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Section 6.2 Modular Curves Background I (John)

Main references are lecture notes by Darmon and Weinstein “introduction to modular forms”.

Definition 6.2.1.

Let

\begin{equation*} \Gamma (N) = \left\{\begin{pmatrix} a \amp b \\ c \amp d \end{pmatrix}\in \SL_2(\ZZ) : \begin{pmatrix} a \amp b \\ c \amp d \end{pmatrix} \equiv \begin{pmatrix} 1 \amp 0 \\ 0 \amp 1 \end{pmatrix} \pmod N\right\}\text{.} \end{equation*}

\(\Gamma\subseteq \SL_2(\ZZ)\) is a congruence subgroup if it contains \(\Gamma (N)\) for some \(N\text{.}\) Some important examples are

\begin{equation*} \Gamma_1 (N) = \left\{\begin{pmatrix} a \amp b \\ c \amp d \end{pmatrix}\in \SL_2(\ZZ) : \begin{pmatrix} a \amp b \\ c \amp d \end{pmatrix} \equiv \begin{pmatrix} 1 \amp \ast \\ 0 \amp 1 \end{pmatrix} \pmod N\right\} \end{equation*}
\begin{equation*} \Gamma_0 (N) = \left\{\begin{pmatrix} a \amp b \\ c \amp d \end{pmatrix}\in \SL_2(\ZZ) : \begin{pmatrix} a \amp b \\ c \amp d \end{pmatrix} \equiv \begin{pmatrix} \ast \amp \ast \\ 0 \amp \ast \end{pmatrix} \pmod N\right\} \end{equation*}
Definition 6.2.2.
\begin{equation*} f\colon \HH \to \CC \end{equation*}

is a modular form of weight \(2k\) for \(\Gamma \) (with character \(\epsilon \)) if

  1. \(f\) is holomorphic on \(\HH\text{.}\)
  2. \(f\) is holomorphic at infinity.
  3. \begin{equation*} f|_{2k} \gamma (z) = f(z)\,\forall\gamma \in \Gamma \end{equation*}
    where
    \begin{equation*} f|_{2k} \gamma (z) = (cz+d)^{-2k}f(\gamma z)\epsilon (d) \end{equation*}
Example 6.2.3.

For \(\Gamma = \SL_2(\ZZ)\) from now on.

\begin{equation*} f(z+1) = f(z) \end{equation*}
\begin{equation*} f\left(-\frac 1z\right) = z^{2k}f(z) \end{equation*}

Using this we can write

\begin{equation*} f(z) = f(q),\,q= e^{2\pi i z}\text{,} \end{equation*}

where \(q\) is a parameter at infinity.

\begin{equation*} f(q) = \sum_{n=0}^\infty a_n q^n\text{.} \end{equation*}
Definition 6.2.4.

A modular form is a cusp form if \(a_0 = 0\text{.}\)

Definition 6.2.5.

\(M_k(\Gamma )\) is the space of weight \(k\) modular forms. \(S_k(\Gamma )\) is the space of weight \(k\) cusp forms.

Example 6.2.6.
\begin{equation*} G_{2k}(z) = \sum_{m,n\in \ZZ}' \frac{1}{(mz+n)^{2k}} \end{equation*}
\begin{equation*} g_{2} = G_4(z)/2\zeta (4) \end{equation*}
\begin{equation*} g_{3} = G_6(z)/2\zeta (6) \end{equation*}

then

\begin{equation*} \Delta = \frac{g_2^3 - g_3^2}{1728} \end{equation*}

is a cusp form of weight 12.

\begin{equation*} (f\Delta) (-1/z) = (cz+d)^{-k}f(z) (cz+d)^{-12} \Delta (z) \end{equation*}
\begin{equation*} = (cz+d)^{-(k+12)}(f\Delta ) (z)\text{.} \end{equation*}

If \(k \lt 0\) , \(f \in M_k\) have \(f^{12}\Delta ^k \in S_0 = 0\text{.}\) \(M_0 =\CC\) corresponds to holomorphic functions on \(\SL_2(\ZZ) \backslash \HH\text{.}\)

\begin{equation*} M_k \to \CC \end{equation*}
\begin{equation*} f(q) = a_0 + a_1 q + \cdots \mapsto a_0 \end{equation*}
\begin{equation*} \dim(M_k / \ker) \le 1 \end{equation*}
\begin{equation*} M_k = S_k + G_k \CC \end{equation*}

we get

\(n\) \(\dim M_k\) \(\dim S_k\)
\(\lt 0\) \(0\) \(0\)
\(0\) \(1\) \(0\)
\(2\) \(0\) \(0\)
\(4\) \(1\) \(0\)
\(6\) \(1\) \(0\)
\(8\) \(1\) \(0\)
\(10\) \(1\) \(0\)
\(12\) \(2\) \(1\)
\(14\) \(1\) \(0\)
\(16\) \(2\) \(1\)
\(18\) \(2\) \(1\)
\(20\) \(2\) \(1\)
\(22\) \(2\) \(1\)
Table 6.2.8. dimensions
Hecke operators.
Definition 6.2.9.

\(\Lambda \) is a lattice if it is a rank 2 \(\ZZ\)-module in \(\CC\) s.t. \(\CC/\Lambda \) is compact.

\begin{equation*} \Lambda = \tau _1\ZZ + \tau _2 \ZZ \end{equation*}
\begin{equation*} \dim_\RR(\tau _1 \RR + \tau _2\RR) = 2 \end{equation*}
\begin{equation*} \Im (\tau _2 / \tau _1 ) \gt 0 \end{equation*}

\(F\) is a homogeneous lattice function of weight \(k\) if it is

\begin{equation*} F\colon R \to \CC \end{equation*}

where \(R\) is the set of lattices, such that

\begin{equation*} F(\lambda \Lambda ) = \lambda ^{-k}F(\Lambda ) \end{equation*}

\(F\) is holomorphic if \(f\colon \HH \to \CC\)

\begin{equation*} f(\tau ) = F(\ZZ + \tau \ZZ) \end{equation*}

is holomorphic on \(\HH\text{.}\)

\begin{equation*} \{\text{holo. homog. wt. }k\text{ lattice fns.}\}\leftrightarrow \{\text{wt. }k\text{ mod. fms.}\} \end{equation*}
\begin{equation*} F\mapsto f_F\colon \tau \mapsto F(\ZZ+ \tau \ZZ) \end{equation*}
\begin{equation*} F_f \mapsfrom f \end{equation*}
\begin{equation*} F_f (\tau _1 \ZZ+ \tau _2 \ZZ) = f(\tau _2/\tau _1)\text{.} \end{equation*}
Definition 6.2.10.

\(F\) is a homogeneous holomorphic weight \(k\) lattice function then

\begin{equation*} T_{n,k} F(\Delta ) = n^{k-1} \sum_{\Lambda ' \subseteq \Lambda ,\,[\Lambda : \Lambda '] = n} F(\Lambda ') \end{equation*}
\begin{equation*} T_{n,k}f = f_{T_{n,k}(F_f)} \end{equation*}
\begin{equation*} T_{n,k} f(z) = n^{k-1} \sum_{\gamma \in \SL_2(\ZZ)\backslash M_n } f(\gamma z)(cz+d)^{-k} \end{equation*}

where \(M_n \subseteq M_2(\ZZ)\) is the set of integer matrices of determinant \(n\text{.}\)

\begin{equation*} f|\alpha \beta = (f|\alpha )|\beta \text{.} \end{equation*}

The fourier expansions of these are given by

\begin{equation*} f(q) = a_0 + a_1 q + \cdots \end{equation*}
\begin{equation*} T_{n,k}f(q) = \sum_{m = 0}^\infty \sum_{d|(m,n)} d^{k-1} a_{nm/d^2} q^m \end{equation*}

fixing \(k\) we have if \((a,b) = 1\)

\begin{equation*} T_aT_b = T_{ab} \end{equation*}

for \(p \) prime

\begin{equation*} T_p T_{p^t} = T_{p^{k+1}} + p^{k-1} T_{p^{t-1}}\text{.} \end{equation*}
Figure 6.2.11. The fundamental domain

These \(T_{n,k}\) operate on \(M_k(\SL_2(\ZZ))\) and we can define:

Definition 6.2.12.

Let \(f\in M_k(\SL_2(\ZZ))\) is an eigenform if it is a simultaneous eigenvector for \(\{T_n\}_{n=1}^\infty \text{.}\) We have

\begin{equation*} \pair - - \colon S_k \times S_k\to \CC \end{equation*}
\begin{equation*} \pair fg = \int_F f(z) \overline{g(z)} y^k \frac{\diff x \diff y}{y^2}\text{.} \end{equation*}
Definition 6.2.14.

\(f\) is a normalized eigenform if

\begin{equation*} a_1(f) =1 \end{equation*}
\begin{equation*} a_n(f) = a_1(T_nf) = a_1(\lambda _n f) = \lambda _na_1(f) = \lambda _n \text{.} \end{equation*}

\(G_k/\zeta (k)\) and \(\Delta \) have rational coefficients.

\begin{equation*} M_k(\QQ) = f_1 \QQ+ \cdots + f_d \QQ \end{equation*}

\(T_n\) operates on \(M_k(\QQ)\) so

\begin{equation*} T_n \hookrightarrow \Mat_d(\QQ) \end{equation*}
\begin{equation*} \mathbf T_k =\QQ[T_1,T_2, \ldots] \hookrightarrow \Mat_d(\QQ) \end{equation*}

\(\forall \phi \in \Hom(\mathbf T_{k}, \overline \QQ)\) have \(\im (\phi )\) lies in a degree \(\le d\) extension. this is totally real because \(T_n\) are self adjoint w.r.t. a Hermitian inner product.

Generalisation.

If \(\Gamma \) is any congruence subgroup

\begin{equation*} M_k(\Gamma ) = \operatorname{Eis}_k(\Gamma ) + S_k(\Gamma ) \end{equation*}
\begin{equation*} M_k(\SL_2(\ZZ)) \subseteq M_k(\Gamma ) \end{equation*}

Given \(d|N\) we have

\begin{equation*} f \in M_k(\Gamma (d)) \end{equation*}

can define dilation

\begin{equation*} g(z) = f(N/d z) \in M_k(\Gamma ) \end{equation*}

for large enough \(k\) we can find a basis of \(M_k(\Gamma (N))\) by taking dilations of products of \(M_a(\Gamma (N))\) for \(a \lt k\) and Hecke operators.

For \((n,N) = 1\text{,}\) \(f\in M_k(\Gamma (N))\text{.}\)

\begin{equation*} T_{n,k} f = n^{k-1} \sum_{\gamma \in \Gamma (N) \backslash M_n} (f|\gamma )(z) \end{equation*}

\(M_n\) is integer upper triangular with determinant \(n\text{.}\) For primes \(l\)

\begin{equation*} l\nmid N,\, T_l f(q) = \sum a_n l q^n + l^{k-1} \sum a_n \langle l \rangle f q^{nl} \end{equation*}
\begin{equation*} l | N,\, T_lf(q) = \sum a_{nl} q^n \end{equation*}
\begin{equation*} \langle l \rangle f(z) = \left(f|_k \begin{pmatrix} l \amp 0 \\ 0 \amp l\inv \end{pmatrix}\right) (z)\text{.} \end{equation*}
Definition 6.2.16.
\begin{equation*} g \in S_k(\Gamma (N)) \end{equation*}

s.t.

\begin{equation*} g(z) = f(d z) \end{equation*}

for some \(f \in S_k(\Gamma (N/d))\) is called an oldform. Newforms are \(f\in S_k(\Gamma (N))\) s.t.

\begin{equation*} \pair fg = 0 \end{equation*}

for all \(g\in S_k(\Gamma (N))^{old}\text{.}\)