1. Modular Forms, Elliptic Curves, and Modular Curves
1.1 First definition and examples
Def: The modular group is $SL_2(\mathbb{Z}) = \left\{ \begin{pmatrix}a & b \\ c & d \\\end{pmatrix}: a, b, c, d \in \mathbb{Z}, ad - bc = 1 \right\}$
- generators: $\begin{pmatrix}1 & 1 \\ 0 & 1 \\\end{pmatrix}, \begin{pmatrix}0 & -1 \\ 1 & 0 \\\end{pmatrix}$
- group action on $\mathbb{H}$: $\forall \gamma = \begin{pmatrix}a & b \\ c & d \\\end{pmatrix} \in SL_2(\mathbb{Z}), \begin{pmatrix}a & b \\ c & d \\\end{pmatrix}(\tau) = \frac{a\tau + b}{c\tau + d}$ for $\tau \in \mathbb{C}$ or $\frac{a}{c}$ for $\tau = \infty$.
- Prop: $Im(\gamma(\tau)) = \frac{Im(\tau)}{\left| c\tau + d\right|^2}, \tau \in \mathbb{H}, \gamma(\tau) \in \mathbb{H}$
- Prop: (association) $(\gamma\gamma^{‘})(\tau) = \gamma(\gamma^{‘}(\tau))$.
Def: Let $G$ be a group and $S$ be a topological $G$-set. Then a closed subset $F$ of $S$ is called a fundamental domain of $G$ in $S$ if $S$ is the union of conjugates of $F$, $i.e., S = \sqcup_{g \in G}gF$, and the intersection of any two conjugates has no interior.
- e.g. fundamental domain for the action of the modular group $\Lambda$ on the upper half-plane $\mathbb{H}$:
Def: Assume $f: \mathbb{H} \to \mathbb{C}$ is meromorphic function, if $f(\gamma(\tau)) = (c\tau + d)^kf(\tau), \forall \gamma \in SL_2(\mathbb{Z}), \tau \in \mathbb{H}$, we call $f$ is weakly modular of weight $k$.
- e.g. weak modularity of weight 0 is $SL_2(\mathbb{Z})$-invariance.
- e.g. $k = 2, f(\gamma(\tau))d(\gamma(\tau)) = f(\tau)d\tau$.
- e.g. $k$ odd number, take $\gamma = -I$, we have $f(\tau) = (-1)^kf(\tau) \Rightarrow f \equiv 0$.
- Prop: $f(\gamma(\tau))$ and $f(\tau)$ has same zeros and poles.
Def: Let $k \in \mathbb{Z}, f : \mathbb{H} \to \mathbb{C}$, if $f$ satisfies:
- (1) $f$ is holomorphic in $\mathbb{H}$;
- (2) $f$ is weakly modular of weight $k$;
- (3) $f$ is holomorphic at $\infty$.
Then we call $f$ modular form of weight $k$.- discussion for (3):
Take $\gamma = \begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix}$, we have $f(\tau + 1) = f(\tau)$.
Function that satisfies $\mathbb{H} \to \mathbb{D}^{‘} = \mathbb{D} \backslash \{0\}, \tau \mapsto e^{2\pi i\tau} = q$ is holomorphic, $\mathbb{Z}$-periodic.
We define $g : \mathbb{D}^{‘} \to \mathbb{C}, q \mapsto f(\frac{\log(q)}{2\pi i})$, we have $f(\tau) = g(e^{2\pi i}) = g(q)$.
By Laurent expansion, $g(q) = \Sigma_{n = -\infty}^{+\infty}a_nq^n, q = e^{2\pi i \tau}$. Since $\left| q\right| = e^{-2\pi Im(\tau)}$, we immediately get that $q \to 0 \iff Im(\tau) \to +\infty, \tau \in \mathbb{H}$.
So $f$ holomorphic at $\infty$ $\Leftrightarrow$ $g$ could be holomorphically extended in $\mathbb{D} \iff f(\tau) = g(q) = \Sigma_{n = 0}^{+\infty}a_nq^n, q = e^{2\pi i\tau}$.
- discussion for (3):
- (4) If $a_0 = 0$, we call $f$ a cusp form.
The set of modular forms of weight $k$ is denoted as $\mathcal{M}_k(SL_2(\mathbb{Z}))$, the set of cusp forms of weight $k$ is denoted as $\mathcal{S}_k(SL_2(\mathbb{Z}))$.
Prop: For $\mathcal{M}_k(SL_2(\mathbb{Z}))$ and $\mathcal{S}_k(SL_2(\mathbb{Z}))$:
- $\mathcal{M}_k(SL_2(\mathbb{Z}))$ is vector space over $\mathbb{C}$, since $f$ is holomorphic at $\infty$, $dim(\mathcal{M}_k(SL_2(\mathbb{Z}))) < \infty$.
- $\oplus_{k \in \mathbb{Z}}\mathcal{M}k(SL_2(\mathbb{Z})) : = \mathcal{M}(SL_2(\mathbb{Z}))$ is a graded ring,
$\oplus{k \in \mathbb{Z}}\mathcal{S}_k(SL_2(\mathbb{Z})) : = \mathcal{S}(SL_2(\mathbb{Z}))$ is an ideal of $\mathcal{M}$.
e.g. zero function is a modular form of every weight; constant function is a modular form of weight 0.
e.g. Eisenstein series: $G_k(\tau) = \Sigma_{(c, d)}^{‘}\frac{1}{(c\tau + d)^k}, k > 2$ even number, $\Sigma_{(c, d)}^{‘}$ means summing up $(c, d) \in \mathbb{Z}^2 \backslash \{(0, 0)\}$.
- $G_k(\tau)$ absolutely converges for all $\tau \in \mathbb{H}$, converges uniformly in any compact subset of $\mathbb{H}$ $\Rightarrow G_k$ is holomorphic in $\mathbb{H}$ and can be rearranged.
$\gamma = \begin{pmatrix}a & b \\ c & d \\\end{pmatrix} \in SL_2(\mathbb{Z})$, $G_k(\gamma(\tau)) = \Sigma^{‘}{(m, n)}\frac{1}{(m(\frac{a\tau + b}{c\tau + d}) + n)^k} = (c\tau + d)^k\Sigma{(m, n)}^{‘}\frac{1}{((ma + nc)\tau + (mb + nd))^k} = (c\tau + d)^kG_k(\tau) \Rightarrow G_k(\tau) \in \mathcal{M}_k(SL_2(\mathbb{Z}))$ where $k > 2$ is even.
- $G_k(\tau)$ absolutely converges for all $\tau \in \mathbb{H}$, converges uniformly in any compact subset of $\mathbb{H}$ $\Rightarrow G_k$ is holomorphic in $\mathbb{H}$ and can be rearranged.