Fix H={x+iy:y>0}.
Modular forms are :
Holomorphic functions on 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 GL2(R) acting on L2(Γ∖GL2(R)).
Irreducible representations of GL2(A) acting on L2(GL2(Q)∖GL2(A).
Representations Gal(¯Q:Q)→GL2(¯Qℓ).
Euler products satisfying a particular functional equation.
Cohomology classes.
Definition. Let g∈GL2(R) such that det. 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')