Modular form

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 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 $\mathcal{F}$.

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 $\gamma \in \Gamma$.

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

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 ${\sum }_{(m,n)\ne (0,0)}(m\tau +n{)}^{-k}$.

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

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.

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.

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