Recently, I have been learning about the basic notions of automorphic forms, and I want to document what I have learned.

I had a very vague understanding of ‘automorphic forms’, and I was (well, and still I am) a little bit intimidated by it. However, finally, I had to delve into the world of these objects.

The only idea I had was that automorphic forms are a vast generalization of modular forms, so that’s where I started.

Preliminaries

Let \(\HH = \{\tau \in \CC : \Im(\tau) > 0\}\) be the upper half plane with action of $\SL_2(\RR)$ by fractional linear transformation \(\begin{pmatrix} a & b \\ c & d \end{pmatrix}.\tau = \dfrac{a\tau+b}{c\tau+d}\).

Let $\Gamma \subset \SL_2(\ZZ)$ be a subgroup of finite index. (These groups are called congruence subgroups.)

We can extend the action of $\Gamma$ to $\HH \cup \QQ \cup {\infty}$ with \(\begin{pmatrix} a & b \\ c & d \end{pmatrix}.\infty = a/c\).

The cusps of $\Gamma$ are the orbits of $\Gamma$ on $\QQ\cup {\infty}$.

For example, let \(\frac{p}{q} \in \QQ\) (the minimal representation), then there exists $a, b$ with $ap-bq =1$, and \(\begin{pmatrix}p & b \\ q & a\end{pmatrix}.\infty = p/q\); that is, $\PSL_2(\ZZ)$ has one cusp at $\infty$ (as a equivalence class of $\QQ\cup \{\infty\}$). The number of (the classes of) cusps can be greater than one, but it is always finite.

We defined the action of $\Gamma$ on $\HH$ above. $\Gamma$ also acts on the functions on $\HH$ using the slash operator, defined as \(f\mapsto f\mid_k\gamma\), to be \((f\mid_k \gamma)(\tau) = (c\tau+d)^{-k}f(\gamma.\tau)\), where \(\gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \Gamma\). Here $(c\tau+d)$ (or its \(k\)-th power) is called the factor of automorphy.

Modular form

Definition (Modular form). A function $f\colon \HH\to \CC$ is said to be a (holomorphic) modular form of weight $k$ and level $\Gamma$, if

  1. Holomorphy: $f$ is holomorphic on $\HH$,
  2. Modularity: \((f|_k\gamma)(\tau) = f(\tau)\) for all \(\gamma \in \Gamma\),
  3. Growth condition: $f$ extends holomorphically to every cusp of $\Gamma$.

We write $M_k(\Gamma)$ the space of modular forms of weight $k$ and level $\Gamma$.

Say \(T = \begin{pmatrix}1 & 1 \\ 0 & 1 \end{pmatrix} \in \Gamma\). Note that $T.\tau = \tau+1$. If \(\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \Gamma\) is a stabilizer of $\infty$, then one can easily check that $c=0$, $a=d=1$ and thus $\Gamma_\infty$, the stabilizer of $\infty$, is generated by $T$.

The modularity condition yields

\[(f|_k T)(\tau) = f(\tau+1) = f(\tau),\]

i.e. it is periodic. One can consider a function $g\colon q\mapsto f\left(\frac{\log q}{2\pi i}\right)$ is well defined and holomorphic on ${q \in \CC : 0<|q|<1}$, so it attains a Laurent series $g(q) = \sum_{n \in \ZZ}a_n q^n$, or equivalently, we can write

\[f(\tau) = \sum_{n\in \ZZ} a_n q^n = \sum_{n\in \ZZ} a_n e^{2\pi i n \tau}.\quad (q = e^{2\pi i \tau})\]

The growth condition is that $f$ is holomorphic at the cusp $\infty$. Note that $q=e^{2\pi i \tau}$ tends to $0$ as $\Im(\tau) \to \infty$, so $f$ is holomorphic at $\infty$ if and only if $a_n = 0$ for all $n<0$. So the $f$ has the following Fourier expansion

\[f(\tau) = \sum_{n\ge0} a_n e^{2\pi i n \tau}.\]

Note that a modular form $f$ is said to be a cusp form when $a_0=0$ (i.e., $f$ vanishes at the cusp $\infty$).

Examples

One of the most natural examples of modular forms arises from the quadratic form. Let $Q$ be a (positive definite integral) quadratic form in $2$ (or, $2k$) variables, for example, $Q(x,y) = x^2+y^2$. Then we can define the Theta function

\[\Theta_Q(\tau) = \sum_{(x,y)\in \ZZ^2} e^{2\pi i Q(x,y)\tau} = 1 +\sum_{n\ge1} r_Q(n) q^n,\]

where $r_Q(n)$ is the number of representations of $n$ by $Q$. Then $\Theta_Q$ is a modular form of weight $2$ (or, $2k$) and some level $\Gamma \leq \SL_2(\ZZ)$.

Another class of examples is the Eisenstein series. Let $k\ge4$ be even, and let \(G_k(\tau) = \sideset{}{'}\sum_{m,n\in \ZZ}\frac{1}{(m\tau+n)^k}\ ,\) where $\sum’$ is the sum over indices which excludes the term $m=n=0$. It is a modular form of weight $k$ and level $\SL_2(\ZZ)$.

The Fourier expansion of $G_k$ is

\[G_k(\tau) = 2\zeta(k) + \frac{2\cdot(2\pi)^k}{\Gamma(k)}\sum_{n\ge1}\sigma_{k-1}(n)q^n,\]

where $\sigma_{r}(n) = \sum_{d\mid n}d^{r}$ is the divisor function. The normalized Eisenstein series $E_k(\tau)$ is defined to have the constant term $1$; i.e. $E_k(\tau) = \dfrac{G_k(\tau)}{2 \zeta(k)}$.

Note that, if $f$ and $g$ are a modular form of weight $k$ and level $\Gamma$, then $f+g$ and $f^n$ is a modular form of weight $k$ and $nk$ respectively (and level $\Gamma$). One can construct a first cusp form from the Eisenstein series; $E_4^3 - E_6^2$, with vanishing constant in its Fourier expansion, is a cusp form of weight $12$ and level $\SL_2(\ZZ)$. We normalize this function to define the Discriminant function,

\[\Delta(\tau) = \frac{1}{1728}\left(E_4^3 - E_6^2\right) = q\prod_{n\ge1}(1-q^n)^{24}.\]

(Some authors defines it as $(2\pi)^{12}q\prod_{n\ge1}(1-q^n)^{24}$.)

$\Delta(\tau)$ is the unique cusp form of weight $12$ and level $\SL_2(\ZZ)$ (up to scalar multiple).

References:

  • Cogdell, James W.; Kim, Henry H.; Murty, M. Ram. Lectures on automorphic L-functions. Fields Inst. Monogr., 20. American Mathematical Society, Providence, RI, 2004.