Fix \(\mathbb{H}= \{x+iy : y>0\}\).
Modular forms are :
Holomorphic functions on \(\mathbb{H}\) with periodicity condition + asymptotics on the boundary.
Generating functions/series for arithmetic functions.
Holomorphic sections of holomorphic line bundles on \(1\)-dimensional complex manifolds.
Algebraic sections of algebraic line bundles on complex projective curves.
Solutions to a PDE satisfying certain boundary conditions.
Irreducible representations of \(\mathrm{GL}_2(\mathbb{R})\) acting on \(L^2(\Gamma\backslash \mathrm{GL}_2(\mathbb{R}))\).
Irreducible representations of \(\mathrm{GL}_2(\mathbb{A})\) acting on \(L^2(\mathrm{GL}_2(\mathbb{Q})\backslash\mathrm{GL}_2(\mathbb{A})\).
Representations \(\mathrm{Gal}(\overline{\mathbb{Q}} :\mathbb{Q})\rightarrow \mathrm{GL}_2(\overline{\mathbb{Q}_\ell})\).
Euler products satisfying a particular functional equation.
Cohomology classes.
Definition. Let \(g\in \mathrm{GL}_2(\mathbb{R})\) such that \(\det(g)>0\). Write \(g=\begin{pmatrix}a&b\\c&d\end{pmatrix}\). For \(z\in \mathbb{H}\) set \[ g\cdot z = \frac{az+b}{cz+d},\] (if \(\det(g)<0\) then \(g\cdot z = \frac{a\overline{z}+b}{c\overline{z}+d}\)).
For \(f : \mathbb{H}\rightarrow \mathbb{C}\) set \((f|_kg)(z) = (\det(g))^{k/2}(cz+d)^{-k}f(g\cdot z)\).
Fix \(\Gamma\subset \mathrm{SL}_2(\mathbb{R})\), lattice, and let \(\chi : \Gamma \rightarrow S^1\) be a character,
A modular forms with respect to \(k\) for \(\Gamma, \chi\) is a function \(f: \mathbb{H}\rightarrow \mathbb{C}\) such that
\(f\) is holomorphic
\(f|_k\gamma = \chi(\gamma)\) for all \(\gamma\in \Gamma\).
at every cusp \(\xi\) of \(\Gamma\), \(|f(z)|\leq Ce^{My_\xi (z)}\)
we say \(f\) is holomorphic at cusps if we can take \(M=0\). We say it is a cusp form if we can take \(M<0\).
Examples. \(\mathcal{O}(q) = \sum_{n\in\mathbb{Z}}q^{n^2}\).
\(\mathcal{O}^k(q) = \sum_{m}n_k(m)q^m\) where \(n_k(m)\) = number of representation of \(m\) as sum of \(k\) squares.
Set \(q=e^{2\pi z}\), if \(\mathrm{Re}(z)<0\) then \(|q|<1\), \(q(z+1)=q(z)\).
\(\mathcal{O}(z) = \sqrt{\frac{1}{2\pi z}}\mathcal{O}(-\frac{1}{4z})\).
The action of \(\Gamma\) on \(\mathbb{H}\times \mathbb{C}\) is \(\gamma\cdot (z,w)= (\gamma\cdot z, (cz+d)^kw)\).
The map \(\Gamma\backslash \mathbb{H}\times\mathbb{C} \rightarrow \Gamma\backslash \mathbb{H}\) defines the line bundle that our holomorphic form is a section of.
The PDE it is solution of comes from the Cauchy-Riemann condition.
Eisenstein series : \(\Lambda \subset \mathbb{C}\) the corresponding Eisenstein series is \(G_k(\Lambda) = \sum_{\omega\in \Lambda\backslash\{0\}}\frac{1}{\omega^k}\). \(G_{k}(r\Lambda) = r^{-k}G_k(\Lambda)\).
Representation-theoritic view
Let \(\Gamma\) be a lattice in \(\mathrm{SL}_2(\mathbb{R})\), \(f\in H_k(\Gamma,\chi)\). For \(g\in \mathrm{SL}_2(\mathbb{R})\) set \(F(g) = (f|_kg)(i)\), \(F : \mathrm{SL}_2(\mathbb{R}\rightarrow \mathbb{C})\).
\(F(\gamma g) = (f|_k\gamma g)(i)= ((f|_k\gamma)||_g)(i) = \chi(\gamma) (f|_kg)(i)=\chi(\gamma)F(g)\).
\(F(gK_{\theta}) = ((f|_kg)|_kK_\theta)(i)= (f|_kg)(i)(-\sin(\theta)i+\cos(\theta))^{-k}=e^{ik\theta}F(g)\),
where \(K_\theta=\begin{pmatrix} \cos(\theta)&\sin(\theta)\\ -\sin(\theta)& \cos(\theta)\end{pmatrix}\).
Now we explore the equivalent condition to ``\(f(z)\) harmonic'' (same as holomorphic, since we just saw we can rotate, and real harmonic = real part of holomorphic function). Define
\[\mathfrak{sl}_2(\mathbb{R}) \cong \left\lbrace X\in M_2(\mathbb{R}) : \mathrm{tr}(X)=0 \right\rbrace= \mathrm{Span}_{\mathbb{R}}(H,X_-,X_+),\] where \(H = \begin{pmatrix}1&0\\0&-1\end{pmatrix}\), \(X_+= \begin{pmatrix}0&1\\0&0\end{pmatrix}\), and \(X_-= \begin{pmatrix}0&0\\1&0\end{pmatrix}\).
If \(X\in \mathfrak{sl}_2(\mathbb{R})\) and \(F: \mathrm{SL}_2(\mathbb{R})\rightarrow \mathbb{C}\), set \((\mathcal{R}(H)F)(g) = \left.\frac{\mathrm{d}}{\mathrm{d}t}\right|_{t=0} F(ge^{tX})\), then \(f\) holomorphic corresponds to \(R(H^2-2H+4X_-X_=)F = \lambda F\), where \(\lambda = \lambda(k)\).
\(W = X_+-X_-\) generator of \(\mathrm{Lie}(\mathrm{SO}(2))\). \(\mathcal{R}(W) f = ikf\).
For \(g\in \mathrm{SL}_2(\mathbb{R})\) \(F : \Gamma\backslash \mathrm{SL}_2(\mathbb{R})\rightarrow \mathbb{C}\), \(x\in \Gamma\backslash\mathrm{SL}_2(\mathbb{R})\) set the right-translation action \[ (R(g) F)(x) = F(xg).\]
\(R(g)\)= functions on \(g\in\Gamma\backslash \mathbb{C}\) is linear \(R(g_1g_2) = R(g_1)R(g_2)\)
Check : \(\omega\) commutes with group \(g\in \mathrm{SL}_2(\mathbb{R})\), i.e. \(\mathrm{Ad}_g\omega=\omega\) for all \(g\in\mathrm{SL}_2(\mathbb{R})\).
Here \(\mathrm{Ad}\) is defined for \(g\in\mathrm{SL}_2(\mathbb{R}),\ X\in\mathfrak{sl}_2(\mathbb{R})\) as \(\mathrm{Ad}_g X = gXg^{-1}\)
We have a function \(F:\Gamma\backslash \mathrm{SL}_2(\mathbb{R})\rightarrow \mathbb{C}\) where
\(F(xK_\theta) = e^{ik\theta}F(x)\)
\(\mathcal{R}(\omega)\overset{\text{def}}{=}\mathcal{R}(H)\mathcal{R}(H)F-2\mathcal{R}(H)F+4\mathcal{R}(X_+)\mathcal{R}(X_-)F\),
and \(R(\omega)R(g)F = R(g)R(\omega)F\)
Corollary. \(\mathrm{span}_\mathbb{C}\left\lbrace R(g)F\right\rbrace_{g\in \mathrm{SL}_2(\mathbb{R})}\) consists of eigenfunctions of \(\omega\) with eigenvector \(\lambda\). FACT : the representation it defines is irreducible.
Note : Have here irreducible subrepresentations of regular representations of \(\mathrm{SL}_2(\mathbb{R})\) on \(\left\lbrace \text{functions on }\Gamma\backslash \mathrm{SL}_2(\mathbb{R}) \right\rbrace\).
Definition. Say \(F : \Gamma\backslash \mathrm{SL}_2(\mathbb{R})\rightarrow \mathbb{C}\) has ``weight \(k\)'' if \(F(X.K_\theta)=e^{ik\theta}F(x)\).
Fact : If \(F\) has weight \(k\), this subrepresentation contains vectors of weights \(k, k+2,k+4,\cdots\), each with multiplicity \(1\).
Definition. A modular form of weight \(k\) is the lowest weight vector in an irreducible representation as above, \(\lambda=\lambda(k)\).
Aside. Let \(G\) be a group, \(K\) a (compact) subgroup.
\(\sigma : K \rightarrow \mathrm{GL}(V)\) a finite-dimensional \(\mathbb{C}\)-representation.
Consider \(\epsilon = \Gamma\backslash G/K\times_K V = \{(\Gamma g,v) : g\in G\ v\in V\}/K \rightarrow B = \Gamma\backslash G/K\) this define a line bundle. Our form arise as section of this type of line bundle.
\(f\) is a cusp form \(\Rightarrow\) \(f\) decays exponentially at cusps.
\(y(\gamma_z) = \frac{y(z)}{|cz+d|^2}\ \Rightarrow\ y^{k/2}(z) |f(z)|\) is \(\Gamma\)-invariant.
\(\|f\|_{\text{peterson}}^2 = \int_{\Gamma\backslash \mathbb{H}}y^k|f(z)|^2 \frac{\mathrm{d}x\mathrm{d}y}{y^2}<\infty\).
\(\|F\|_{L^2(\Gamma\backslash\mathrm{SL}_2(\mathbb{R}))} = \|f\|_{\text{peterson}}\).
Suppose \(\Gamma(N)\subset\Gamma\subset \mathrm{SL}_2(\mathbb{Z})\), \(\Gamma(N) = \mathrm{Ker}(\mathrm{SL}_2(\mathbb{Z})\rightarrow \mathrm{SL}_2(\mathbb{Z}/N\mathbb{Z}))\).
\[T_pf(\Lambda) = \sum_{p\Lambda<\Lambda'<\Lambda}f(\Lambda') \]