Various Eisenstein series

For the first series of post I want to talk about real-analytic Eisenstein series and their vector-valued generalization. The motivation is to understand the ensemble average/3D quantum gravity aspects of this paper, which has been on my bucket list for while now.

Here’s a rough list of topics:

Various Eisenstein series.
Representation theoretic definition of Eisenstein series and continuous spectrum of automorphic forms.
Siegel-Weil formulas.
Vector-valued generalizations.

Other things to write about: Hecke operators.

In this post we will review all the different type of Eisenstein series, from the simplest holomorphic Eisenstein series all the way up to weight $k$ real-analytic Eisenstein series. Much of it is from here

In this post we will take our time to review through various version of Eisenstein series. Here’s the summarization:

Holomorphic Eisenstein seires $E_k(\tau)$: labeled by $k \in 2\mathbb{N}$, this is holomorphic, transforms as weight $k$ modular form under the $Mp_2(\mathbb{Z})$ transform, and holomorphic in $\infty$.
Real-analytic Eisenstein series $E(\tau, s)$: they are labeled by $s$ in the upper complex plane (???), real-analytic, transform as weight $0$ under $Mp_2(\mathbb{Z})$, and are Eigenvalues of the Laplacian \begin{equation} \Delta = -y^2(\partial_x^2 + \partial_y^2) \end{equation} with eigenvalue $s(1-s)$. For $s = \frac{1}{2} + it$ family they show up in the spectral theory of modular function, as they are the continuous series.
Weight $k$ real-analytic Eisenstein series $E_k(s, \tau)$: This is the combination of the two above. They are harmonic weak Maass forms, satisfies the weight $k$ Laplacian equation \begin{equation} \Delta_k = -y^2(\partial_x^2 + \partial_y^2)+ iky(\partial_x + i \partial_y) \end{equation} QUESTION: what about the spectral theory of here? do we still use $s = \frac{1}{2}+it$?

Holomorphic Eisenstein series

Recall the group $\SLZ$ as well as its double cover $\MpZ$. Basically $\MpZ$ allows us to take square root of $c \tau + d$. –> There is a canonical action of $\SLZ$ on the upper half plane, which has coordinate $\tau$. $\MpZ$ has generators $S$ and $T$, with $T$ generates the subgroup $\Gamma_{\infty}$ which is the fixed point of $y = \Im(\tau)$.

For $M = \begin{pmatrix}a & b \\ c & d \end{pmatrix}$, we have this important equation: \begin{equation}\label{eq:imaginary-under-M} \Im(M\tau) = \Im(\frac{a \tau + b}{c \tau + d}) = \frac{\Im{(a \tau + b)(c \bar{\tau} + d)}}{|c\tau + d|^2} = \frac{\Im(ad \tau + bc \bar{\tau})}{|c\tau + d|^2} = \frac{\Im(\tau)}{|c\tau + d|^2} \end{equation} From

Let us first recall the definition of a modular form:

Definition: Let $k$ be an half integer. A modular form of weight $k$ is a holomorphic function: $\mbb{H} \to \mbb{C}$ satisfying the following:

$f$ transforms under the action of $\MPZ$ by
\begin{equation} f(M \cdot \tau) = (c \tau + d)^k f(\tau). \end{equation}
$f$ is holomorphic at $\infty$. That is, let $q = e^{2\pi i \tau}$. Consider the Fourier expansion \begin{equation} f(\tau) = \sum a_n q^n, \end{equation} $f$ is holomorphic at $\infty$ if all coefficients $a_n$ are zero for $n < 0$.

Such a holomorphic form is a cusp form if $a_0 = 0$. Note that condition (1) above is equivalent to \begin{equation} f(\tau + 1) = f(\tau), \quad f(-1/\tau) = \tau^k f(\tau). \end{equation} One can show that for all $k$, the space of weight $k$ modular forms is finite dimensional. With the first non-trivial one at $k = 4$, where it is one-dimensional spanned by the Eisenstein series $E_2$, which we will now define.

Definition: Let $k$ be an integer, then the Eisenstein series $E_k(\tau)$ is defined as \begin{equation} E_k(\tau) = \sum_{c, d \in \mbb{Z}^2-(0,0)} \frac{1}{(m + n \tau)^{2k}} \end{equation} This sum converges absolutely when $k \geq 2$. It is straightfoward to see that for $k \geq 2$, $E_k(\tau)$ is a modular form with weight $k$.

Example:

$E_4(\tau) = 1 + 240 \sum_{n=1}^{\infty} \frac{n^3q^n}{1-q^n}, \quad E_4^2 = E_8$.

Real-analytic Eisenstein series

Now we move onto real-analytic Eisenstein series. They are Maass forms, which are real-analytic generalization of modular functions (i.e. modular functions of weight $0$).

Definition: A complex-valued smooth function $f \colon \mbb{H} \to \mbb{C}$ is a Maass form if

We have \begin{equation} f(M \cdot \tau) = f(\tau). \end{equation}
$f$ is an eigenfunction of the hyperbolic Laplacian: \begin{equation} \Delta_k = -y^2 (\partial_x^2 + \partial_y^2 ) .\end{equation}
$f$ has moderate growth at the cusp, i.e., there exists $N$ such that $f(\tau) =O(y^N)$ as $y \to \infty$.

We call $f$ weak Maass form if it satifies (1) and (2) above, and the following growth condition:

3’. $f$ have at most linear exponential growth at the cusp, i.e., exists $C> 0$ such that $f(\tau) = O(e^{Cy})$ as $y \to \infty$.

More generally, when we are dealing with cases where there are more than one cusp, for example working with higher level $N$, we are going to have growth condition on all cusps.

The idea of real-analytic Eisenstein series is simple: if you want a modular function, you can sum up $\MpZ$ images of a function $f$. However, we can do something a bit smarter, take the function $y^{s}$, it is automatically invariant under $T$ transforms, therefore we just need to sum up its image under $\Gamma_{\infty}\backslash\MpZ$, which is in correspondence with pairs of integers $c,d$ with the $(c,d) = 1$.

Definition: Fix $s \in \mbb{C}$, then \begin{equation} E_s(\tau) = \sum_{\gamma \in \Gamma_{\infty}\backslash \MpZ}\frac{1}{2|c\tau + d|^{-2s}} = \sum_{c \geq 0, (c,d) = 1} y^s|_{\gamma} \end{equation} Note that we avoided doubling by setting $c \geq 0$. Also sometimes people do not write the factor $y^s$.

This sum converges when $\Re(s) \geq 1$, however, it has an analytic continuation everywhere in the $s$ plane: \begin{equation}\label{eq:Es-analytic-def} E_s(\tau ) = y^s + \frac{\Lambda(1-s)}{\Lambda(s)}y^{1-s} + \sum_{j=1}^{\infty} \frac{4 \sigma_{2s-1}j \sqrt{y}K_{s-\frac{1}{2}}(2 \pi j y)} {\Lambda(s)j^{s-1/2}} \, \cos(2\pi j x). \end{equation} Here $\sigma_{2s-1}(j)$ is the divisor function, $K$ is the modified Bessel function of the second kind, and $\Lambda$ is the completed zeta function: \begin{equation} \Lambda(s) = \pi^{-s}\zeta(2s)\Gamma(s), \end{equation} and it satisfies the functional equation: \begin{equation} \Lambda(s) = \Lambda(\frac{1}{2} - s) \end{equation} From \eqref{eq:Es-analytic-def} we have \begin{equation} \Lambda(s)E_s(\tau) = \Lambda(1-s)E_{1-s}(\tau). \end{equation}

$E_s(\tau)$ has the following properties:

The Eisenstein series $E_s(\tau)$ is analytic in $\tau$.
$E_s(\tau)$ is meromorphic in $s$ with a unique pole of residue $3/\pi$ at $s = 1$ (for all $s$ in $\mbb{H}$), and infinitely many poles in the strip of $0 < \Re(s)< \frac{1}{2}$ at $\rho/2$, where $\rho$ is a non-trivial zero of the Riemann zeta-function $\zeta$.
For any $s$, $E_s(\tau)$ is a weak Maass form of weight $0$, and is an eigenvector of the Laplacian $\Delta_0 = \Delta$ with eigenvalue $s(1-s)$.
Note that $E_s(\tau)$ are not square integrable over the fundamental domain $\mbb{H}/\SLZ$.

These $E_s(\tau)$, for $s = \frac{1}{2} + it$, will also play a crucial in the harmonic analysis of modular functions on $\mbb{H}$, which we will study in the [next post]{???}.

Real analytic weight $k$ Eisenstein series

Now we introduce weight $k$ Maass forms, which are higher weight generalizations of Maass forms.

Definition: A function $f \colon \mbb{H} \to \mbb{C}$ is a Maass form of weight $k$ if

We have \begin{equation} f(M \cdot \tau) = (\frac{c\bar{\tau}+d}{|c \tau + d|})^k f(\tau). \end{equation} NOTE THAT SOME PEOPLE ALSO DIVIDE BY
$f$ is an eigenfunction of the hyperbolic Laplacian: \begin{equation} \Delta_k = -y^2 (\partial_x^2 + \partial_y^2 ) + iky \partial_x .\end{equation}
$f$ has moderate growth at the cusp, i.e., there exists $N$ such that $f(\tau) =O(y^N)$ as $y \to \infty$.

Now we are going to define real-analytic weight $k$ Eisenstein series Definition: ???? WHat is it

The real-analytic Eisenstein seires of weight $k$ is \begin{equation} \sum_{c \geq 0, (c,d) = 1} (c \tau + d)^{-k} | \end{equation}

Real analytic weight $(p,q)$ Eisenstein series

Our next goal is to generalize this to weight

References:

https://arxiv.org/abs/2311.00699
https://math.berkeley.edu/~btw/small-eisenstein.pdf,
Borcherds Products on O(2,l) and Chern Classes of Heegner Divisors Section I

Various Eisenstein series

Holomorphic Eisenstein series

Real-analytic Eisenstein series

Real analytic weight \(k\) Eisenstein series

Real analytic weight \((p,q)\) Eisenstein series

References:

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