Modular function

A modular function of weight $k$ for a subgroup $\Gamma \le S{L}_2(\mathbb{Z})$ is a meromorphic function on the complex upper half-plane that satisfiesfor $\gamma =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in \Gamma$ and is meromorphic at every cusp.

For a congruence subgroup $\Gamma$, the compact modular curve isAt an ordinary point it inherits a coordinate from $\mathfrak{h}$; at an elliptic point of stabilizer order $e$ a coordinate is the $e$th power of a local disk coordinate; and at a cusp of width $h$ a coordinate is $e^{2\pi i\tau /h}$.

Let $\mu =[PS{L}_2(\mathbb{Z}):\overline{\Gamma }]$, let $e_2,e_3$ count elliptic orbits of orders two and three, and let $c$ be the number of cusps. Then

The modular curve $X_0(3)$ has index four, two cusps, no elliptic orbit of order two, and one elliptic orbit of order three. Its genus is therefore zero.

A cusp of a finite-index subgroup $\Gamma \le S{L}_2(\mathbb{Z})$ is an orbit in $\Gamma \backslash {\mathbb{P}}^1(\mathbb{Q})$. If $\sigma \in S{L}_2(\mathbb{Z})$ sends infinity to a cusp, behavior there is studied through $f{|}_k\sigma$.

The width of a cusp represented by $\sigma \in S{L}_2(\mathbb{Z})$ is the least positive $h$ such that $\sigma T^h{\sigma }^{-1}$ lies in the subgroup, up to its center. Its local parameter is $q_h=e^{2\pi i\tau /h}$.

A modular function is holomorphic at a cusp when its expansion in the corresponding local parameter has no negative powers. It vanishes at the cusp when that expansion has zero constant term.

A modular form is a holomorphic function on the complex upper half-plane that transforms with a fixed weight under a congruence subgroup and is holomorphic at every cusp.

The Fricke matrix $W_N=\left(\begin{array}{cc}0& -1\\ N& 0\end{array}\right)$ normalizes ${\Gamma }_1(N)$ and ${\Gamma }_0(N)$ under the appropriate determinant-normalized slash action. It exchanges the cusps zero and infinity.

The modular group is $S{L}_2(\mathbb{Z})$ acting on the complex upper half-plane. Its central element $-I$ acts trivially on points but contributes the factor $(-1{)}^k$ in weight $k$.

The standard fundamental domain isEvery orbit of the modular group meets $\mathcal{F}$.

Given $\tau \in \mathfrak{h}$, choose a primitive pair $(c,d)$ minimizing $|c\tau +d|$ and apply a modular matrix with bottom row $(c,d)$. This maximizes the imaginary part within the orbit. Translation puts the real part in $[-1/2,1/2]$, and inversion then shows that the imaginary part is at least $\sqrt{3}/2$.

For $\gamma =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$, the weight-$k$ slash operator isA modular form of weight $k$ and level $\Gamma$ satisfies $f{|}_k\gamma =f$ for every $\gamma \in \Gamma$.

For $\gamma =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$, the automorphy factor is $j(\gamma ,\tau )=c\tau +d$. It obeys the cocycle identity

Translation invariance at a cusp gives a Fourier expansion in its local parameter $q$. Holomorphy at the cusp excludes negative powers; vanishing of the constant term defines a cusp form.

A cusp form is a modular form that vanishes at every cusp. At the cusp at infinity for the full modular group it has an expansion ${\sum }_{n\ge 1}{a}_n{q}^n$.

For a weight-$k$ modular form on the full modular group, the quantityis modular invariant. For a cusp form it tends to zero at the cusp and is therefore bounded on a fundamental domain.

If $f(\tau )={\sum }_{n\ge 1}{a}_n{q}^n$ is a weight-$k$ level-one cusp form, boundedness of its invariant norm givesIntegrate $f(x+iy)e^{-2\pi inx}$ over one period and choose $y=1/n$.

Let even $k\ge 4$ and let $g(\tau )={\sum }_{n\ge 0}{b}_n{q}^n$ be a level-one modular form of weight $k$. If $b_n=O(n^{k/2})$, then $b_0=0$ and $g$ is a cusp form. Otherwise subtract the matching Eisenstein series; its coefficients grow like ${\sigma }_{k-1}(n)$, contradicting the bound at prime indices.

For even $k\ge 12$,The dimension is zero for odd $k$ and for $k<12$.

For $f(\tau )={\sum }_{n\ge 1}{a}_n{q}^n$, termwise integration initially givesA Fricke transformation converts the behavior near zero into cusp decay at infinity, giving analytic continuation and a functional equation for the completed L-function.

The modular discriminant is the normalized weight-twelve level-one cusp form

For every real $s$, the rapidly convergent integralis the analytic continuation of $(2\pi {)}^{-s}\Gamma (s)L(\Delta ,s)$. The product for $\Delta$ makes the integrand positive, so $L(\Delta ,s)>0$ for real $s>0$.

For weight-$k$ cusp forms on $\Gamma$, the Petersson inner product isCusp decay makes the integral convergent.

The Rankin–Selberg method represents Dirichlet series built from automorphic forms as integrals against Eisenstein series and studies them by unfolding those integrals.

For cusp forms $f=\sum a_n{q}^n$ and $g=\sum b_n{q}^n$, their Rankin–Selberg convolution in the elementary normalization is $L(f,g,s)={\sum }_{n\ge 1}{a}_n\overline{b_n}{n}^{-s}$.

If $f$ and $g$ have respective weights $k$ and $l$, and $r=k-l>2$, unfolding the weight-$r$ Eisenstein series gives

An Eisenstein series is a modular form constructed by summing a weight factor over a parabolic coset space. For even $k>2$, the level-one holomorphic Eisenstein series is a scalar multiple of ${\sum }_{(m,n)\ne (0,0)}(m\tau +n{)}^{-k}$.

For even $k\ge 4$, the normalized level-one Eisenstein series has expansion

The identity $E_8=E_4^2$ gives

If $N+1=\dim M_k(S{L}_2(\mathbb{Z}))$, there is a basis $f_0,\dots ,f_N$ with integral Fourier coefficients andStart from the integral forms ${\Delta }^i{E}_4^a{E}_6^b$ and use integer elimination on their unitriangular leading coefficients.

If the first nonconstant coefficient of $E_k$ has reduced denominator divisible by a prime $p$, the integral echelon basis produces an integral cusp form whose Fourier coefficients are congruent modulo $p$ to ${\sigma }_{k-1}(n)$.

For $k>2$ and $(x,y)\in (\mathbb{Z}/N\mathbb{Z}{)}^2$,is a modular form of weight $k$ for the principal congruence subgroup $\Gamma (N)$. Right multiplication of $(x,y)$ by $\gamma \in S{L}_2(\mathbb{Z})$ describes its slash transformation.

For the full modular group, a cusp form is orthogonal under the Petersson inner product to every holomorphic Eisenstein series of the same weight. Unfolding reduces the integral to the constant Fourier coefficient of the cusp form, which is zero.

The seriesis quasimodular: $E_2(-1/\tau )={\tau }^2{E}_2(\tau )+6\tau /(\pi i)$.

For $\mathrm{Re}s>1$,is an absolutely convergent modular-invariant function.

For the normalization $(T_{p}f)(\tau )=p^{-1}(f(p\tau )+{\sum }_{b\mathrm{mod}p}f((\tau +b)/p))$, one has

For even $k$ and $\mathrm{Re}s>(2-k)/2$, defineIt is generally nonholomorphic and transforms with weight $k$.

For fixed $\tau$, Mellin transformation of a weighted Gaussian theta series expresses $G_k(\tau ,s)$ as a gamma factor times a Mellin integral. Splitting at one and applying Poisson summation to the small-time part continues it to all $s\in \mathbb{C}$; for positive even $k$, the reciprocal gamma factor cancels the apparent poles.

The absolute value of the summand indexed by a primitive bottom row $(c,d)$ isThe exponent exceeds two, so comparison with the lattice sum over $(c,d)\in {\mathbb{Z}}^2\setminus \{0\}$ proves absolute and locally uniform convergence.

If $f$ is a weight-$k$ modular form, thenis modular invariant. The factors transform by $j(\gamma ,\tau {)}^k$, ${\overline{j(\gamma ,\tau )}}^k$, and $|j(\gamma ,\tau ){|}^{-2k}$.

For a prime $p$, one normalization of the weight-$k$ Hecke operator at level one isIf $f={\sum }_n{a}_n{q}^n$, thenwhere $a_{n/p}=0$ unless $p$ divides $n$.

Let a level-one modular function be holomorphic on the upper half-plane. If the span of $f,T_{p}f,T_p^{2}f,\dots$ is finite-dimensional, then $f$ is holomorphic at infinity. Indeed, a pole of order $N$ would make $T_p^{r}f$ have pole order $p^{r}N$, producing linearly independent functions.

A meromorphic modular form obeys the modular transformation law and is meromorphic on the upper half-plane and at the cusps.

For a nonzero meromorphic modular form $f$ of weight $k$,where the sum takes one representative from each modular-group orbit.

The Klein j-invariant is the weight-zero level-one modular functionIt is holomorphic on the upper half-plane and has a simple pole at infinity.

The function $j(\tau )/j(2\tau )$ has weight zero and level ${\Gamma }_1(2)={\Gamma }_0(2)$. It has a simple zero at the cusp at infinity and a simple pole at the cusp zero.

A theta function is a holomorphic function formed by summing an exponential quadratic form over a lattice.

The Jacobi theta functionsatisfies $\theta (\tau +2)=\theta (\tau )$ and $\theta (-1/\tau )=(-i\tau {)}^{1/2}\theta (\tau )$.

For $|q|<1$ and $z\ne 0$, the Jacobi triple product is

The theta group is ${\Gamma }_{\theta }=\Gamma (2)\cup \Gamma (2)S$. It is generated by $T^2$ and $S$ and has index three in the modular group.

The Jacobi triple product givesEvery factor is nonzero for $|q|<1$, and the product converges to a nonzero limit, so $\theta$ has no zero in the upper half-plane.

Let $\mathcal{L}$ be the set of complex lattices. A weight-$k$ lattice function satisfies $F(\lambda \Lambda )={\lambda }^{-k}F(\Lambda )$. Evaluating at ${\Lambda }_{\tau }=\mathbb{Z}\tau +\mathbb{Z}$ identifies such functions with weight-$k$ modular-invariant functions on the upper half-plane.

A Gamma 1 level structure on a complex lattice $\Lambda$ is a point of exact order $N$ in $\mathbb{C}/\Lambda$. Similarity classes of such pairs are parametrized by ${\Gamma }_1(N)\backslash \mathfrak{h}$.

A lattice has $p+1$ index-$p$ overlattices. All preserve the exact order of a Gamma 1 level structure of order $N$ when $p\nmid N$; exactly $p$ do so when $p\mid N$.

For an even nonnegative integer $k$,Poisson summation and the Fourier eigenfunction identity for $(x+iy{)}^k{e}^{-\pi (x^2+y^2)}$ give

One standard normalization isUnder the lattice model this becomes the usual weight-$k$ Hecke operator.