}\to \mathbb {C}$ from theupper half-planesuch that two conditions are satisfied:

Automorphy condition: For any $\gamma \in \Gamma$ there is the equality^{[note 1]} $f(\gamma (z))=(cz+d)^{k}f(z)$
Growth condition: For any $\gamma \in {\text{SL}}_{2}(\mathbb {Z} )$ the function $(cz+d)^{-k}f(\gamma (z))$ is bounded for ${\text{im}}(z)\to \infty$

where ${\textstyle \gamma (z)={\frac {az+b}{cz+d}}}$ and the function ${\textstyle \gamma }$ is identified with the matrix ${\textstyle \gamma ={\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in {\text{SL}}_{2}(\mathbb {Z} ).\,}$ The identification of such functions with such matrices causes composition of such functions to correspond to matrix multiplication. In addition, it is called acusp formif it satisfies the following growth condition:

Cuspidal condition: For any $\gamma \in {\text{SL}}_{2}(\mathbb {Z} )$ the function $(cz+d)^{-k}f(\gamma (z))\to 0$ as ${\text{im}}(z)\to \infty$

As sections of a line bundle[edit]

Modular forms can also be interpreted as sections of a specificline bundleonmodular varieties.For $\Gamma \subset {\text{SL}}_{2}(\mathbb {Z} )$ a modular form of level $\Gamma$ and weight $k$ can be defined as an element of

$f\in H^{0}(X_{\Gamma },\omega ^{\otimes k})=M_{k}(\Gamma )$

where $\omega$ is a canonical line bundle on themodular curve

$X_{\Gamma }=\Gamma \backslash ({\mathcal {H}}\cup \mathbb {P} ^{1}(\mathbb {Q} ))$

The dimensions of these spaces of modular forms can be computed using theRiemann–Roch theorem.^[3]The classical modular forms for $\Gamma ={\text{SL}}_{2}(\mathbb {Z} )$ are sections of a line bundle on themoduli stack of elliptic curves.

Modular function[edit]

A modular function is a function that is invariant with respect to the modular group, but without the condition that $f (z)$ beholomorphicin the upper half-plane (among other requirements). Instead, modular functions aremeromorphic:they are holomorphic on the complement of a set of isolated points, which are poles of the function.

Modular forms for SL(2, Z)[edit]

Standard definition[edit]

A modular form of weight $k$ for themodular group

{\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\}

is acomplex-valuedfunction $f$ on theupper half-plane $H = {z \in C, Im (z) > 0},$ satisfying the following three conditions:

$f$ is aholomorphic functionon $H$ .
For any $z \in H$ and any matrix in $SL(2, Z)$ as above, we have:
$f\left({\frac {az+b}{cz+d}}\right)=(cz+d)^{k}f(z)$
$f$ is required to be bounded as $z \to i \infty$ .

Remarks:

The weight $k$ is typically a positive integer.
For odd $k$ ,only the zero function can satisfy the second condition.
The third condition is also phrased by saying that $f$ is "holomorphic at the cusp", a terminology that is explained below. Explicitly, the condition means that there exist some $M,D>0$ such that $\operatorname {Im} (z)>M\implies |f(z)|<D$ ,meaning $f$ is bounded above some horizontal line.
The second condition for

S={\begin{pmatrix}0&-1\\1&0\end{pmatrix}},\qquad T={\begin{pmatrix}1&1\\0&1\end{pmatrix}}

reads

f\left(-{\frac {1}{z}}\right)=z^{k}f(z),\qquad f(z+1)=f(z)

respectively. Since

S

and

T

generatethe modular group

SL(2, Z)

,the second condition above is equivalent to these two equations.

Since $f (z + 1) = f (z)$ ,modular forms areperiodic functions,with period $1$ ,and thus have aFourier series.

Definition in terms of lattices or elliptic curves[edit]

A modular form can equivalently be defined as a functionFfrom the set oflatticesin $C$ to the set ofcomplex numberswhich satisfies certain conditions:

If we consider the lattice $Λ = Z α + Z z$ generated by a constant $α$ and a variable $z$ ,then $F (Λ)$ is ananalytic functionof $z$ .
If $α$ is a non-zero complex number and $α Λ$ is the lattice obtained by multiplying each element of $Λ$ by $α$ ,then $F (α Λ) = α - k F (Λ)$ where $k$ is a constant (typically a positive integer) called theweightof the form.
Theabsolute valueof $F (Λ)$ remains bounded above as long as the absolute value of the smallest non-zero element in $Λ$ is bounded away from 0.

The key idea in proving the equivalence of the two definitions is that such a function $F$ is determined, because of the second condition, by its values on lattices of the form $Z + Z τ$ ,where $τ \in H$ .

Examples[edit]

I. Eisenstein series

The simplest examples from this point of view are theEisenstein series.For each even integer $k > 2$ ,we define $G k (Λ)$ to be the sum of $λ - k$ over all non-zero vectors $λ$ of $Λ$ :

G_{k}(\Lambda )=\sum _{0\neq \lambda \in \Lambda }\lambda ^{-k}.

Then $G k$ is a modular form of weight $k$ .For $Λ = Z + Z τ$ we have

G_{k}(\Lambda )=G_{k}(\tau )=\sum _{(0,0)\neq (m,n)\in \mathbf {Z} ^{2}}{\frac {1}{(m+n\tau )^{k}}},

and

{\begin{aligned}G_{k}\left(-{\frac {1}{\tau }}\right)&=\tau ^{k}G_{k}(\tau ),\\G_{k}(\tau +1)&=G_{k}(\tau ).\end{aligned}}

The condition $k > 2$ is needed forconvergence;for odd $k$ there is cancellation between $λ - k$ and $(- λ) - k$ ,so that such series are identically zero.

II. Theta functions of even unimodular lattices

Aneven unimodular lattice $L$ in $R n$ is a lattice generated by $n$ vectors forming the columns of a matrix of determinant 1 and satisfying the condition that the square of the length of each vector in $L$ is an even integer. The so-calledtheta function

\vartheta _{L}(z)=\sum _{\lambda \in L}e^{\pi i\Vert \lambda \Vert ^{2}z}

converges when Im(z) > 0, and as a consequence of thePoisson summation formulacan be shown to be a modular form of weight $n /2$ .It is not so easy to construct even unimodular lattices, but here is one way: Let $n$ be an integer divisible by 8 and consider all vectors $v$ in $R n$ such that $2 v$ has integer coordinates, either all even or all odd, and such that the sum of the coordinates of $v$ is an even integer. We call this lattice $L n$ .When $n = 8$ ,this is the lattice generated by the roots in theroot systemcalledE₈.Because there is only one modular form of weight 8 up to scalar multiplication,

\vartheta _{L_{8}\times L_{8}}(z)=\vartheta _{L_{16}}(z),

even though the lattices $L 8 \times L 8$ and $L 16$ are not similar.John Milnorobserved that the 16-dimensionaltoriobtained by dividing $R 16$ by these two lattices are consequently examples ofcompact Riemannian manifoldswhich areisospectralbut notisometric(seeHearing the shape of a drum.)

III. The modular discriminant

TheDedekind eta functionis defined as

\eta (z)=q^{1/24}\prod _{n=1}^{\infty }(1-q^{n}),\qquad q=e^{2\pi iz}.

whereqis the square of thenome.Then themodular discriminant $Δ(z) = (2π) 12 η (z) 24$ is a modular form of weight 12. The presence of 24 is related to the fact that theLeech latticehas 24 dimensions.A celebrated conjectureofRamanujanasserted that when $Δ(z)$ is expanded as a power series in q, the coefficient of $q p$ for any prime $p$ has absolute value $\leq 2 p 11/2$ .This was confirmed by the work ofEichler,Shimura,Kuga,Ihara,andPierre Deligneas a result of Deligne's proof of theWeil conjectures,which were shown to imply Ramanujan's conjecture.

The second and third examples give some hint of the connection between modular forms and classical questions in number theory, such as representation of integers byquadratic formsand thepartition function.The crucial conceptual link between modular forms and number theory is furnished by the theory ofHecke operators,which also gives the link between the theory of modular forms andrepresentation theory.

Modular functions[edit]

When the weightkis zero, it can be shown usingLiouville's theoremthat the only modular forms are constant functions. However, relaxing the requirement thatfbe holomorphic leads to the notion ofmodular functions.A functionf:H→Cis called modular if it satisfies the following properties:

fismeromorphicin the openupper half-planeH
For every integermatrix ${\begin{pmatrix}a&b\\c&d\end{pmatrix}}$ in themodular group $Γ$ , $f\left({\frac {az+b}{cz+d}}\right)=f(z)$ .
The second condition implies thatfis periodic, and therefore has aFourier series.The third condition is that this series is of the form

f(z)=\sum _{n=-m}^{\infty }a_{n}e^{2i\pi nz}.

It is often written in terms of $q=\exp(2\pi iz)$ (the square of thenome), as:

f(z)=\sum _{n=-m}^{\infty }a_{n}q^{n}.

This is also referred to as theq-expansion off(q-expansion principle). The coefficients $a_{n}$ are known as the Fourier coefficients off,and the numbermis called the order of the pole offat i∞. This condition is called "meromorphic at the cusp", meaning that only finitely many negative-ncoefficients are non-zero, so theq-expansion is bounded below, guaranteeing that it is meromorphic atq= 0.^{[note 2]}

Sometimes a weaker definition of modular functions is used – under the alternative definition, it is sufficient thatfbe meromorphic in the open upper half-plane and thatfbe invariant with respect to a sub-group of the modular group of finite index.^[4]This is not adhered to in this article.

Another way to phrase the definition of modular functions is to useelliptic curves:every lattice Λ determines anelliptic curveC/Λ overC;two lattices determineisomorphicelliptic curves if and only if one is obtained from the other by multiplying by some non-zero complex number $α$ .Thus, a modular function can also be regarded as a meromorphic function on the set of isomorphism classes of elliptic curves. For example, thej-invariantj(z) of an elliptic curve, regarded as a function on the set of all elliptic curves, is a modular function. More conceptually, modular functions can be thought of as functions on themoduli spaceof isomorphism classes of complex elliptic curves.

A modular formfthat vanishes at $q = 0$ (equivalently, $a 0 = 0$ ,also paraphrased as $z = i \infty$ ) is called acusp form(SpitzenforminGerman). The smallestnsuch that $a n \neq 0$ is the order of the zero offat $i \infty$ .

Amodular unitis a modular function whose poles and zeroes are confined to the cusps.^[5]

Modular forms for more general groups[edit]

The functional equation, i.e., the behavior offwith respect to $z\mapsto {\frac {az+b}{cz+d}}$ can be relaxed by requiring it only for matrices in smaller groups.

The Riemann surfaceG\H^∗[edit]

Let $G$ be a subgroup of $SL(2, Z)$ that is of finiteindex.Such a group $G$ actsonHin the same way as $SL(2, Z)$ .Thequotient topological spaceG\Hcan be shown to be aHausdorff space.Typically it is not compact, but can be compactified by adding a finite number of points calledcusps.These are points at the boundary ofH,i.e. inQ∪{∞},^{[note 3]}such that there is a parabolic element of $G$ (a matrix withtrace±2) fixing the point. This yields a compact topological spaceG\H^∗.What is more, it can be endowed with the structure of aRiemann surface,which allows one to speak of holo- and meromorphic functions.

Important examples are, for any positive integerN,either one of thecongruence subgroups

{\begin{aligned}\Gamma _{0}(N)&=\left\{{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in {\text{SL}}(2,\mathbf {Z} ):c\equiv 0{\pmod {N}}\right\}\\\Gamma (N)&=\left\{{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in {\text{SL}}(2,\mathbf {Z} ):c\equiv b\equiv 0,a\equiv d\equiv 1{\pmod {N}}\right\}.\end{aligned}}

ForG= Γ₀(N) or $Γ(N)$ ,the spacesG\HandG\H^∗are denotedY₀(N) andX₀(N) andY(N),X(N), respectively.

The geometry ofG\H^∗can be understood by studyingfundamental domainsforG,i.e. subsetsD⊂Hsuch thatDintersects each orbit of the $G$ -action onHexactly once and such that the closure ofDmeets all orbits. For example, thegenusofG\H^∗can be computed.^[6]

Definition[edit]

A modular form for $G$ of weightkis a function onHsatisfying the above functional equation for all matrices in $G$ ,that is holomorphic onHand at all cusps of $G$ .Again, modular forms that vanish at all cusps are called cusp forms for $G$ .TheC-vector spaces of modular and cusp forms of weightkare denoted $M k (G)$ and $S k (G)$ ,respectively. Similarly, a meromorphic function onG\H^∗is called a modular function for $G$ .In caseG= Γ₀(N), they are also referred to as modular/cusp forms and functions oflevelN.For $G = Γ(1) = SL(2, Z)$ ,this gives back the afore-mentioned definitions.

Consequences[edit]

The theory of Riemann surfaces can be applied toG\H^∗to obtain further information about modular forms and functions. For example, the spaces $M k (G)$ and $S k (G)$ are finite-dimensional, and their dimensions can be computed thanks to theRiemann–Roch theoremin terms of the geometry of the $G$ -action onH.^[7]For example,

\dim _{\mathbf {C} }M_{k}\left({\text{SL}}(2,\mathbf {Z} )\right)={\begin{cases}\left\lfloor k/12\right\rfloor &k\equiv 2{\pmod {12}}\\\left\lfloor k/12\right\rfloor +1&{\text{otherwise}}\end{cases}}

where $\lfloor \cdot \rfloor$ denotes thefloor functionand $k$ is even.

The modular functions constitute thefield of functionsof the Riemann surface, and hence form a field oftranscendence degreeone (overC). If a modular functionfis not identically 0, then it can be shown that the number of zeroes offis equal to the number ofpolesoffin theclosureof thefundamental regionR_Γ.It can be shown that the field of modular function of levelN(N≥ 1) is generated by the functionsj(z) andj(Nz).^[8]

Line bundles[edit]

The situation can be profitably compared to that which arises in the search for functions on theprojective spaceP(V): in that setting, one would ideally like functionsFon the vector spaceVwhich are polynomial in the coordinates ofv≠ 0 inVand satisfy the equationF(cv) =F(v) for all non-zeroc.Unfortunately, the only such functions are constants. If we allow denominators (rational functions instead of polynomials), we can letFbe the ratio of twohomogeneouspolynomials of the same degree. Alternatively, we can stick with polynomials and loosen the dependence onc,lettingF(cv) =c^kF(v). The solutions are then the homogeneous polynomials of degree $k$ .On the one hand, these form a finite dimensional vector space for eachk,and on the other, if we letkvary, we can find the numerators and denominators for constructing all the rational functions which are really functions on the underlying projective space P(V).

One might ask, since the homogeneous polynomials are not really functions on P(V), what are they, geometrically speaking? Thealgebro-geometricanswer is that they aresectionsof asheaf(one could also say aline bundlein this case). The situation with modular forms is precisely analogous.

Modular forms can also be profitably approached from this geometric direction, as sections of line bundles on the moduli space of elliptic curves.

Rings of modular forms[edit]

For a subgroup $Γ$ of the $SL(2, Z)$ ,the ring of modular forms is thegraded ringgenerated by the modular forms of $Γ$ .In other words, if $M k (Γ)$ be the ring of modular forms of weight $k$ ,then the ring of modular forms of $Γ$ is the graded ring $M(\Gamma )=\bigoplus _{k>0}M_{k}(\Gamma )$ .

Rings of modular forms of congruence subgroups of $SL(2, Z)$ are finitely generated due to a result ofPierre DeligneandMichael 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.

More generally, there are formulas for bounds on the weights of generators of the ring of modular forms and its relations for arbitraryFuchsian groups.

Types[edit]

Entire forms[edit]

Iffisholomorphicat the cusp (has no pole atq= 0), it is called anentire modular form.

Iffis meromorphic but not holomorphic at the cusp, it is called anon-entire modular form.For example, thej-invariantis a non-entire modular form of weight 0, and has a simple pole at i∞.

New forms[edit]

New formsare a subspace of modular forms^[9]of a fixed level $N$ which cannot be constructed from modular forms of lower levels $M$ dividing $N$ .The other forms are calledold forms.These old forms can be constructed using the following observations: if $M\mid N$ then $\Gamma _{1}(N)\subseteq \Gamma _{1}(M)$ giving a reverse inclusion of modular forms $M_{k}(\Gamma _{1}(M))\subseteq M_{k}(\Gamma _{1}(N))$ .

Cusp forms[edit]

Acusp formis a modular form with a zero constant coefficient in its Fourier series. It is called a cusp form because the form vanishes at all cusps.

Generalizations[edit]

There are a number of other usages of the term "modular function", apart from this classical one; for example, in the theory ofHaar measures,it is a function $Δ(g)$ determined by the conjugation action.

Maass formsarereal-analytic eigenfunctionsof theLaplacianbut need not beholomorphic.The holomorphic parts of certain weak Maass wave forms turn out to be essentially Ramanujan'smock theta functions.Groups which are not subgroups of $SL(2, Z)$ can be considered.

Hilbert modular formsare functions innvariables, each a complex number in the upper half-plane, satisfying a modular relation for 2×2 matrices with entries in atotally real number field.

Siegel modular formsare associated to largersymplectic groupsin the same way in which classical modular forms are associated to $SL(2, R)$ ;in other words, they are related toabelian varietiesin the same sense that classical modular forms (which are sometimes calledelliptic modular formsto emphasize the point) are related to elliptic curves.

Jacobi formsare a mixture of modular forms and elliptic functions. Examples of such functions are very classical - the Jacobi theta functions and the Fourier coefficients of Siegel modular forms of genus two - but it is a relatively recent observation that the Jacobi forms have an arithmetic theory very analogous to the usual theory of modular forms.

Automorphic formsextend the notion of modular forms to generalLie groups.

Modular integrals of weight $k$ are meromorphic functions on the upper half plane of moderate growth at infinity whichfail to be modular of weight $k$ by a rational function.

Automorphic factorsare functions of the form $\varepsilon (a,b,c,d)(cz+d)^{k}$ which are used to generalise the modularity relation defining modular forms, so that

f\left({\frac {az+b}{cz+d}}\right)=\varepsilon (a,b,c,d)(cz+d)^{k}f(z).

The function $\varepsilon (a,b,c,d)$ is called the nebentypus of the modular form. Functions such as theDedekind eta function,a modular form of weight 1/2, may be encompassed by the theory by allowing automorphic factors.

History[edit]

The theory of modular forms was developed in four periods:

In connection with the theory ofelliptic functions,in the early nineteenth century
ByFelix Kleinand others towards the end of the nineteenth century as the automorphic form concept became understood (for one variable)
ByErich Heckefrom about 1925
In the 1960s, as the needs of number theory and the formulation of themodularity theoremin particular made it clear that modular forms are deeply implicated.

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

In 1994Andrew Wilesused modular forms to proveFermat’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 realquadratic fields.In 2023 elliptic curves were proven to be modular over about half of imaginary quadratic fields, including fields formed by combining therational numberswith thesquare rootof integers down to −5.^[1]

Notes[edit]

^Some authors use different conventions, allowing an additional constant depending only on $\gamma$ ,see e.g."DLMF: §23.15 Definitions ‣ Modular Functions ‣ Chapter 23 Weierstrass Elliptic and Modular Functions".dlmf.nist.gov.Retrieved2023-07-07.
^Ameromorphicfunction can only have a finite number of negative-exponent terms in its Laurent series, its q-expansion. It can only have at most apoleatq= 0, not anessential singularityas exp(1/q) has.
^Here, a matrix ${\begin{pmatrix}a&b\\c&d\end{pmatrix}}$ sends ∞ toa/c.

Citations[edit]

^^a ^bVan Wyk, Gerhard (July 2023)."Elliptic Curves Yield Their Secrets in a New Number System".Quanta.
^Lan, Kai-Wen."Cohomology of Automorphic Bundles"(PDF).Archived(PDF)from the original on 1 August 2020.
^Milne."Modular Functions and Modular Forms".p. 51.
^Chandrasekharan, K. (1985).Elliptic functions.Springer-Verlag.ISBN 3-540-15295-4.p. 15
^Kubert, Daniel S.;Lang, Serge(1981),Modular units,Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 244, Berlin, New York:Springer-Verlag,p. 24,ISBN 978-0-387-90517-4,MR 0648603,Zbl 0492.12002
^Gunning, Robert C. (1962),Lectures on modular forms,Annals of Mathematics Studies, vol. 48,Princeton University Press,p. 13
^Shimura, Goro (1971),Introduction to the arithmetic theory of automorphic functions,Publications of the Mathematical Society of Japan, vol. 11, Tokyo: Iwanami Shoten,Theorem 2.33, Proposition 2.26
^Milne, James (2010),Modular Functions and Modular Forms(PDF),p. 88,Theorem 6.1.
^Mocanu, Andreea."Atkin-Lehner Theory of $\Gamma _{1}(N)$ -Modular Forms "(PDF).Archived(PDF)from the original on 31 July 2020.

References[edit]

Apostol, Tom M.(1990),Modular functions and Dirichlet Series in Number Theory,New York:Springer-Verlag,ISBN 0-387-97127-0
Diamond, Fred; Shurman, Jerry Michael (2005),A First Course in Modular Forms,Graduate Texts in Mathematics, vol. 228, New York:Springer-Verlag,ISBN 978-0387232294Leads up to an overview of the proof of themodularity theorem.
Gelbart, Stephen S.(1975),Automorphic Forms on Adèle Groups,Annals of Mathematics Studies, vol. 83, Princeton, N.J.:Princeton University Press,MR 0379375.Provides an introduction to modular forms from the point of view of representation theory.
Hecke, Erich(1970),Mathematische Werke,Göttingen:Vandenhoeck & Ruprecht
Rankin, Robert A. (1977),Modular forms and functions,Cambridge:Cambridge University Press,ISBN 0-521-21212-X
Ribet, K.; Stein, W.,Lectures on Modular Forms and Hecke Operators
Serre, Jean-Pierre(1973),A Course in Arithmetic,Graduate Texts in Mathematics, vol. 7, New York:Springer-Verlag.Chapter VII provides an elementary introduction to the theory of modular forms.
Skoruppa, N. P.;Zagier, D.(1988), "Jacobi forms and a certain space of modular forms",Inventiones Mathematicae,94,Springer:113,Bibcode:1988InMat..94..113S,doi:10.1007/BF01394347
Behold Modular Forms, the ‘Fifth Fundamental Operation’ of Math

[3] Some authors use different conventions, allowing an additional constant depending only on $\gamma$ ,see e.g."DLMF: §23.15 Definitions ‣ Modular Functions ‣ Chapter 23 Weierstrass Elliptic and Modular Functions".dlmf.nist.gov.Retrieved2023-07-07.

[5] Ameromorphicfunction can only have a finite number of negative-exponent terms in its Laurent series, its q-expansion. It can only have at most apoleatq= 0, not anessential singularityas exp(1/q) has.

[8] Here, a matrix ${\begin{pmatrix}a&b\\c&d\end{pmatrix}}$ sends ∞ toa/c.

[:0-1] Van Wyk, Gerhard (July 2023)."Elliptic Curves Yield Their Secrets in a New Number System".Quanta.

[2] Lan, Kai-Wen."Cohomology of Automorphic Bundles"(PDF).Archived(PDF)from the original on 1 August 2020.

[4] Milne."Modular Functions and Modular Forms".p. 51.

[6] Chandrasekharan, K. (1985).Elliptic functions.Springer-Verlag.ISBN 3-540-15295-4.p. 15

[7] Kubert, Daniel S.;Lang, Serge(1981),Modular units,Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 244, Berlin, New York:Springer-Verlag,p. 24,ISBN 978-0-387-90517-4,MR 0648603,Zbl 0492.12002

[9] Gunning, Robert C. (1962),Lectures on modular forms,Annals of Mathematics Studies, vol. 48,Princeton University Press,p. 13

[10] Shimura, Goro (1971),Introduction to the arithmetic theory of automorphic functions,Publications of the Mathematical Society of Japan, vol. 11, Tokyo: Iwanami Shoten,Theorem 2.33, Proposition 2.26

[11] Milne, James (2010),Modular Functions and Modular Forms(PDF),p. 88,Theorem 6.1.

[12] Mocanu, Andreea."Atkin-Lehner Theory of $\Gamma _{1}(N)$ -Modular Forms "(PDF).Archived(PDF)from the original on 31 July 2020.

[1]

[2]

[note 1]

[3]

[note 2]

[4]

[5]

[note 3]

[6]

[7]

[8]

[9]

Definition[edit]

As sections of a line bundle[edit]

Modular function[edit]

Modular forms for SL(2, Z)[edit]

Standard definition[edit]

Definition in terms of lattices or elliptic curves[edit]

Examples[edit]

Modular functions[edit]

Modular forms for more general groups[edit]

The Riemann surfaceG\H∗[edit]

Definition[edit]

Consequences[edit]

Line bundles[edit]

Rings of modular forms[edit]

Types[edit]

Entire forms[edit]

New forms[edit]

Cusp forms[edit]

Generalizations[edit]

History[edit]

See also[edit]

Notes[edit]

Citations[edit]

References[edit]

The Riemann surfaceG\H^∗[edit]