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Inmathematics, amodular form is aholomorphic function on thecomplex upper half-plane,${\displaystyle \mathcal{H}}$, that roughly satisfies afunctional equation with respect to thegroup action of themodular group and a growth condition. The theory of modular forms has origins incomplex analysis, with important connections withnumber theory. Modular forms also 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 groups that transform nicely with respect to the action of certaindiscrete subgroups, generalizing the example of the modular group${\displaystyle {\mathrm{S}\mathrm{L}}_2(\mathbb{Z})\subset {\mathrm{S}\mathrm{L}}_2(\mathbb{R})}$. Every modular form is attached to aGalois representation.^[1]

The term "modular form", as a systematic description, is usually attributed toErich Hecke. The importance of modular forms across multiple field of mathematics has been humorously represented in a possibly apocryphal quote attributed toMartin Eichler describing modular forms as being the fifth fundamental operation in mathematics, after addition, subtraction, multiplication and division.^[2]

In general,^[3] given a subgroup offinite index (called anarithmetic group), amodular form of level${\displaystyle \mathrm{\Gamma }}$ and weight${\displaystyle k}$ is aholomorphic function${\displaystyle f:\mathcal{H}\to \mathbb{C}}$ from theupper half-plane satisfying the following two conditions:

• Automorphy condition: for any${\displaystyle \gamma \in \mathrm{\Gamma }}$, we have${\displaystyle f(\gamma (z))=(cz+d{)}^{k}f(z)}$ ,^{[note 1]} and

• Growth condition: for any${\displaystyle \gamma \in {\text{SL}}_2(\mathbb{Z})}$, the function${\displaystyle (cz+d{)}^{-k}f(\gamma (z))}$ is bounded for${\displaystyle \text{im}(z)\to \mathrm{\infty }}$.

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

• Cuspidal condition: For any${\displaystyle \gamma \in {\text{SL}}_2(\mathbb{Z})}$, we have${\displaystyle (cz+d{)}^{-k}f(\gamma (z))\to 0}$ as${\displaystyle \text{im}(z)\to \mathrm{\infty }}$.

Note that${\displaystyle \gamma }$ is a matrix

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

Modular forms can also be interpreted as sections of a specificline bundle onmodular varieties. For a modular form of level${\displaystyle \mathrm{\Gamma }}$ and weight${\displaystyle k}$ can be defined as an element of

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

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

${\displaystyle X_{\mathrm{\Gamma }}=\mathrm{\Gamma }\mathrm{\setminus }(\mathcal{H}\cup {\mathbb{P}}^1(\mathbb{Q})).}$

The dimensions of these spaces of modular forms can be computed using theRiemann–Roch theorem.^[4] The classical modular forms for${\displaystyle \mathrm{\Gamma }={\text{SL}}_2(\mathbb{Z})}$ are sections of a line bundle on themoduli stack of elliptic curves.

A modular function is a function that is invariant with respect to the modular group, but without the condition that it beholomorphic in 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.

A modular form of weight${\displaystyle k}$ for themodular group

is a function${\displaystyle f}$ on theupper half-plane${\displaystyle \mathcal{H}=\{z\in \mathbb{C}\mid \mathrm{Im}(z)>0\}}$ satisfying the following three conditions:

1. ${\displaystyle f}$ isholomorphic on${\displaystyle \mathcal{H}}$.
2. For any${\displaystyle z\in \mathcal{H}}$ and any matrix in${\displaystyle \text{SL}(2,\mathbb{Z})}$, we have

   ${\displaystyle f\left(\frac{az+b}{cz+d}\right)=(cz+d{)}^{k}f(z)}$.

3. ${\displaystyle f}$ is bounded as${\displaystyle \mathrm{Im}(z)\to \mathrm{\infty }}$.

Remarks:

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

reads

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

respectively. Since${\displaystyle S}$ and${\displaystyle T}$generate the group${\displaystyle \text{SL}(2,\mathbb{Z})}$, the second condition above is equivalent to these two equations.

• Since${\displaystyle f(z+1)=f(z)}$, modular forms areperiodic functions with period1, and thus have aFourier series.

A modular form can equivalently be defined as a functionF from the set oflattices inC to the set ofcomplex numbers which satisfies certain conditions:

1. If we consider the latticeΛ =Zα +Zz generated by a constantα and a variablez, thenF(Λ) is ananalytic function ofz.
2. Ifα is a non-zero complex number andαΛ is the lattice obtained by multiplying each element ofΛ byα, thenF(αΛ) =α^−kF(Λ) wherek is a constant (typically a positive integer) called theweight of the form.
3. Theabsolute value ofF(Λ) remains bounded above as long as the absolute value of the smallest non-zero element inΛ is bounded away from 0.

The key idea in proving the equivalence of the two definitions is that such a functionF is determined, because of the second condition, by its values on lattices of the formZ +Zτ, whereτ ∈H.

I. Eisenstein series

The simplest examples from this point of view are theEisenstein series. For each even integerk > 2, we defineG_k(Λ) to be the sum ofλ^−k over all non-zero vectorsλ ofΛ:

${\displaystyle G_k(\mathrm{\Lambda })=\sum _{0\ne \lambda \in \mathrm{\Lambda }}{\lambda }^{-k}.}$

ThenG_k is a modular form of weightk. ForΛ =Z +Zτ we have

${\displaystyle G_k(\mathrm{\Lambda })=G_k(\tau )=\sum _{(0,0)\ne (m,n)\in {\mathbf{Z}}^2}\frac{1}{(m+n\tau {)}^k},}$

and

The conditionk > 2 is needed forconvergence; for oddk there is cancellation betweenλ^−k and(−λ)^−k, so that such series are identically zero.

II. Theta functions of even unimodular lattices

Aneven unimodular latticeL inRⁿ is a lattice generated byn vectors forming the columns of a matrix of determinant 1 and satisfying the condition that the square of the length of each vector inL is an even integer. The so-calledtheta function

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

converges when Im(z) > 0, and as a consequence of thePoisson summation formula can be shown to be a modular form of weightn/2. It is not so easy to construct even unimodular lattices, but here is one way: Letn be an integer divisible by 8 and consider all vectorsv inRⁿ such that2v has integer coordinates, either all even or all odd, and such that the sum of the coordinates ofv is an even integer. We call this latticeL_n. Whenn = 8, this is the lattice generated by the roots in theroot system calledE₈. Because there is only one modular form of weight 8 up to scalar multiplication,

${\displaystyle {\vartheta }_{L_8\times L_8}(z)={\vartheta }_{L_{16}}(z),}$

even though the latticesL₈ ×L₈ andL₁₆ are not similar.John Milnor observed that the 16-dimensionaltori obtained by dividingR¹⁶ by these two lattices are consequently examples ofcompactRiemannian manifolds which areisospectral but notisometric (seeHearing the shape of a drum.)

III. The modular discriminant

TheDedekind eta function is defined as

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

whereq is the square of thenome. Then themodular discriminantΔ(z) = (2π)¹²η(z)²⁴ is a modular form of weight 12. The presence of 24 is related to the fact that theLeech lattice has 24 dimensions.A celebrated conjecture ofRamanujan asserted that whenΔ(z) is expanded as a power series in q, the coefficient ofq^p for any primep has absolute value≤ 2p^11/2. This was confirmed by the work ofEichler,Shimura,Kuga,Ihara, andPierre Deligne as 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 forms and 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.

When the weightk is zero, it can be shown usingLiouville's theorem that the only modular forms are constant functions. However, relaxing the requirement thatf be holomorphic leads to the notion ofmodular functions. A functionf :H →C is called modular if it satisfies the following properties:

• f ismeromorphic in the openupper half-planeH
• For every integermatrix in themodular groupΓ,${\displaystyle f\left(\frac{az+b}{cz+d}\right)=f(z)}$.
• The second condition implies thatf is periodic, and therefore has aFourier series. The third condition is that this series is of the form

${\displaystyle f(z)=\sum _{n=-m}^{\mathrm{\infty }}{a}_n{e}^{2i\pi nz}.}$

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

${\displaystyle f(z)=\sum _{n=-m}^{\mathrm{\infty }}{a}_n{q}^n.}$

This is also referred to as theq-expansion off (q-expansion principle). The coefficients${\displaystyle a_n}$ are known as the Fourier coefficients off, and the numberm is called the order of the pole off at i∞. This condition is called "meromorphic at the cusp", meaning that only finitely many negative-n coefficients 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 thatf be meromorphic in the open upper half-plane and thatf be invariant with respect to a sub-group of the modular group of finite index.^[5] 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 determineisomorphic elliptic curves if and only if one is obtained from the other by multiplying by some non-zero complex numberα. 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 space of isomorphism classes of complex elliptic curves.

A modular formf that vanishes atq = 0 (equivalently,a₀ = 0, also paraphrased asz =i∞) is called acusp form (Spitzenform inGerman). The smallestn such thata_n ≠ 0 is the order of the zero off ati∞.

Amodular unit is a modular function whose poles and zeroes are confined to the cusps.^[6]

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

LetG be a subgroup ofSL(2,Z) that is of finiteindex. Such a groupGacts onH in the same way asSL(2,Z). Thequotient topological spaceG\H can be shown to be aHausdorff space. Typically it is not compact, but can becompactified 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 ofG (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

ForG = Γ₀(N) orΓ(N), the spacesG\H andG\H^∗ are denotedY₀(N) andX₀(N) andY(N),X(N), respectively.

The geometry ofG\H^∗ can be understood by studyingfundamental domains forG, i.e. subsetsD ⊂H such thatD intersects each orbit of theG-action onH exactly once and such that the closure ofD meets all orbits. For example, thegenus ofG\H^∗ can be computed.^[7]

A modular form forG of weightk is a function onH satisfying the above functional equation for all matrices inG, that is holomorphic onH and at all cusps ofG. Again, modular forms that vanish at all cusps are called cusp forms forG. TheC-vector spaces of modular and cusp forms of weightk are denotedM_k(G) andS_k(G), respectively. Similarly, a meromorphic function onG\H^∗ is called a modular function forG. In caseG = Γ₀(N), they are also referred to as modular/cusp forms and functions oflevelN. ForG = Γ(1) = SL(2,Z), this gives back the afore-mentioned definitions.

The theory of Riemann surfaces can be applied toG\H^∗ to obtain further information about modular forms and functions. For example, the spacesM_k(G) andS_k(G) are finite-dimensional, and their dimensions can be computed thanks to theRiemann–Roch theorem in terms of the geometry of theG-action onH.^[8] For example,

where${\displaystyle \lfloor \cdot \rfloor }$ denotes thefloor function and${\displaystyle k}$ is even.

The modular functions constitute thefield of functions of the Riemann surface, and hence form a field oftranscendence degree one (overC). If a modular functionf is not identically 0, then it can be shown that the number of zeroes off is equal to the number ofpoles off in theclosure of 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).^[9]

The situation can be profitably compared to that which arises in the search for functions on theprojective space P(V): in that setting, one would ideally like functionsF on the vector spaceV which are polynomial in the coordinates ofv ≠ 0 inV and 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 letF be the ratio of twohomogeneous polynomials 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 degreek. On the one hand, these form a finite dimensional vector space for each k, and on the other, if we letk vary, 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-geometric answer is that they aresections of asheaf (one could also say aline bundle in 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.

For a subgroupΓ of theSL(2,Z), the ring of modular forms is thegraded ring generated by the modular forms ofΓ. In other words, ifM_k(Γ) is the vector space of modular forms of weightk, then the ring of modular forms ofΓ is the graded ring${\displaystyle M(\mathrm{\Gamma })=\underset{k>0}{⨁}{M}_k(\mathrm{\Gamma })}$.

Rings of modular forms of congruence subgroups ofSL(2,Z) are finitely generated due to a result ofPierre Deligne andMichael 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.

New forms are a subspace of modular forms^[10] of a fixed level${\displaystyle N}$ which cannot be constructed from modular forms of lower levels${\displaystyle M}$ dividing${\displaystyle N}$. The other forms are calledold forms. These old forms can be constructed using the following observations: if${\displaystyle M\mid N}$ then${\displaystyle {\mathrm{\Gamma }}_1(N)\subseteq {\mathrm{\Gamma }}_1(M)}$ giving a reverse inclusion of modular forms${\displaystyle M_k({\mathrm{\Gamma }}_1(M))\subseteq M_k({\mathrm{\Gamma }}_1(N))}$.

Acusp form is 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.

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 functionΔ(g) determined by the conjugation action.

Maass forms arereal-analyticeigenfunctions of theLaplacian but 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 ofSL(2,Z) can be considered.

Hilbert modular forms are functions inn variables, 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 forms are associated to largersymplectic groups in the same way in which classical modular forms are associated toSL(2,R); in other words, they are related toabelian varieties in the same sense that classical modular forms (which are sometimes calledelliptic modular forms to emphasize the point) are related to elliptic curves.

Jacobi forms are 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 forms extend the notion of modular forms to generalLie groups.

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

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

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

The function${\displaystyle \epsilon (a,b,c,d)}$ 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.

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The theory of modular forms was developed in four periods:

• In connection with the theory ofelliptic functions, in the early nineteenth century
• ByFelix Klein and others towards the end of the nineteenth century as the automorphic form concept became understood (for one variable)
• ByErich Hecke from about 1925
• In the 1960s, as the needs of number theory and the formulation of themodularity theorem in 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 Langlands built 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 Wiles used 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 numbers with thesquare root of integers down to −5.^[1]

• Wiles's proof of Fermat's Last Theorem

• 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

• Modular forms
• Analytic number theory
• Special functions

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