Modular form

In mathematics, a modular form is a (complex) analytic function on the upper half-plane, $\,{\mathcal {H}}\,$ , that roughly satisfies a functional equation with respect to the 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.

Modular form theory is a special case of the more general theory of automorphic forms, which are functions defined on Lie groups that transform nicely with respect to the action of certain discrete subgroups, generalizing the example of the modular group $\mathrm {SL} _{2}(\mathbb {Z} )\subset \mathrm {SL} _{2}(\mathbb {R} )$ . Every modular form is attached to a Galois representation.^[1]

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

Definition

In general,^[2] given a subgroup $\Gamma \subset {\text{SL}}_{2}(\mathbb {Z} )$ of finite index, called an arithmetic group, a modular form of level $\Gamma$ and weight $k$ is a holomorphic function $f:{\mathcal {H}}\to \mathbb {C}$ from the upper half-plane such 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 ${\textstyle \gamma (z)=(az+b)/(cz+d)}$ 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 a cusp form if 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

Modular forms can also be interpreted as sections of a specific line bundle on modular 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 the modular 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 the Riemann–Roch theorem.^[3] The classical modular forms for $\Gamma ={\text{SL}}_{2}(\mathbb {Z} )$ 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 $f (z)$ be holomorphic in the upper half-plane (among other requirements). Instead, modular functions are 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 $k$ for the modular 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 a complex-valued function $f$ on the upper half-plane $H = {z \in C, Im (z) > 0},$ satisfying the following three conditions:

$f$ is a holomorphic function on $H$ .
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

generate the modular group

SL(2, Z)

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

Since $f (z + 1) = f (z)$ , modular forms are periodic functions, with period $1$ , and thus have a Fourier series.

Definition in terms of lattices or elliptic curves

A modular form can equivalently be defined as a function F from the set of lattices in $C$ to the set of complex numbers which 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 an analytic function of $z$ .
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 the weight of the form.
The absolute value of $F (Λ)$ 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

I. Eisenstein series

The simplest examples from this point of view are the Eisenstein 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 for convergence; 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

An even 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-called theta 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 the Poisson summation formula can 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 the root system called E₈. 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 Milnor observed that the 16-dimensional tori obtained by dividing $R 16$ by these two lattices are consequently examples of compact Riemannian manifolds which are isospectral but not isometric (see Hearing the shape of a drum.)

III. The modular discriminant

The Dedekind eta function is defined as

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

where q is the square of the nome. Then the modular 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 the Leech lattice has 24 dimensions. A celebrated conjecture of Ramanujan asserted 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 of Eichler, Shimura, Kuga, Ihara, and Pierre Deligne as a result of Deligne's proof of the Weil 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 by quadratic forms and the partition function. The crucial conceptual link between modular forms and number theory is furnished by the theory of Hecke operators, which also gives the link between the theory of modular forms and representation theory.

Modular functions

When the weight k is zero, it can be shown using 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 if it satisfies the following properties:

f is meromorphic in the open upper half-plane H
For every integer matrix ${\begin{pmatrix}a&b\\c&d\end{pmatrix}}$ in the modular group $Γ$ , $f\left({\frac {az+b}{cz+d}}\right)=f(z)$ .
The second condition implies that f is periodic, and therefore has a Fourier 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 the nome), as:

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

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

Sometimes a weaker definition of modular functions is used – under the alternative definition, it is sufficient that f be meromorphic in the open upper half-plane and that f be 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 use elliptic curves: every lattice Λ determines an elliptic curve C/Λ over C; two lattices determine isomorphic elliptic 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, the j-invariant j(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 the moduli space of isomorphism classes of complex elliptic curves.

A modular form f that vanishes at $q = 0$ (equivalently, $a 0 = 0$ , also paraphrased as $z = i \infty$ ) is called a cusp form (Spitzenform in German). The smallest n such that $a n \neq 0$ is the order of the zero of f at $i \infty$ .

A modular unit is a modular function whose poles and zeroes are confined to the cusps.^[5]

Modular forms for more general groups

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

The Riemann surface G\H^∗

Let $G$ be a subgroup of $SL(2, Z)$ that is of finite index. Such a group $G$ acts on H in the same way as $SL(2, Z)$ . The quotient topological space G\H can be shown to be a Hausdorff space. Typically it is not compact, but can be compactified by adding a finite number of points called cusps. These are points at the boundary of H, i.e. in Q∪{∞},^{[note 3]} such that there is a parabolic element of $G$ (a matrix with 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 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}}

For G = Γ₀(N) or $Γ(N)$ , the spaces G\H and G\H^∗ are denoted Y₀(N) and X₀(N) and Y(N), X(N), respectively.

The geometry of G\H^∗ can be understood by studying fundamental domains for G, i.e. subsets D ⊂ H such that D intersects each orbit of the $G$ -action on H exactly once and such that the closure of D meets all orbits. For example, the genus of G\H^∗ can be computed.^[6]

Definition

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

Consequences

The theory of Riemann surfaces can be applied to G\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 the Riemann–Roch theorem in terms of the geometry of the $G$ -action on H.^[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 the floor function and $k$ is even.

The modular functions constitute the field of functions of the Riemann surface, and hence form a field of transcendence degree one (over C). If a modular function f is not identically 0, then it can be shown that the number of zeroes of f is equal to the number of poles of f in the closure of the fundamental region R_Γ.It can be shown that the field of modular function of level N (N ≥ 1) is generated by the functions j(z) and j(Nz).^[8]

Line bundles

The situation can be profitably compared to that which arises in the search for functions on the projective space P(V): in that setting, one would ideally like functions F on the vector space V which are polynomial in the coordinates of v ≠ 0 in V and satisfy the equation F(cv) = F(v) for all non-zero c. Unfortunately, the only such functions are constants. If we allow denominators (rational functions instead of polynomials), we can let F be the ratio of two homogeneous polynomials of the same degree. Alternatively, we can stick with polynomials and loosen the dependence on c, letting F(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 each k, and on the other, if we let k 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? The algebro-geometric answer is that they are sections of a sheaf (one could also say a line 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.

Rings of modular forms

For a subgroup $Γ$ of the $SL(2, Z)$ , the ring of modular forms is the graded ring generated by the modular forms of $Γ$ . In other words, if $M k (Γ)$ is the vector space 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 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.

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

Types

New forms

New forms are 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 called old 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

A cusp 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.

Generalizations

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

Maass forms are real-analytic eigenfunctions of the Laplacian but need not be holomorphic. The holomorphic parts of certain weak Maass wave forms turn out to be essentially Ramanujan's mock theta functions. Groups which are not subgroups of $SL(2, Z)$ can be considered.

Hilbert modular forms are functions in n variables, each a complex number in the upper half-plane, satisfying a modular relation for 2×2 matrices with entries in a totally real number field.

Siegel modular forms are associated to larger symplectic groups in the same way in which classical modular forms are associated to $SL(2, R)$ ; in other words, they are related to abelian varieties in the same sense that classical modular forms (which are sometimes called elliptic 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 general Lie groups.

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

Automorphic factors are 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 the Dedekind eta function, a modular form of weight 1/2, may be encompassed by the theory by allowing automorphic factors.

History

The theory of modular forms was developed in four periods:

In connection with the theory of elliptic functions, in the early nineteenth century
By Felix Klein and others towards the end of the nineteenth century as the automorphic form concept became understood (for one variable)
By Erich Hecke from about 1925
In 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.

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.^[1]

Notes

^ 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. Retrieved 2023-07-07.
^ A meromorphic function can only have a finite number of negative-exponent terms in its Laurent series, its q-expansion. It can only have at most a pole at q = 0, not an essential singularity as exp(1/q) has.
^ Here, a matrix ${\begin{pmatrix}a&b\\c&d\end{pmatrix}}$ sends ∞ to a/c.

Citations

^ ^a ^b Van 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

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 978-0387232294 Leads up to an overview of the proof of the modularity 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. Retrieved 2023-07-07.

[5] A meromorphic function can only have a finite number of negative-exponent terms in its Laurent series, its q-expansion. It can only have at most a pole at q = 0, not an essential singularity as exp(1/q) has.

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

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Modular form

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