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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*}
Remark 5.2.2.

If \(-I \in \Gamma\) and \(k\) odd

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

so in this setting we only have interesting behaviour for even \(k\text{.}\)

If \(\Gamma\) is a congruence subgroup of level \(N\) we have

\begin{equation*} \begin{pmatrix} 1\amp N \\ 0 \amp 1 \end{pmatrix} \in \Gamma \end{equation*}

gives you a \(q\)-expansion

\begin{equation*} q= e^{2\pi i z} \end{equation*}
\begin{equation*} f(z) = \sum_{m\in \ZZ} a_m q^{m/N}\text{.} \end{equation*}

\(f\) is holomorphic at \(\infty\) if \(a_m = 0\) for \(m \lt 0\text{.}\)

\(f\) is holomorphic at all cusps if \(f(\gamma z)(cz+d)^k\) is holomorphic at \(\infty\) for all \(\gamma \in \SL_2(\ZZ)\text{.}\)

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 \(\alpha = 1/p\text{?}\)

  1. Find an element \(\gamma \in \SL_2(\ZZ)\) s.t. \(\gamma(\infty) = \alpha\text{.}\)
  2. Compute
    \begin{equation*} \delta(x) = \gamma \begin{pmatrix} 1\amp x \\ 0 \amp 1 \end{pmatrix} \gamma\inv \end{equation*}
  3. 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.

\begin{equation*} \dim(S_2(\Gamma)) = g(X(\Gamma)) \end{equation*}

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 \(n\)th 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 \(R_\lambda \Lambda = \lambda\Lambda\text{.}\)

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..

Hecke operators on lattices Given \(\Lambda' \subseteq_n \Lambda\) there is an integer matrix of determinant \(n\) taking one basis to the other. Have a correspondence

\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*}

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*}

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*}

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 \(\SL_2(\ZZ)\text{.}\)