Section 5.2 Modular forms (Asra)
¶Last time we saw the j-function, which was \SL_2(\ZZ)-invariant, this is quite a strong condition, and in fact j is pretty much all we get under this condition. So instead we weaken this somewhat to some other variance property.
If w = f(z) \diff z on \HH and f(z) is meromorphic. \gamma \in \Gamma then
\begin{equation*}
f(\gamma z) d(\gamma z) = f(\gamma z) d\left( \frac{az+b}{cz+d} \right)
\end{equation*}
\begin{equation*}
= f(\gamma z) \left( \frac{\cdots \diff }{(cz+d)^2} \right)
\end{equation*}
so we get a condition
\begin{equation*}
f(\gamma z) = (cz+d)^2 f(z)
\end{equation*}
this is how we come to:
Definition 5.2.1.
A holomorphic function f\colon \HH \to \CC is a weakly modular function for \Gamma of weight k if
\begin{equation*}
f(\gamma z) = (cz+d)^k f(z) \forall \gamma = \begin{pmatrix} a\amp b \\ c \amp d \end{pmatrix} \in \Gamma\text{.}
\end{equation*}
Example 5.2.3.
Cusps for \Gamma_0(p)\text{,} we know we have \infty\text{,} what is the orbit of this?
\begin{equation*}
\gamma\in \Gamma_0(p),\, \gamma = \begin{pmatrix} a\amp b \\ cp \amp d \end{pmatrix}
\end{equation*}
\begin{equation*}
\gamma \infty = \frac{a}{cp}
\end{equation*}
so anything with a p in the denominator is equivalent to \infty\text{,} what about the rest?
\begin{equation*}
\gamma 0 = \frac{b}{d}, \, \gcd(b,d) =1\text{,}
\end{equation*}
so we have two cusps.
Definition 5.2.4. Modular forms.
A modular form is a weakly modular function that is holomorphic at all the cusps.
Example 5.2.5.
Eisenstein series
\begin{equation*}
G_k(z) = \sum'_{m,n\in \ZZ} \frac{1}{(mz+n)^k}
\end{equation*}
is a modular form of weight k \gt 2 for \SL_2(\ZZ)\text{.}
\begin{equation*}
\lim_{\im z\to \infty} G_k(z) = \lim_{\im z\to \infty} \sum'_{m,n\in \ZZ} \frac{1}{(mz+n)^k} = \sum'_{n\in \ZZ} \frac{1}{n^k} = 2\zeta(k)\text{.}
\end{equation*}
So here the function does not vanish at 0.
Definition 5.2.6. Cusp forms.
A cusp form is a modular form that vanishes at all cusps.
Given a cusp it will be stabilised by some
\begin{equation*}
\begin{pmatrix} 1\amp h \\ 0 \amp 1 \end{pmatrix}
\end{equation*}
call the smallest such h for a given cusp the width of the cusp.
Example 5.2.7.
Let's find the width of a cusp in \Gamma_0(qp) we have
\begin{equation*}
\begin{pmatrix} 1\amp 1 \\ 0 \amp 1 \end{pmatrix}
\end{equation*}
so the width of \infty is 1.
What about \alpha = 1/p\text{?}
- Find an element \gamma \in \SL_2(\ZZ) s.t. \gamma(\infty) = \alpha\text{.}
- Compute
\begin{equation*}
\delta(x) = \gamma \begin{pmatrix} 1\amp x \\ 0 \amp 1 \end{pmatrix} \gamma\inv
\end{equation*}
- Find the smallest x such that \delta(x) = \Gamma_0(pq)
\begin{equation*}
\gamma = \begin{pmatrix} 1\amp 0 \\ p \amp 1 \end{pmatrix},\, \gamma(\infty) = \frac 1p
\end{equation*}
\begin{equation*}
\begin{pmatrix} 1\amp 0 \\ p \amp 1 \end{pmatrix} \begin{pmatrix} 1\amp x \\ 0 \amp 1 \end{pmatrix} \begin{pmatrix} 1\amp 0 \\ -p \amp 1 \end{pmatrix} = \begin{pmatrix} 1 - px \amp x \\ -p^2 \amp px+ 1 \end{pmatrix}
\end{equation*}
Example 5.2.8. A cusp form.
Let \Delta(\tau) = g_2(\tau)^3 - 27g_3(\tau)^2,\,g_2(\tau) = 60G_4(\tau),\,g_3(\tau) =140G_6(\tau) \Delta(\tau) has weight 12 for \SL_2(\ZZ)\text{.} This vanishes at \infty because
\begin{equation*}
\zeta(4) = \frac{\pi^4}{90}
\end{equation*}
\begin{equation*}
\zeta(6) = \frac{\pi^6}{945}
\end{equation*}
also
\begin{equation*}
j(z) = \frac{g_2(\tau)^3}{\Delta(\tau)}
\end{equation*}
so \Delta(\tau) vanishes at \infty because g_2(\tau) doesn't and j(z) has a simple pole at \infty\text{.}
M_k(\Gamma) as the space of modular forms of weight k for \Gamma is a \CC -v.s. S_k(\Gamma) as the space of cusp forms of weight k for \Gamma is a \CC -v.s.
Theorem 5.2.9.
M_k(\Gamma) and S_k(\Gamma) are finite dimensional
\begin{equation*}
\dim(M_k(\Gamma)) =
\begin{cases}
0 \amp \text{ if } k \le -1\\
1 \amp \text{ if } k = 0\\
(k-1)(g-1) + v_\infty \frac k2 + \sum_p [\frac k2 (1- \frac{1}{e_p})] \amp \text{ if } k \ge 2
\end{cases}
\end{equation*}
where g is the genus of X(\Gamma) v_\infty is the number of inequivalent cusps P are the elliptic points \lb \cdot \rb is the integer part
\begin{equation*}
\dim(S_k(\Gamma)) =
\begin{cases}
0 \amp \text{ if } k \le 0 \\
(k-1)(g-1) + v_\infty (\frac k2 - 1) + \sum_p [\frac k2 (1- \frac{1}{e_p})] \amp \text{ if } k \ge 2
\end{cases}
\end{equation*}
\begin{equation*}
\dim(S_2(\Gamma)) = g(X(\Gamma))
\end{equation*}
Proposition 5.2.10.
If f \in S_2(\Gamma) then f(z) \diff z is a holomorphic differential.
Given an elliptic curve
\begin{equation*}
E/ \CC = \CC/\Lambda
\end{equation*}
\begin{equation*}
E \to E'
\end{equation*}
\begin{equation*}
\CC/\Lambda \to \CC/\Lambda'\text{,}
\end{equation*}
studying degree n isogenies, is like studying index n sublattices
Definition 5.2.11. Hecke operators.
n \ge 1 then T(n) is the nth Hecke operator acting on
\begin{equation*}
\divisors (\mathcal L)
\end{equation*}
by
\begin{equation*}
T(n) \Lambda = \sum_{\Lambda' \subseteq \Lambda,\,[\Lambda : \Lambda'] = n} (\Lambda')
\end{equation*}
Definition 5.2.12.
Let \lambda \in \CC^\times the homothety operator R_\lambda is R_\lambda \Lambda = \lambda\Lambda\text{.}
Theorem 5.2.13.
\begin{equation*}
R_\lambda R_\mu = R_{\lambda\mu}
\end{equation*}
\begin{equation*}
R_\lambda T(n) = T(n) R_{\lambda}
\end{equation*}
\begin{equation*}
T(nm) = T(n) T(m),\,\gcd(n,m) =1
\end{equation*}
\begin{equation*}
T(p^e)T(p) = T(p^{e+1 }) + p T(p^{e-1}) R_p
\end{equation*}
Proof.
Of 4.
\(\Lambda \in \mathcal L\) for \(\Lambda' \subseteq \Lambda \) index \(p^{e+1}\) have
\begin{equation*}
a(\Lambda') = \# \{ \Gamma : \Lambda' \subseteq \Lambda \subseteq_p \Lambda \}
\end{equation*}
\begin{equation*}
b(\Lambda') = 1 \text{ if } \Lambda' \subseteq p\Lambda
\end{equation*}
now
\begin{equation*}
T(p^e)T(p) \Lambda = T(p^e) \sum_{\Gamma \subseteq_p \Lambda } (\Gamma) = \sum_{\Gamma \subseteq_p \Lambda } \sum_{\Lambda' \subseteq_{p^e} \Gamma} (\Lambda') =\sum_{\Lambda' \subseteq_{p^e} \Gamma} a(\Lambda') (\Lambda')
\end{equation*}
\begin{equation*}
T(p^{e+1} ) \Lambda = \sum_{\Lambda'\subseteq_{p^{e+1}} \Lambda} (\Lambda')
\end{equation*}
\begin{equation*}
T(p^{e-1} )R_p \Lambda = T(p^{e-1}) (p \Lambda) = \sum_{\Lambda'' \subseteq_{p^{e-1}} p\Lambda } (\Lambda'') = \sum_{\Lambda' \subseteq_{p^{e+1}} \Lambda} b(\Lambda') (\Lambda')
\end{equation*}
Split into cases, do some maths..
\begin{equation*}
\{ \alpha \in M_2(\ZZ) : \det(\alpha) = n \} \leftrightarrow \{ \Lambda ' : \Lambda ' \subseteq_n \Lambda\}
\end{equation*}
representatives in Hermite normal form
\begin{equation*}
S_n = \{\begin{pmatrix} a\amp b \\ 0 \amp d\end{pmatrix} : ad= n,\, a,d\gt 0\, 0\le b \lt d\}
\end{equation*}
Corollary 5.2.14.
Let \Lambda \in \mathcal L\text{,} \Lambda = \ZZ w_1 + \ZZ w_2 then T(n) acts as follows
\begin{equation*}
T(n) \Lambda = \sum_{ ad= n,\, a,d\gt 0\, 0\le b \lt d} \ZZ(aw_1 + bw_2) + \ZZ dw_2 = \sum_{\alpha \in S_n} \alpha \Lambda
\end{equation*}
Corollary 5.2.15.
For p prime T(p)\text{:}
\begin{equation*}
T(p) \Lambda = \ZZ pw_1 + \ZZ w_2 +\sum_{0 \le b \lt p} \ZZ(w_1 + bw_2) + \ZZ pw_2\text{.}
\end{equation*}
The Hecke operators act on modular forms f(\tau) by reinterpreting weakly modular functions of weight k as functions on lattices that have a weight k action under homothety.
This boils down to
\begin{equation*}
(T_k(n) f)(\tau) = n^{k-1} \sum_{ ad= n,\, a,d\gt 0\, 0\le b \lt d} d^{-k} f\left( \frac{ a\tau +b}{d}\right)
\end{equation*}
Corollary 5.2.16.
For p prime
\begin{equation*}
(T_k(p) f)(\tau) = p^{k-1}f(pz) +\frac 1p \sum_{0 \le b \lt p} f\left( \frac {z + b}{p} \right)\text{.}
\end{equation*}
We have an action on fourier expansions
\begin{equation*}
f(\psi) = \sum_{m\in \ZZ} a_m q^m
\end{equation*}
\begin{equation*}
T_k(p)f(\tau) = p^{k-1} \sum_{m\in \ZZ} a_m q^{pm} + \frac{1}{p} \sum_{b=0}^{p-1} \left( \sum_{m\in \ZZ} a_m e^{2\pi i m (z+b)/p}\right)
\end{equation*}
\begin{equation*}
= p^{k-1} \sum_{m\in \ZZ} a_m q^{pm} + \frac{1}{p} \sum_{m\in \ZZ} a_me^{2\pi i m z/p} \sum_{b=0}^{p-1} \underbrace{e^{2\pi i m b/p}}_{p\text{ if }p|m,0\text{ otw}}
\end{equation*}
\begin{equation*}
= p^{k-1} \sum_{m\in \ZZ} a_m q^{pm} +\sum_{m\in \ZZ} a_{pm}q^m
\end{equation*}
Corollary 5.2.17.
\begin{equation*}
a_1(T_p(f)) = a_p(f)
\end{equation*}
If f\in S_k(\Gamma_0(1)) is an eigenfunction for these operators we can normalise so that a_1(f) = 1\text{.}
\begin{equation*}
T(m)T(n) = T(mn)
\end{equation*}
\begin{equation*}
a_ma_n = a_{mn}
\end{equation*}
\begin{equation*}
a_{p^r} = a_p a_{p^{r-1}} + p^{k-1} a_{p^{r+1}}
\end{equation*}
Definition 5.2.18. Petersson inner product.
The Petersson inner product of two cusp forms f,g\in S_k(\SL_2(\ZZ)) is defined to be
\begin{equation*}
\pair{f}{g} = \int_{\mathcal D} f \bar g y^{k-2} \diff x \diff y
\end{equation*}
where \mathcal D is a fundamental domain for \SL_2(\ZZ)\text{.}
Proposition 5.2.19.
Let f,g\in S_k(\SL_2(\ZZ))\text{,} n\in \NN then
\begin{equation*}
\pair{T(n) f}{g} = \pair{f}{T(n)g}\text{.}
\end{equation*}