# Algebraic proof of modular form inequalities for optimal sphere packings

Seewoo Lee

###### Abstract.

We give algebraic proofs of Viazovska and Cohn-Kumar-Miller-Radchenko-
Viazovska’s modular form inequalities for 8 and 24-dimensional optimal sphere
packings.

##  1\. Introduction

In their celebrated papers [20, 3], Viazovska and
Cohn–Kumar–Miller–Radchenko–Viazovska proved the optimality of $E_{8}$ and
Leech lattice sphere packings in dimensions $8$ and $24$, respectively. Their
proof is based on the Cohn and Elkies’ linear programming bound [2], which
states that the existence of a function on $\mathbb{R}^{d}$ satisfying certain
inequalities implies an upper bound on the sphere packing densities. The
conjectural _magic functions_ in dimensions $8$ and $24$ are constructed by
Viazovska and Cohn et al. as integral transforms of certain (quasi)modular
forms. When the dimension is $8$, the required inequalities for the function
and its Fourier transform reduce to the following inequalities on modular
forms.

######  Theorem 1.1 (Viazovska [20]).

Define111Note that the original inequality is written in terms of
$\psi_{I}(z)=z^{2}\psi_{S}(-1/z)$, but we found that the above form is more
convenient for us to work with.

$\displaystyle\phi_{0}$
$\displaystyle=1728\frac{(E_{2}E_{4}-E_{6})^{2}}{E_{4}^{3}-E_{6}^{2}},$ (1)
$\displaystyle\psi_{S}$
$\displaystyle=-128\left(\frac{\Theta_{3}^{4}+\Theta_{2}^{4}}{\Theta_{4}^{8}}+\frac{\Theta_{2}^{4}-\Theta_{4}^{4}}{\Theta_{3}^{8}}\right),$
(2)

Then

$\displaystyle\phi_{0}(it)-\frac{36}{\pi^{2}}\psi_{S}(it)$ $\displaystyle>0,$
(3) $\displaystyle\phi_{0}(it)+\frac{36}{\pi^{2}}\psi_{S}(it)$
$\displaystyle>0,$ (4)

for all $t>0$.

(See Section 2 for the definitions of the terms appearing above.) Similarly,
the case of dimension $24$ reduces to the following three inequalities. Here
we abuse notations so that two $\psi_{S}$ in (2) and (6) are different.

######  Theorem 1.2 (Cohn et al. [3]).

Define

$\displaystyle\varphi$
$\displaystyle=-\frac{49E_{2}^{2}E_{4}^{3}-25E_{2}^{2}E_{6}^{2}-48E_{2}E_{4}^{2}E_{6}-25E_{4}^{4}+49E_{4}E_{6}^{2}}{\Delta^{2}},$
(5) $\displaystyle\psi_{S}$
$\displaystyle=-\frac{\Theta_{2}^{20}(2\Theta_{2}^{8}+7\Theta_{2}^{4}\Theta_{4}^{4}+7\Theta_{4}^{8})}{\Delta^{2}}.$
(6)

Then for all $t>0$,

$\displaystyle\varphi(it)+\frac{432}{\pi^{2}}\psi_{S}(it)$ $\displaystyle<0,$
(7) $\displaystyle\varphi(it)-\frac{432}{\pi^{2}}\psi_{S}(it)$
$\displaystyle>0.$ (8)

and for all $t\geq 1$,

$t^{10}\left(\varphi\left(\frac{i}{t}\right)-\frac{432}{\pi^{2}}\psi_{S}\left(\frac{i}{t}\right)\right)\geq\frac{725760}{\pi}e^{2\pi
t}\left(t-\frac{10}{3\pi}\right).$ (9)

In dimension $8$, the inequality (3) is easier to prove than (4) since we have
$\phi_{0}(it)>0$ and $-\psi_{S}(it)>0$ separately. However, (4) is trickier
and also unnatural in the sense that we need to compare two modular forms with
different weights, and it also includes a bizarre constant
$\frac{36}{\pi^{2}}$ (note that both $\phi_{0}$ and $\psi_{S}$ have rational
Fourier coefficients). The third inequality (9) is much more complicated due
to the non-modular terms (polynomial and exponential). The original proofs in
[20, 3] use approximations of Fourier coefficients and numerical analysis, and
it is natural to ask whether there’s a more conceptual and simpler proof of
the inequalities. Recently, Romik proposed an alternative proof of (3) and (4)
[14], based on various identities among modular forms and special values of
them at $z=i$ (which can be expressed in terms of $\pi$ and $\Gamma(1/4)$).

In this article, we provide simple and _algebraic_ proofs of theorems 1.1 and
1.2 which do not require any approximations or numerical analysis. For (4) and
(8), the idea is simple: the _ratio_ of two forms with the input $z=it$ is a
_monotone function_ in $t$, and the _limit_ as $t\to 0^{+}$ equals the
constants $\frac{36}{\pi^{2}}$ or $\frac{432}{\pi^{2}}$ appearing in Theorem
(1.1) and Theorem (1.2) (see Proposition 5.2, 5.1, 6.5, and 6.4). Figure 1
shows this pheonomenon for $d=8$, where $F$ and $G$ are certain normalization
of $\phi_{0}$ and $\psi_{S}$ respectively (see (56) and (57)). We can compute
the limit easily using the modular transformation laws of $F$ and $G$, and
they coincide with the above constants. Also, the monotonicity of the quotient
is equivalent to

$F^{\prime}(it)G(it)-F(it)G^{\prime}(it)>0,$

(where the derivatives are defined as in (13)), which is now a _homogeneous_
inequality. For (9), we first reduce it to a simpler inequality by replacing
the exponential term with $\Delta$ (Lemma 6.7). After rewriting the inequality
as (76), we observe that the resulting function is also monotone (Figure 3 and
Proposition 6.8).

To prove these inequalities, we develop a general theory of quasimodular forms
that are nonnegative on the imaginary axis or have nonnegative Fourier
coefficients (Section 3). In particular, we study how the positivity behaves
under (anti-)(Serre-)derivatives, which are simple but surprisingly useful.
Combined with the differential equations that the modular forms satisfy
((65),(66), (72), and (73)), our theory gives short proofs of the
inequalities, by considering their Serre derivatives of appropriate weight
(Corollary 3.7, Proposition 5.2, Lemma 6.1, Proposition 6.5, and Proposition
6.8). Along the way, we found that the modular forms are closely connected to
the _extremal quasimodular forms_ defined by Kaneko and Koike [10]. The
extremal forms are conjectured to have nonnegative Fourier coefficients, and
can be defined recursively when the depth is less than $5$ [10, 7]. As a
byproduct, we prove their conjecture in the case of depth $1$ (Corollary 4.3).

Figure 1. Graph of the quotient $F(it)/G(it)$ as a function in $t>0$. Here
$F=\Delta\phi_{0}$ and $G=-2\Delta\psi_{S}$.

Our proofs are based on several nontrivial identities among modular forms.
These can be checked directly with SageMath [19], since the rings of
quasimodular forms are polynomial rings over certain generators, and we only
need to check identities for the corresponding polynomials. Also, we can
easily factor a given modular form or write it as a (nonnegative) linear
combination of other modular forms using SageMath, for example (44), (46),
(47), (65), (66), (72), and (73). Codes are available in the GitHub repository
https://github.com/seewoo5/posqmf, checking all the identities appearing in
the paper.

### Acknowledgement

We thank YoungJu Choie, Henry Cohn, Dan Romik, and Sug Woo Shin for the
valuable comments and discussions. Especially, Romik’s talk on his work [14]
at UC Berkeley RTG seminar motivated this project.

##  2\. Preliminaries

###  2.1. Quasimodular forms

For any function $f:\mathbb{H}\to\mathbb{C}$ defined on the complex upper half
plane $\mathbb{H}=\\{z\in\mathbb{C}:\Im z>0\\}$ and integer $k$, we define the
action of $\gamma\in\mathrm{SL}_{2}(\mathbb{Z})$ as

$(f|_{k}\gamma)(z):=(cz+d)^{-k}f\left(\frac{az+b}{cz+d}\right),\quad\gamma=\begin{bmatrix}a&b\\\
c&d\end{bmatrix}\in\mathrm{SL}_{2}(\mathbb{Z}).$

We denote $T=\left[\begin{smallmatrix}1&1\\\ 0&1\end{smallmatrix}\right]$ and
$S=\left[\begin{smallmatrix}0&-1\\\ 1&0\end{smallmatrix}\right]$ that generate
$\mathrm{SL}_{2}(\mathbb{Z})$. Let $q=e^{2\pi iz}$ for $z\in\mathbb{H}$.
Define the Eisenstein series of weight $2,4,6$ as

$\displaystyle E_{2}$ $\displaystyle=1-24\sum_{n\geq 1}\sigma_{1}(n)q^{n},$
$\displaystyle E_{4}$ $\displaystyle=1+240\sum_{n\geq 1}\sigma_{3}(n)q^{n},$
$\displaystyle E_{6}$ $\displaystyle=1-504\sum_{n\geq 1}\sigma_{5}(n)q^{n},$

where $\sigma_{a}(n):=\sum_{d|n}d^{a}$. The last two Eisenstein series
$E_{4},E_{6}$ are genuine modular forms. However, $E_{2}$ is not a modular
form - it is a _quasi_ modular form of weight 2 and level 1. They obey the
following transformation laws:

$\displaystyle(E_{2}|_{2}S)(z)=z^{-2}E_{2}\left(-\frac{1}{z}\right)$
$\displaystyle=E_{2}(z)-\frac{6i}{\pi z},$ (10)
$\displaystyle(E_{4}|_{4}S)(z)=z^{-4}E_{4}\left(-\frac{1}{z}\right)$
$\displaystyle=E_{4}(z),$ (11)
$\displaystyle(E_{6}|_{6}S)(z)=z^{-6}E_{6}\left(-\frac{1}{z}\right)$
$\displaystyle=E_{6}(z).$ (12)

The (graded) ring of quasimodular forms is isomorphic to a polynomial ring in
3 variables with generators $E_{2}$, $E_{4}$, and $E_{6}$ [1, §5.1], and we
define the _depth_ of a quasimodular form as the highest degree of $E_{2}$ in
its expression as a polynomial in $E_{2}$, $E_{4}$, and $E_{6}$. It is closed
under the differentiation

$F^{\prime}=DF:=\frac{1}{2\pi
i}\frac{\mathrm{d}F}{\mathrm{d}z}=q\frac{\mathrm{d}F}{\mathrm{d}q},\quad\sum_{n}a_{n}q^{n}\mapsto\sum_{n}na_{n}q^{n}$
(13)

which increases depth by $1$ and weight by $2$. For the Eisenstein series, we
have Ramanujan’s identities [1, Proposition 15, p. 49]

$\displaystyle E_{2}^{\prime}$ $\displaystyle=\frac{E_{2}^{2}-E_{4}}{12},$
(14) $\displaystyle E_{4}^{\prime}$
$\displaystyle=\frac{E_{2}E_{4}-E_{6}}{3},$ (15) $\displaystyle
E_{6}^{\prime}$ $\displaystyle=\frac{E_{2}E_{6}-E_{4}^{2}}{2}.$ (16)

We write
$\mathcal{QM}_{w}^{s}=\mathcal{QM}_{w}^{s}(\mathrm{SL}_{2}(\mathbb{Z}))$ for
the space of quasimodular forms of weight $w$ and depth $\leq s$, and
$\mathcal{M}_{w}:=\mathcal{QM}_{w}^{0}$ for the space of genuine modular forms
of weight $w$. We denote $\Delta:=(E_{4}^{3}-E_{6}^{2})/1728$ for the
discriminant form, which is the unique normalized cusp form of weight 12 and
level $\mathrm{SL}_{2}(\mathbb{Z})$. It can be expressed as an infinite
product

$\Delta(z)=q\prod_{n\geq 1}(1-q^{n})^{24}=\eta(z)^{24}$

where $\eta(z):=q^{1/24}\prod_{n\geq 1}(1-q^{n})$ is the Dedekind eta function
($q^{1/24}=e^{\pi iz/12}$).

###  2.2. Jacobi’s thetanull functions

Jacobi’s _thetanull_ functions are defined as

$\displaystyle\Theta_{2}(z)$
$\displaystyle=\theta_{10}(z)=\sum_{n\in\mathbb{Z}}q^{\frac{1}{2}(n+\frac{1}{2})^{2}}$
$\displaystyle\Theta_{3}(z)$
$\displaystyle=\theta_{00}(z)=\sum_{n\in\mathbb{Z}}q^{\frac{n^{2}}{2}}$
$\displaystyle\Theta_{4}(z)$
$\displaystyle=\theta_{01}(z)=\sum_{n\in\mathbb{Z}}(-1)^{n}q^{\frac{n^{2}}{2}}.$

Here we choose $q^{1/2}=e^{\pi iz}$. These are weight $1/2$ modular forms of
level
$\Gamma(2)=\\{\gamma\in\mathrm{SL}_{2}(\mathbb{Z}):\gamma\equiv\left[\begin{smallmatrix}1&0\\\
0&1\end{smallmatrix}\right]\,(\mathrm{mod}\,2)\\}$. Although the definition of
half-integral weight modular forms is quite subtle (for example, see [18]), we
can ignore such subtleties since we will only care about 4-th powers of these
forms, which are modular forms of weight $2$ and level $\Gamma(2)$. We will
denote them as $H_{2}$, $H_{3}$, and $H_{4}$, which admit Fourier expansions

$\displaystyle H_{2}(z)$ $\displaystyle:=\Theta_{2}^{4}(z)=2\sum_{n\geq
0}r_{4}(2n+1)q^{n+\frac{1}{2}},$ (17) $\displaystyle H_{3}(z)$
$\displaystyle:=\Theta_{3}^{4}(z)=1+\sum_{n\geq 1}r_{4}(n)q^{\frac{n}{2}},$
(18) $\displaystyle H_{4}(z)$ $\displaystyle:=\Theta_{4}^{4}(z)=1+\sum_{n\geq
1}(-1)^{n}r_{4}(n)q^{\frac{n}{2}},$ (19)

where $r_{4}(k):=\\{\mathbf{x}\in\mathbb{Z}^{4}:\|\mathbf{x}\|^{2}=k\\}$. They
transform under $\mathrm{SL}_{2}(\mathbb{Z})$ as:

$\displaystyle(H_{2}|_{2}T)(z)=-H_{2}(z),\quad(H_{3}|_{2}T)(z)$
$\displaystyle=H_{4}(z),\quad(H_{4}|_{2}T)(z)=H_{3}(z),$ (20)
$\displaystyle(H_{2}|_{2}S)(z)=-H_{4}(z),\quad(H_{3}|_{2}S)(z)$
$\displaystyle=-H_{3}(z),\quad(H_{4}|_{2}S)(z)=-H_{2}(z).$ (21)

Also, we have the Jacobi identity $H_{3}=H_{2}+H_{4}$. These functions are
related to the Eisenstein series and the discriminant form as ([1, p. 29])

$\displaystyle E_{4}$
$\displaystyle=\frac{1}{2}(H_{2}^{2}+H_{3}^{2}+H_{4}^{2})=H_{2}^{2}+H_{2}H_{4}+H_{4}^{2}$
(22) $\displaystyle E_{6}$
$\displaystyle=\frac{1}{2}(H_{2}+H_{3})(H_{3}+H_{4})(H_{4}-H_{2})=\frac{1}{2}(H_{2}+2H_{4})(2H_{2}+H_{4})(H_{4}-H_{2})$
(23) $\displaystyle\Delta$ $\displaystyle=\frac{1}{256}(H_{2}H_{3}H_{4})^{2}.$
(24)

We also have the following identities relating weight 2 Eisenstein series and
theta functions (See Exercises of [15, Chapter 5]):

$\displaystyle H_{2}(z)$
$\displaystyle=\frac{1}{3}\left(E_{2}\left(\frac{z+1}{2}\right)-E_{2}\left(\frac{z}{2}\right)\right)$
(25) $\displaystyle H_{3}(z)$
$\displaystyle=\frac{1}{3}\left(4E_{2}(2z)-E_{2}\left(\frac{z}{2}\right)\right)$
(26) $\displaystyle H_{4}(z)$
$\displaystyle=\frac{1}{3}\left(4E_{2}(2z)-E_{2}\left(\frac{z+1}{2}\right)\right).$
(27)

Note that $E_{2}$ also satisfies the following identity

$E_{2}(z)=\frac{1}{6}\left(4E_{2}(2z)+E_{2}\left(\frac{z}{2}\right)+E_{2}\left(\frac{z+1}{2}\right)\right).$
(28)

Combining (25), (27), and (28), we obtain

$H_{2}(z)+2H_{4}(z)=4E_{2}(2z)-2E_{2}(z).$ (29)

###  2.3. Serre derivative

For a (positive) integer $k$ and a quasimodular form $F$, the weight $k$ Serre
derivative $\partial_{k}F$ of $F$ is given by

$\partial_{k}F:=F^{\prime}-\frac{k}{12}E_{2}F.$

$\partial_{k}F$ is _a priori_ quasimodular form of weight $w+2$ and depth
$s+1$, but when $F\in\mathcal{QM}_{w}^{s}$ and $k=w-s$, Kaneko and Koike
proved that it preserves depth of quasimodular forms [10, Proposition 3.3].
Serre derivative is equivariant under the $\mathrm{SL}_{2}(\mathbb{Z})$-action
in the sense that

$\partial_{k}(F|_{k}\gamma)=(\partial_{k}F)|_{k+2}\gamma,\quad\forall\gamma\in\mathrm{SL}_{2}(\mathbb{Z}).$

Here is a list of special Serre derivatives of Eisenstein series and Jacobi
thetanull functions:

$\partial_{1}E_{2}=-\frac{1}{12}E_{4},\quad\partial_{4}E_{4}=-\frac{1}{3}E_{6},\quad\partial_{6}E_{6}=-\frac{1}{2}E_{4}^{2}$
(30)

and

$\displaystyle\partial_{2}(H_{2})$
$\displaystyle=\frac{1}{6}(H_{2}^{2}+2H_{2}H_{4}),\quad\partial_{2}(H_{3})=\frac{1}{6}(H_{2}^{2}-H_{4}^{2}),\quad\partial_{2}(H_{4})=-\frac{1}{6}(2H_{2}H_{4}+H_{4}^{2}).$
(31)

Serre derivative satisfies the product rule

$\partial_{w_{1}+w_{2}}(FG)=(\partial_{w_{1}}F)G+F(\partial_{w_{2}}G).$ (32)

In particular, we have

$\displaystyle\partial_{w}(E_{2}F)$
$\displaystyle=E_{2}(\partial_{w-1}F)-\frac{1}{12}E_{4}F,$ (33)
$\displaystyle\partial_{w}(E_{4}F)$
$\displaystyle=E_{4}(\partial_{w-4}F)-\frac{1}{3}E_{6}F,$ (34)
$\displaystyle\partial_{w}(E_{6}F)$
$\displaystyle=E_{6}(\partial_{w-6}F)-\frac{1}{2}E_{4}^{2}F,$ (35)

which are useful for the computations. Also, combining the product rule with
the Serre derivatives of theta functions, we have the following formula: For
$a,b\geq 0$, we have

$\partial_{2a+2b}(H_{2}^{a}H_{4}^{b})=\frac{1}{6}H_{2}^{a}H_{4}^{b}((a-2b)H_{2}+(2a-b)H_{4}).$
(36)

At last, we often denote the $r$-fold Serre derivatives as

$\partial_{k}^{r}F:=\partial_{k+2(r-1)}\partial_{k+2(r-2)}\cdots\partial_{k}F,\quad\partial_{k}^{0}F:=F.$

###  2.4. Linear programming bounds for optimal sphere packings and modular
form inequalities

Cohn–Elkies’ linear programming bound for sphere packings [5] reads as
follows.

######  Theorem 2.1 (Cohn–Elkies [2, Theorem 3.2]).

Let $f:\mathbb{R}^{d}\to\mathbb{R}$ be an admissible function satisfying the
following three conditions for some $r>0$:

  1. (1)

$f(\mathbf{0})=\widehat{f}(\mathbf{0})>0$;

  2. (2)

$f(\mathbf{x})\leq 0$ for $\|\mathbf{x}\|\geq r$;

  3. (3)

$\widehat{f}(\mathbf{x})\geq 0$ for all $\mathbf{x}\in\mathbb{R}^{d}$.

Then the optimal density $\Delta_{d}$ of sphere packings in $\mathbb{R}^{d}$
is bounded above by

$\Delta_{d}\leq\left(\frac{r}{2}\right)^{d}\frac{\pi^{d/2}}{(d/2)!}.$

Hence one can prove optimality of certain sphere packings by finding _magic
functions_ with the corresponding radius $r$. For example, when $d=1$, the
function

$f(x)=\frac{1}{1-x^{2}}\left(\frac{\sin(\pi x)}{\pi x}\right)^{2}$

satisfies all the conditions above for $r=1$, and yields the (trivial) bound
$\Delta_{1}\leq 1$. Based on their numerical experiments, Cohn and Elkies
conjectured the existence of magic functions in dimension $2$, $8$, and $24$
giving the optimal bounds correspond to the best known packings. After few
years, Viazovska [20] found an elegant construction of a magic function in
dimension $8$ that proves optimality of the $E_{8}$ lattice sphere packing,
and it only took one more week for her and other colleagues to find a similar
magic function in dimension $24$ (for Leech lattice) [3]. Sphere packing
problem can be also viewed as a special case of potential minimization
problem, and more general question on the universal optimality of $E_{8}$ and
Leech lattices is recently resolved by the same authors via similar methods
[4]. Especially, they proved Fourier interpolation formula for these
dimensions, which imply _uniqueness of magic functions_ in dimensions $8$ and
$24$ up to a constant (see Theorem 1.7 and Corollary 1.8 of [4]). Existence of
a magic function in dimension $2$ is still wide open.

Viazovska and Cohn et al.’s constructions are based on ingenious use of
(quasi)modular forms. They first decompose it as a sum of $(\pm 1)$-Fourier
eigenfunctions $f=f_{+}+f_{-}$ (hence $\widehat{f}_{+}=f_{+}$ and
$\widehat{f}_{-}=-f_{-}$), and assume that they have a following form:

$f_{\pm}(\mathbf{x})=\sin^{2}\left(\frac{\pi\|\mathbf{x}\|^{2}}{2}\right)\int_{0}^{\infty}\varphi_{\pm}(it)e^{-\pi\|\mathbf{x}\|^{2}t}\mathrm{d}t$

for $x\in\mathbb{R}^{d}$, where $\varphi_{\pm}$ is function defined on
$\mathbb{H}$. Here we can assume that $f$ is radial (by averaging over spheres
centered at the origin), and the $\sin^{2}$ factor is added to enforce roots
at the desired radii, i.e. the lengths of the vectors in $E_{8}$ or Leech
lattices (if $f$ is a magic function, then both $f$ and $\widehat{f}$ should
have zeros at the nonzero lattices points, which follows from the proof of
Theorem 2.1). Surprisingly, under this assumption, they proved that $f_{\pm}$
is a ($\pm 1$)-Fourier eigenfunction if and only if $\varphi_{\pm}$ behaves as
“modular forms”. For example, when $d=8$, we have
$\varphi_{\pm}(t)=t^{2}\psi_{\pm}(i/t)$ with222Here we normalized in a
slightly different way. We have $f(0)=\widehat{f}(0)=\frac{5}{4\pi}$. This
normalization will be also used in Section 5.

$\displaystyle\psi_{+}$
$\displaystyle=-\phi_{0}=-\frac{(E_{2}E_{4}-E_{6})^{2}}{\Delta},$
$\displaystyle\psi_{-}$
$\displaystyle=2\psi_{S}=-\frac{18}{\pi^{2}}\frac{\Theta_{2}^{12}(2\Theta_{2}^{8}+5\Theta_{2}^{4}\Theta_{4}^{4}+5\Theta_{4}^{8})}{\Delta}.$

The corresponding integrals only converge for $\|\mathbf{x}\|>\sqrt{2}$, and
one needs to analytically continue to $0\leq\|\mathbf{x}\|\leq\sqrt{2}$. Under
this construction, the corresponding inequalities for $f$ and $\widehat{f}$
reduces to the inequalities between $\psi_{+}$ and $\psi_{-}$, which are
essentially the inequalities (3) and (4). Viazovska proved the inequalities by
approximating the functions and reducing it to the finite calculations, which
can be checked by interval arithmetic with computer calculations. $d=24$ case
is similar but more involved, and requires more complicated and careful
computer analysis.

In [14], Romik proposed an alternative proof of (3) and (4). His proof of (4)
uses explicit values of modular forms at $z=i$, such as

$E_{2}(i)=\frac{3}{\pi},\quad
E_{4}(i)=\frac{3\Gamma(1/4)^{8}}{64\pi^{6}},\quad E_{6}(i)=0.$ (37)

(Note that the first and third identity directly follows from (10) and (12),
respectively.) He treated $0<t<1$ and $t\geq 1$ cases separately. Proofs for
the both cases utlize monotonicity of certain modular forms on the imaginary
axis and reducing the inequalities to comparisons of the values of the
functions at $t=1$. This gives a simpler proof of the inequality (4) that
requires no interval arithmetic.

##  3\. Positive quasimodular forms

In this section, we study properties of quasimodular forms that are positive
on the imaginary axis or have nonnegative Fourier coefficients. We say that a
quasimodular form $F\in\mathcal{QM}_{w}^{s}$ is _positive_ if it takes
nonnegative real values on the (positive) imaginary axis, i.e. $F(it)\geq 0$
for all $t>0$. We denote as
$\mathcal{QM}_{w}^{s,+}\subset\mathcal{QM}_{w}^{s}$ for the subset of positive
quasimodular forms. We define _completely positive_ quasimodular forms as
those with nonnegative (real) Fourier coefficients, i.e. $F(z)=\sum_{n\geq
n_{0}}a_{n}q^{n}$ with $a_{n}\geq 0$ for all $n\geq n_{0}$. We denote the set
of such forms of weight $w$ and depth $s$ as $\mathcal{QM}_{w}^{s,++}$. Both
$\mathcal{QM}_{w}^{s,+}$ and $\mathcal{QM}_{w}^{s,++}$ are _convex cones_ in
$\mathcal{QM}_{w}^{s}$, i.e. closed under nonnegative linear combination.
Clearly we have $\mathcal{QM}_{w}^{s,++}\subset\mathcal{QM}_{w}^{s,+}$, and
the inclusion is strict in general (e.g. $\Delta(z)=q\prod_{n\geq
1}(1-q^{n})^{24}\in\mathcal{QM}_{12}^{0,+}$ but
$\Delta(z)=q-24q^{2}+252q^{3}-1472q^{4}+\cdots\not\in\mathcal{QM}_{12}^{0,++}$).

###  3.1. Derivative and positivity

Differentiation preserves complete positivity of quasimodular cusp forms,
since it changes the $n$-th Fourier coefficient from $a_{n}$ to $na_{n}$.

######  Proposition 3.1.

Let $F\in\mathcal{QM}_{w}^{s}$ and $F^{\prime}\in\mathcal{QM}_{w+2}^{s+1}$.
Assume $F$ is a cusp form. Then $F\in\mathcal{QM}_{w}^{s,++}$ if and only if
$F^{\prime}\in\mathcal{QM}_{w+2}^{s+1,++}$.

Positivity (not necessarily complete) is not preserved under derivatives in
general. For example, the discriminant form $\Delta$ is positive due to its
product expansion, but its derivative $\Delta^{\prime}=E_{2}\Delta$ has a
unique simple zero on the imaginary axis ($\lim_{t\to\infty}E_{2}(it)=1$ and
$\lim_{t\to 0^{+}}E_{2}(it)=-\infty$). However, the positivity is preserved
under _antiderivatives._

######  Proposition 3.2.

Let $X\in\mathcal{QM}_{w}^{s}$ be a cusp form. If
$X^{\prime}\in\mathcal{QM}_{w+2}^{s+1,+}$, then $X\in\mathcal{QM}_{w}^{s,+}$.

###### Proof.

Let $x(t):=X(it)$ for $t>0$. If $X^{\prime}\in\mathcal{QM}_{w+2}^{s+1,+}$,
then $\frac{\mathrm{d}x}{\mathrm{d}t}=-2\pi X^{\prime}(it)\leq 0$ and $x(t)$
is monotone decreasing for all $t$. Hence $x(t)\geq\lim_{u\to\infty}x(u)=0$. ∎

Complete positivity can be characterized as positivity of (higher)
derivatives. To prove this, we need the following version of Bernstein’s
theorem.333This theorem was the motivation for the naming of _complete_
positivity.

######  Theorem 3.3 (Bernstein).

Let $g:(0,\infty)\to\mathbb{R}$ be a smooth function. Then the following are
equivalent:

  1. (1)

$g(t)$ is a _completely monotone function_ , i.e. $(-1)^{n}g^{(n)}(t)\geq 0$
for all $n\geq 0$ and $t>0$.

  2. (2)

There exists a nonnegative measure $\mu$ on $(0,\infty)$ such that

$g(t)=\int_{0}^{\infty}e^{-tu}\mathrm{d}\mu(u)$

where the integral converges absolutely.

A proof can be found in [16, Theorem 4.8, page 40]. Note that the original
version of the Bernstein’s theorem consider functions on $[0,\infty)$ and
finite measures, and this is a slightly generalized version of it.

######  Proposition 3.4.

$F\in\mathcal{QM}_{w}^{s}$ is completely positive if and only if its
derivatives are all positive.

###### Proof.

This is a direct corollary of Theorem 3.3. Observe that
$\frac{\mathrm{d}^{k}}{\mathrm{d}t^{k}}F(it)=(-2\pi)^{k}F^{(k)}(it)$, and the
series $f(t)=\sum_{n\geq n_{0}}a_{n}e^{-2\pi nt}$ is the Laplace transform of
the measure $\mu=\sum_{n\geq n_{0}}a_{n}\delta_{2\pi n}$. Although the measure
$\mu$ is not finite, the integral

$f(t)=\int_{0}^{\infty}e^{-tu}\mathrm{d}\mu(u)$

converges absolutely (by the polynomial growth of the coefficients [17,
Theorem 5, page 94]) and we can apply Theorem 3.3. ∎

###  3.2. Serre derivative and positivity

By using a standard technique of solving first order ordinary differential
equations, we can express the _anti-_ Serre derivative of a given form as a
certain integral transform with a kernel given in terms of Dedekind eta
function.

######  Proposition 3.5.

Let $G$ be a quasimodular form and $k\geq 0$ be an integer. Let $F$ be another
quasimodular form satisfying $\partial_{k}F=G$. If we put $f(t):=F(it)$ and
$g(t)=G(it)$ for $t>0$, then for given $t_{0}>0$,

$f(t)=\left(\frac{\eta(it)}{\eta(it_{0})}\right)^{2k}f(t_{0})+2\pi\int_{t}^{t_{0}}\left(\frac{\eta(it)}{\eta(iu)}\right)^{2k}g(u)\mathrm{d}u.$
(38)

If $F$ admits a Fourier expansion $F(z)=\sum_{n\geq n_{0}}a_{n}q^{n}$ with
$n_{0}>\frac{k}{12}$, then we further have

$f(t)=2\pi\int_{t}^{\infty}\left(\frac{\eta(it)}{\eta(iu)}\right)^{2k}g(u)\mathrm{d}u.$
(39)

###### Proof.

The equation $\partial_{k}F=G$ is equivalent to

$-\frac{1}{2\pi}\frac{\mathrm{d}f}{\mathrm{d}t}-\frac{k}{12}E_{2}(it)f(t)=g(t)$

and this implies

$\frac{\mathrm{d}}{\mathrm{d}t}\left[\exp\left(\int_{t_{0}}^{t}\frac{2\pi
k}{12}E_{2}(iu)\mathrm{d}u\right)f(t)\right]=-2\pi\exp\left(\int_{t_{0}}^{t}\frac{2\pi
k}{12}E_{2}(iu)\mathrm{d}u\right)g(t).$

Since $(\log\Delta)^{\prime}=\frac{\Delta^{\prime}}{\Delta}=E_{2}$ and
$\Delta^{1/24}=\eta$, we can rewrite the integrals as

$\frac{\mathrm{d}}{\mathrm{d}t}\left[\left(\frac{\eta(it)}{\eta(it_{0})}\right)^{-2k}f(t)\right]=-2\pi\left(\frac{\eta(it)}{\eta(it_{0})}\right)^{-2k}g(t).$

Integrating both sides from $t$ to $t_{0}$ gives (38). For the second
assertion, we have

$\lim_{t_{0}\to\infty}\frac{f(t_{0})}{\eta(t_{0})^{2k}}=\lim_{t_{0}\to\infty}\frac{e^{-2\pi
n_{0}t_{0}}\left(\sum_{n\geq
n_{0}}a_{n}e^{-2\pi(n-n_{0})t_{0}}\right)}{e^{-\frac{2\pi
kt_{0}}{12}}\prod_{n\geq 1}(1-e^{-2\pi nt_{0}})^{2k}}=0$

provided that $n_{0}>\frac{k}{12}$. Then taking the limit of (38) as
$t_{0}\to\infty$ gives (39). ∎

######  Proposition 3.6.

Let $F\in\mathcal{QM}_{w}^{s}$ be a quasimodular form. If
$(\partial_{w-s}F)(it)\geq 0$ for $0<t<t_{0}$ and $F(it_{0})>0$ for some
$t_{0}>0$, then $F(it)>0$ for all $0<t\leq t_{0}$.

###### Proof.

This directly follows from (38) and the positivity of the integrands. ∎

As a corollary, _anti-Serre derivative_ preserves positivity.

######  Corollary 3.7.

Let $F\in\mathcal{QM}_{w}^{s}$ be a quasimodular form. If
$\partial_{w-s}F\in\mathcal{QM}_{w+2}^{s,+}$ and $F(it)>0$ for _sufficiently
large $t>0$_, then $F\in\mathcal{QM}_{w}^{s,+}$. Especially, the assumption
holds if all the Fourier coefficients of $F$ are real and the first nonzero
Fourier coefficient of $F$ is positive.

###### Proof.

For the last assertion, when $F$ has a Fourier expansion $F(z)=\sum_{n\geq
n_{0}}a_{n}q^{n}$ with $a_{n_{0}}>0$, then

$e^{2\pi n_{0}t}F(it)=a_{n_{0}}+e^{-2\pi t}\sum_{n\geq
n_{0}+1}a_{n}e^{-2\pi(n-n_{0}-1)t}$

and $\lim_{t\to\infty}e^{2\pi n_{0}t}F(it)=a_{n_{0}}>0$, so $F(it)>0$ for
sufficiently large $t$. ∎

######  Remark 3.8.

We can choose arbitrary (even non-integer) weight for the Serre derivative in
Corollary 3.7 and the same result still remains true, and it is not neccessary
to take $k=w-s$. Also, Proposition 3.6 holds for more general class of
functions (e.g. differentiable and locally integrable) on $(0,\infty)$ with

$\partial_{k}:=D-\frac{k}{12}E_{2}(it)=-\frac{1}{2\pi}\frac{\mathrm{d}}{\mathrm{d}t}-\frac{k}{12}E_{2}(it),$

and this version will be used in the proof of the inquality (9) later.

In general, the Serre derivative does not preserve complete positivity, e.g.
$E_{4}$ is completely positive but
$\partial_{4}E_{4}=-\frac{E_{6}}{3}=-\frac{1}{3}+168q+\cdots$ is not. However,
when the vanishing order at the cusp is sufficiently large, than it actually
does.

######  Proposition 3.9.

Let $F=\sum_{n\geq n_{0}}a_{n}q^{n}\in\mathcal{QM}_{w}^{s,++}$. For $k\geq 0$
and $n\geq k/12$, the $n$-th coefficient of $\partial_{k}F$ is nonnegative.
Especially, if $n_{0}\geq k/12\geq 0$, then $\partial_{k}F$ is also completely
positive.

###### Proof.

One can directly check that the Fourier expansion of Serre derivative is

$\displaystyle\partial_{k}F$
$\displaystyle=\partial_{k}\left(a_{n_{0}}q^{n_{0}}+a_{n_{0}+1}a^{n_{0}+1}+a_{n_{0}+2}q^{n_{0}+2}+\cdots\right)$
$\displaystyle=\left(n_{0}a_{n_{0}}q^{n_{0}}+(n_{0}+1)a_{n_{0}+1}q^{n_{0}+1}+\cdots\right)$
$\displaystyle\qquad-\frac{k}{12}(1-24q-72q^{2}-96q^{3}-\cdots)\left(a_{n_{0}}q^{n_{0}}+a_{n_{0}+1}a^{n_{0}+1}+a_{n_{0}+2}q^{n_{0}+2}+\cdots\right)$
$\displaystyle=\left(n_{0}-\frac{k}{12}\right)a_{n_{0}}q^{n_{0}}+\left(\left(n_{0}+1-\frac{k}{12}\right)a_{n_{0}+1}+2ka_{n_{0}}\right)q^{n_{0}+1}+\cdots$
$\displaystyle\qquad+\left(\left(n_{0}+m-\frac{k}{12}\right)a_{n_{0}+m}+2k\sum_{j=1}^{m}\sigma_{1}(m+1-j)a_{n_{0}+j-1}\right)q^{n_{0}+m}+\cdots.$

Hence if $n_{0}\geq k/12$ and $a_{j}\geq 0$ for all $j\geq n_{0}$, the Fourier
coefficients of $\partial_{k}F$ are also all nonnegative. ∎

###  3.3. Level and positivity

We also consider (completely) positive quasimodular forms of higher level. For
completely positive forms, we will only consider the $q$-expansions at the
cusp $i\infty$, although there are several cusps for a congruence subgroup
$\Gamma\subset\mathrm{SL}_{2}(\mathbb{Z})$ in general. One easy way to
construct (completely) positive quasimodular forms of level $\Gamma_{0}(N)$ is
using _old forms_.

######  Proposition 3.10.

Let $F(z)\in\mathcal{QM}_{w}^{s}(\mathrm{SL}_{2}(\mathbb{Z}))$ be a positive
(resp. completely positive) quasimodular form of weight $w$, depth $s$, and
level $\mathrm{SL}_{2}(\mathbb{Z})$. Then for any $N\in\mathbb{Z}_{\geq 1}$,
the form $G(z):=F(Nz)\in\mathcal{QM}_{w}^{s}(\Gamma_{0}(N))$ is also a
positive (resp. completely positive) quasimodular form.

###### Proof.

If $f$ has a $q$-expansion $f(z)=\sum_{n\geq n_{0}}a_{n}q^{n}$, then
$g(it)=f(iNt)$ and $g(z)=\sum_{n\geq n_{0}}a_{n}q^{Nn}$, and the proposition
immediately follows. ∎

##  4\. Extremal quasimodular forms

###  4.1. Definitions and examples

In [10], Kaneko and Koike defined and studied _extremal quasimodular forms_ ,
which are the quasimodular forms of given depth with maximum possible order of
zeros at infinity. In other words, for given weight $w$ and depth $s$, a
quasimodular form $f\in\mathcal{QM}_{w}^{s}\backslash\mathcal{QM}_{w}^{s-1}$
is _extremal_ if, for $m=\dim_{\mathbb{C}}\mathcal{QM}_{w}^{s}$, the first $m$
Fourier coefficients of $f=\sum_{n\geq 0}a_{n}q^{n}$ are

$a_{0}=a_{1}=\cdots=a_{m-2}=0,a_{m-1}\neq 0.$

The authors conjectured existence and uniqueness (up to a constant) of
extremal forms for each (even) weight $w$ and depth $s$ (satisfying $0\leq
s\leq w/2,s\neq w/2-1$), and give examples in case of depth $1$ and $2$ that
are defined recursively and satisfying certain differential equations.
Pellarin [13] established the conjecture for $s\leq 4$, and Grabner [7]
extended Kaneko–Koike’s result and construct differential equations satisfied
by depth $\leq 4$ extremal quasimodular forms. For these depths, we will
denote the normalized (i.e. the first nonzero Fourier coefficient is zero)
extremal form of weight $w$ and depth $s$ as $X_{w,s}$. Kaneko and Koike also
conjectured that the Fourier coefficients of extremal forms of depth $\leq 4$
are all positive [10, Conjecture 2], and Grabner [8] proved the conjecture
_for all but finitely many coefficients_. The proof uses Jenkins and Rouse’s
explicit version of Deligne’s bound [9].

We can check complete positivity of certain low-weight extremal forms from
Ramanujan’s identities:

######  Lemma 4.1.

The following modular forms are extremal and completely positive:

$\displaystyle X_{4,2}$
$\displaystyle=\frac{1}{288}(E_{4}-E_{2}^{2})\in\mathcal{QM}_{4}^{2,++},$
$\displaystyle X_{6,1}$
$\displaystyle=\frac{1}{720}(E_{2}E_{4}-E_{6})\in\mathcal{QM}_{6}^{1,++},$
$\displaystyle X_{8,1}$
$\displaystyle=\frac{1}{1008}(E_{4}^{2}-E_{2}E_{6})\in\mathcal{QM}_{8}^{1,++}.$

###### Proof.

Extremality are mentioned in [10]. Complete positiveness directly follows from
Ramanujan’s identities,

$\displaystyle X_{4,2}=\frac{1}{288}(E_{4}-E_{2}^{2})$
$\displaystyle=-\frac{1}{24}E_{2}^{\prime}=\sum_{n\geq 1}n\sigma_{1}(n)q^{n},$
$\displaystyle X_{6,1}=\frac{1}{720}(E_{2}E_{4}-E_{6})$
$\displaystyle=\frac{1}{240}E_{4}^{\prime}=\sum_{n\geq 1}n\sigma_{3}(n)q^{n},$
$\displaystyle X_{8,1}=\frac{1}{1008}(E_{4}^{2}-E_{2}E_{6})$
$\displaystyle=-\frac{1}{504}E_{6}^{\prime}=\sum_{n\geq
1}n\sigma_{5}(n)q^{n}.$

∎

More examples can be found in Appendix B, Table 1.

###  4.2. Kaneko–Koike’s conjecture for depth $1$

In this subsection, we prove the conjecture of Kaneko and Koike [10,
Conjecture 2] in the case of depth $1$.

For even $w\geq 6$, let $X_{w}=X_{w,1}$ be the unique normalized extremal
quasimodular form of weight $w$ and depth $1$. We have
$X_{6}=(E_{2}E_{4}-E_{6})/720$ and $X_{w}$’s satisfy the following recursive
formula for $w\geq 6$ with $6|w$ [10, 7]:

$\displaystyle X_{w+2}$ $\displaystyle=\frac{12}{w+1}\partial_{w-1}X_{w}$ (40)
$\displaystyle X_{w+4}$ $\displaystyle=E_{4}X_{w}$ (41) $\displaystyle
X_{w+6}$
$\displaystyle=\frac{w+6}{72(w+1)(w+5)}\left(E_{4}\partial_{w-1}X_{w}-\frac{w+1}{12}E_{6}X_{w}\right)$
$\displaystyle=\frac{w+6}{864(w+5)}\left(E_{4}X_{w+2}-E_{6}X_{w}\right).$ (42)

The vanishing order of $X_{w}$ at the cusp is $\lfloor\frac{w}{6}\rfloor$.
Also, $X_{w}$ is a solution of the following differential equation (for $6|w$)

$X_{w}^{\prime\prime}-\frac{w}{6}E_{2}X_{w}^{\prime}+\frac{w(w-1)}{144}(E_{2}^{2}-E_{4})X_{w}=0$
(43)

Following new recurrence relation (44) can be easily shown from the above
identities, which immediately implies the complete positivity of $X_{w}$’s.

######  Theorem 4.2.

Let $X_{w}=X_{w,1}$ be the unique normalized extremal quasimodular form of
weight $w$ and depth $1$. For $6|w$ and $w\geq 12$, we have

$\displaystyle X_{w}^{\prime}$
$\displaystyle=\frac{5w}{72}X_{6}X_{w-4}+\frac{7w}{72}X_{8}X_{w-6}.$ (44)

###### Proof.

(40)w-4 is equivalent to

$X_{w-4}^{\prime}=\frac{w-5}{12}E_{2}X_{w-4}+\frac{w-7}{12}E_{4}X_{w-6}$ (45)

and differentiating (42)w-6 gives

$\displaystyle X_{w}^{\prime}$
$\displaystyle=\frac{w}{864(w-1)}(E_{4}^{\prime}X_{w-4}+E_{4}X_{w-4}^{\prime}-E_{6}^{\prime}X_{w-6}-E_{6}X_{w-6}^{\prime})\qquad$
$\cdots$(42)w-6
$\displaystyle=\frac{w}{864(w-1)}\left(\frac{E_{2}E_{4}-E_{6}}{3}X_{w-4}+\frac{w-5}{12}E_{2}E_{4}X_{w-4}+\frac{w-7}{12}E_{4}^{2}X_{w-6}\right.\qquad$
$\cdots$(45)w
$\displaystyle\quad\left.-\frac{E_{2}E_{6}-E_{4}^{2}}{2}X_{w-6}-E_{6}\left(\frac{w-5}{12}X_{w-4}+\frac{w-7}{12}E_{2}X_{w-6}\right)\right)\qquad$
$\cdots$(40)w-6
$\displaystyle=\frac{w}{864(w-1)}\left(\frac{w-1}{12}(E_{2}E_{4}-E_{6})X_{w-4}+\frac{w-1}{12}(E_{4}^{2}-E_{2}E_{6})X_{w-6}\right)$
$\displaystyle=\frac{5w}{72}X_{6}X_{w-4}+\frac{7w}{72}X_{8}X_{w-6}.$

∎

######  Corollary 4.3.

The Kaneko–Koike’s conjecture is true for depth $1$ extremal forms.

###### Proof.

First, the conjecture holds for $X_{6}$ and $X_{8}$ by Lemma 4.1, and
$X_{10}=E_{4}X_{6}$ shows that $X_{10}$ is also completely positive. Now,
assume that $X_{k}$ is completely positive for $k\leq w-2$. Then (44)w implies
that $X_{w}$ is also completely positive. Combining it with Proposition 3.9
(recall that the vanishing order of $X_{w}$ at the cusp is
$\frac{w}{6}>\frac{w-1}{12}$) shows $X_{w+2}\in\mathcal{QM}_{w+2}^{1,++}$, and
we also get $X_{w+4}\in\mathcal{QM}_{w+4}^{1,++}$ from (41), since $E_{4}$ has
nonnegative Fourier coefficients. ∎

######  Remark 4.4.

We also have the following relations (for $6|w$):

$\displaystyle X_{w+2}^{\prime}$
$\displaystyle=\frac{5w}{72}X_{6}X_{w-2}+\frac{7w}{12}X_{8}X_{w-4},$ (46)
$\displaystyle X_{w+4}^{\prime}$
$\displaystyle=240X_{6}X_{w}+\frac{7w}{72}X_{8}X_{w-2}+\frac{5w}{72}X_{10}X_{w-4}$
(47)

which can be proven similarly and also provide alternative proof of Corollary
4.3.

###  4.3. Extremal forms of depth $2$

For even $w\geq 4$ and $w\neq 6$, the depth $2$ (normalized) extremal forms
$X_{w,2}$ satisfy the following recurrence relations [7]444There’s a minor
error in [6]. We need to replace $w^{2}$ with $(w+4)^{2}$ on the numerator to
make sure that $X_{w+4,2}$ is normalized, i.e. its leading coefficient is $1$.
We fix this in (48).: $X_{4,2}=\frac{E_{4}-E_{2}^{2}}{288}$ and

$\displaystyle X_{w+4,2}$
$\displaystyle=\frac{3(w+4)^{2}}{16(w+1)(w+2)^{2}(w+3)}\left(\frac{w(w+1)}{36}E_{4}X_{w,2}-\partial_{w-2}^{2}X_{w,2}\right)$
(48) $\displaystyle X_{w+2,2}$
$\displaystyle=\frac{6}{w+1}\partial_{w-2}X_{w,2}.$ (49)
$\displaystyle=\frac{3w^{2}}{16(w^{2}-1)(w-6)^{2}}\left(\frac{(w-4)(w-5)}{36}E_{4}X_{w-2,2}-\partial_{w-4}^{2}X_{w-2,2}\right)$
(50)

The vanishing order of $X_{w,2}$ at the cusp is $\lfloor\frac{w}{4}\rfloor$.
Also, $X_{w}$ (for $4|w$) is a solution of the differential equation

$X_{w,2}^{\prime\prime\prime}-\frac{w}{4}E_{2}X_{w,2}^{\prime\prime}+\frac{w(w-1)}{4}E_{2}^{\prime}X_{w,2}^{\prime}-\frac{w(w-1)(w-2)}{24}E_{6}X_{w,2}=0.$
(51)

Unfortunately, we could not find a similar recurrence relations as (44) in the
case of depth $2$. However, we found the following (exceptional) identities
that verify the conjecture for depth $2$ and weight $\leq 14$, which can be
checked directly.

######  Proposition 4.5.

We have the following identities:

$\displaystyle X_{8,2}^{\prime}$ $\displaystyle=2X_{4,2}X_{6,1},$ (52)
$\displaystyle X_{10,2}^{\prime}$
$\displaystyle=\frac{8}{9}X_{4,2}X_{8,1}+\frac{10}{9}X_{6,1}^{2},$ (53)
$\displaystyle X_{12,2}^{\prime}$ $\displaystyle=3X_{6,1}X_{8,2},$ (54)
$\displaystyle X_{14,2}^{\prime}$ $\displaystyle=3X_{4,2}X_{12,1}.$ (55)

Especially, $X_{w,2}$ is completely positive for $w\leq 14$.

###### Proof.

Complete positivity follows from the identities and Proposition 3.1. ∎

##  5\. 8-dimensional inequalities

Now we are ready to give a new proof of Theorem 1.1. Define

$\displaystyle F(z)$ $\displaystyle=(E_{2}(z)E_{4}(z)-E_{6}(z))^{2}$ (56)
$\displaystyle G(z)$
$\displaystyle=H_{2}(z)^{3}(2H_{2}(z)^{2}+5H_{2}(z)H_{4}(z)+5H_{4}(z)^{2}).$
(57)

Then we can check $F(z)=\Delta(z)\phi_{0}(z)$ and
$G(z)=-2\Delta(z)\psi_{S}(z)$, where the second identity on $G(z)$ follows
from the Jacobi identity and (24). Since $\Delta(it)>0$, the inequalities (3)
and (4) are equivalent to

$\displaystyle F(it)+\frac{18}{\pi^{2}}G(it)$ $\displaystyle>0,$ (58)
$\displaystyle F(it)-\frac{18}{\pi^{2}}G(it)$ $\displaystyle<0.$ (59)

As already mentioned in [14], $F(it)=9E_{4}^{\prime}(it)^{2}=9(240\sum_{n\geq
1}n\sigma_{3}(n)e^{-2\pi nt})^{2}>0$, and $G(it)>0$ follows from
$\Theta_{2}(it)>0$ and $\Theta_{4}(it)>0$. Note that $F$ can be also written
as $F=720^{2}X_{6,1}^{2}$.

As we mentioned before, our proof of the “hard” inequality (59) is based on
the following observations (Figure 1).

######  Proposition 5.1.

$\lim_{t\to 0^{+}}\frac{F(it)}{G(it)}=\frac{18}{\pi^{2}}.$ (60)

######  Proposition 5.2.

The function $t\mapsto\frac{F(it)}{G(it)}$ is strictly decreasing on $t>0$.

It is clear that the inequality (59) follows from Proposition 5.1 and 5.2.

###### Proof of Proposition 5.1.

We have

$\lim_{t\to 0^{+}}\frac{F(it)}{G(it)}=\lim_{t\to\infty}\frac{F(i/t)}{G(i/t)}.$

By using the transformation laws of Eisenstein series and the thetanull
functions (10)-(12) and (21), we get

$\displaystyle F\left(\frac{i}{t}\right)$
$\displaystyle=t^{12}F(it)-\frac{12t^{11}}{\pi}(E_{2}(it)E_{4}(it)-E_{6}(it))E_{4}(it)+\frac{36t^{10}}{\pi^{2}}E_{4}(it)^{2},$
$\displaystyle G\left(\frac{i}{t}\right)$
$\displaystyle=t^{10}H_{4}(it)^{3}(2H_{4}(it)^{2}+5H_{4}(it)H_{2}(it)+5H_{2}(it)^{2}).$

Since $F$, $E_{2}E_{4}-E_{6}$ and $H_{2}$ are cusp forms, we have
$\lim_{t\to\infty}t^{k}A(it)=0$ when $A(z)$ is one of these forms and $k\geq
0$. From $\lim_{t\to\infty}E_{4}(it)=1=\lim_{t\to\infty}H_{4}(it)$, we get

$\displaystyle\lim_{t\to\infty}\frac{F(i/t)}{G(i/t)}$
$\displaystyle=\lim_{t\to\infty}\frac{t^{12}F(it)-\frac{12t^{11}}{\pi}(E_{2}(it)E_{4}(it)-E_{6}(it))E_{4}(it)+\frac{36t^{10}}{\pi^{2}}E_{4}(it)^{2}}{t^{10}H_{4}(it)^{3}(2H_{4}(it)^{2}+5H_{4}(it)H_{2}(it)+5H_{2}(it)^{2})}$
$\displaystyle=\lim_{t\to\infty}\frac{t^{2}F(it)-\frac{12t}{\pi}(E_{2}(it)E_{4}(it)-E_{6}(it))E_{4}(it)+\frac{36}{\pi^{2}}E_{4}(it)^{2}}{H_{4}(it)^{3}(2H_{4}(it)^{2}+5H_{4}(it)H_{2}(it)+5H_{2}(it)^{2})}$
$\displaystyle=\frac{18}{\pi^{2}}.$

∎

###### Proof of Proposition 5.2.

It is enough to show that the derivative

$\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{F(it)}{G(it)}\right)=-2\pi\frac{F^{\prime}(it)G(it)-F(it)G^{\prime}(it)}{F(it)G(it)}$

is negative, which is equivalent to the positivity of (recall that $F(it)>0$
and $G(it)>0$)

$F^{\prime}G-FG^{\prime}=(\partial_{10}F)G-F(\partial_{10}G)=:\mathcal{L}_{1,0}$
(61)

that has weight $24$, level $\Gamma(2)$, and depth $2$. Here we give two
different proofs of $\mathcal{L}_{1,0}>0$.

_First proof_. Using (33)-(35) and (36), one can check that the inequality
(61) nicely factors as

$(H_{2}+H_{4})^{2}H_{4}^{2}(E_{2}E_{4}-E_{6})\left(E_{4}-\frac{1}{2}E_{2}(H_{2}+2H_{4})\right)>0$
(62)

for $z=it$ with $t>0$. Since the first three factors are all positive, it is
enough to prove that the last factor is positive, i.e.

$E_{4}(it)-\frac{1}{2}E_{2}(it)(H_{2}(it)+2H_{4}(it))>0.$ (63)

Using (29), we can rewrite the inequality (63) as

$E_{4}(it)-E_{2}(it)(2E_{2}(2it)-E_{2}(it))>0,$

and the left hand side can be written as

$(E_{4}(it)-E_{4}(2it))+(E_{4}(2it)-E_{2}(2it)^{2})+(E_{2}(it)-E_{2}(2it))^{2}$
(64)

where each term is positive ($E_{4}(it)$ is monotone decreasing in $t$,
$E_{4}-E_{2}^{2}=3E_{4}^{\prime}$, and the square is nonnegative).

_Second proof_. We can directly check that $F$ and $G$ satisfy the following
differential equations:

$\displaystyle\partial_{10}^{2}F$ $\displaystyle=\frac{5}{6}E_{4}F+a\Delta
X_{4,2},$ (65) $\displaystyle\partial_{10}^{2}G$
$\displaystyle=\frac{5}{6}E_{4}G-b\Delta H_{2},$ (66)

where $a=172800$ and $b=640$. Combining these with (32), we get

$\partial_{22}\mathcal{L}_{1,0}=(\partial_{10}^{2}F)G-F(\partial_{10}^{2}G)=\Delta(aX_{4,2}G+bH_{2}F)>0.$
(67)

Since $\mathcal{L}_{1,0}$ has a Fourier expansion

$\mathcal{L}_{1,0}=5308416000q^{\frac{7}{2}}+50960793600q^{\frac{9}{2}}-528718233600q^{\frac{11}{2}}+O(q^{\frac{13}{2}})$

and the first nonzero coefficient is positive, the positivity follows from
Corollary 3.7. ∎

######  Remark 5.3.

For $k\in\mathbb{Z}$, let $L_{2,k}:=\partial_{k}^{2}-\frac{k(k+2)}{144}E_{4}$
be the type $(k,k+4)$ modular linear differential operator defined in [12]
that is originated from [11]. Then the identities (65) and (66) show
$L_{2,10}F>0$ and $L_{2,10}G<0$.

##  6\. 24-dimensional inequalities

We give a similar proof of Theorem 1.2. The _easy_ inequality (7) is not easy
as (3), but it almost directly follows from Corollary 3.7. The idea for the
proof of the inequality (8) is the same as that of (4), and (9) is more
involved. We will abuse notation and write $F=-\Delta^{2}\varphi$ and
$G=-\Delta^{2}\psi_{S}$ for the numerators of (5) and (6). Then the
inequalities (7), (8) and (9) are equivalent to

$\displaystyle F(it)+\frac{432}{\pi^{2}}G(it)$ $\displaystyle>0,$ (68)
$\displaystyle F(it)-\frac{432}{\pi^{2}}G(it)$ $\displaystyle<0,$ (69)
$\displaystyle
t^{10}\left(-\frac{F(i/t)}{\Delta(i/t)^{2}}+\frac{432}{\pi^{2}}\frac{G(i/t)}{\Delta(i/t)^{2}}\right)$
$\displaystyle\geq\frac{725760}{\pi}e^{2\pi t}\left(t-\frac{10}{3\pi}\right).$
(70)

###  6.1. “Easy” inequality (68)

As in the $d=8$ case, we can prove (68) by proving the _easy_ inequalities
$F(it)>0$ and $G(it)>0$ separately. Especially, the second inequality directly
follows from its expression (which is already mentioned in [3, p. 1028]).
However, the other inequality $F(it)>0$ is less trivial, although it follows
from Corollary 3.7 and the following surprising identity.

######  Lemma 6.1.

We have

$\partial_{14}F=6706022400\cdot X_{6,1}X_{12,1}\in\mathcal{QM}_{18}^{2,++}.$

###### Proof.

The identity follows from direct calculations. (Complete) positivity follows
from Theorem 4.2. ∎

######  Corollary 6.2.

$F(it)>0$ for all $t>0$.

###### Proof.

This follows from Lemma 6.1 and Corollary 3.7. Note that $F$ has a Fourier
expansion

$F=3657830400q^{3}+138997555200q^{4}+2567796940800q^{5}+O(q^{6})$

and its first nonzero Fourier coefficient is positive. ∎

######  Remark 6.3.

$F$ is a constant multiple of $f_{16}$ that appears in the family of
Feigenbaum–Grabner–Hardin [6, Proposition 5.1]. The authors already proved
that the functions $f_{w}$ are _completely_ positive for $w\leq 94$ [6, Remark
6.3], and they conjectured that all the forms in the family are completely
positive [6, Conjecture 1]. However, their proof uses approximations based on
Jenkins and Rouse’s explicit bound on Fourier coefficients [9].

###  6.2. “Hard” inequality (69)

Figure 2. Graph of the quotient $F(it)/G(it)$ as a function in $t>0$.

The previous approach we used in the proof of (59) also works for (69), as we
can expect from Figure 2. It follows from the following two propositions.

######  Proposition 6.4.

$\lim_{t\to 0^{+}}\frac{F(it)}{G(it)}=\frac{432}{\pi^{2}}.$ (71)

######  Proposition 6.5.

The function $t\mapsto\frac{F(it)}{G(it)}$ is strictly decreasing on $t>0$.

The computation of the limit in Proposition (6.4) follows the same lines as
the proof of Proposition 5.1, and we omit the details.

###### Proof of Proposition 6.5.

The idea is similar to that of 5.2 (second proof). It is enough to show that

$F^{\prime}G-FG^{\prime}=(\partial_{14}F)G-F(\partial_{14}G)=:\mathcal{L}_{1,0}$

is positive, which has weight $32$, level $\Gamma(2)$, and depth $2$. $F$ and
$G$ satisfy the following differential equations similar to (65) and (66):

$\displaystyle\partial_{14}^{2}F$ $\displaystyle=\frac{14}{9}E_{4}F+c\Delta
X_{8,2}$ (72) $\displaystyle\partial_{14}^{2}G$
$\displaystyle=\frac{14}{9}E_{4}G$ (73)

where $c=5486745600$. If we take the Serre derivative $\partial_{30}$, by
(32), (72), and (73), we have

$\partial_{30}\mathcal{L}_{1,0}=\mathcal{L}_{2,0}:=(\partial_{14}^{2}F)G-F(\partial_{14}^{2}G)=c\Delta
X_{8,2}G>0$ (74)

and $\mathcal{L}_{1,0}$ has a Fourier expansion

$\mathcal{L}_{1,0}=13424296093286400q^{\frac{11}{2}}+494781198866841600q^{\frac{13}{2}}+O(q^{\frac{15}{2}})$

where the first nonzero Fourier coefficient is positive. Hence Corollary 3.7
applies and we get the positivity of $\mathcal{L}_{1,0}$. ∎

######  Remark 6.6.

As in $d=8$ case, (72) and (73) are about the action of the operator
$L_{2,14}$ of type $(14,18)$ in [11, 12] on $F$ and $G$. Especially, $G$ is a
solution of the linear differential equation $L_{2,14}G=0$.

###  6.3. “Harder” inequality (70)

The last inequality (70) is more involved than the previous inequalities
because of the presence of the non-modular terms $t^{10}$ and $e^{2\pi t}$. We
first replace the exponential term $e^{2\pi t}$ with $1/\Delta$ using the
following inequality.

######  Lemma 6.7.

For $t>0$, we have

$\Delta(it)<e^{-2\pi t}.$ (75)

###### Proof.

This directly follows from the product formula of $\Delta$,

$\Delta(it)=e^{-2\pi t}\prod_{n\geq 1}(1-e^{-2\pi nt})^{24}<e^{-2\pi t}.$

∎

The inequality (70) is true for $1\leq t\leq\frac{10}{3\pi}$, since the left
hand side (resp. the right hand side) is nonnegative (resp. nonpositive) on
this range (by (69)). Hence it is enough to show for $t>\frac{10}{3\pi}$. On
this range, we can bound the right hand side of (70) with (75)

$\frac{725760}{\pi}e^{2\pi
t}\left(t-\frac{10}{3\pi}\right)<\frac{725760}{\pi}\frac{1}{\Delta(it)}\left(t-\frac{10}{3\pi}\right)=\frac{725760}{\pi}\frac{t^{12}}{\Delta(i/t)}\left(t-\frac{10}{3\pi}\right)$

and the inequality reduces to

$\displaystyle
t^{10}\left(-\frac{F(i/t)}{\Delta(i/t)^{2}}+\frac{432}{\pi^{2}}\frac{G(i/t)}{\Delta(i/t)^{2}}\right)$
$\displaystyle>\frac{725760}{\pi}\frac{t^{12}}{\Delta(i/t)}\left(t-\frac{10}{3\pi}\right)$
$\displaystyle\Leftrightarrow\frac{1}{t^{2}}\left(-\frac{F(i/t)}{\Delta(i/t)}+\frac{432}{\pi^{2}}\frac{G(i/t)}{\Delta(i/t)}\right)$
$\displaystyle>\frac{725760}{\pi}\left(t-\frac{10}{3\pi}\right)$

for $t>\frac{10}{3\pi}$. If we replace $t$ by $1/t$, the last inequality
becomes

$\displaystyle
t^{3}\left(-\frac{F(it)}{\Delta(it)}+\frac{432}{\pi^{2}}\frac{G(it)}{\Delta(it)}\right)$
$\displaystyle>\frac{725760}{\pi}\left(1-\frac{10t}{3\pi}\right)$
$\displaystyle\Leftrightarrow\frac{432}{\pi^{2}}-\frac{F(it)}{G(it)}$
$\displaystyle>\frac{725760\Delta(it)}{G(it)}\left(\frac{1}{\pi
t^{3}}-\frac{10}{3\pi^{2}t^{2}}\right)$ (76)

for $0<t<\frac{3\pi}{10}$. From Proposition 6.5, we know that the left hand
side of (76) is monotone increasing in $t$. Our main observation is that the
difference between two sides of (76) is also monotone increasing (Figure 3).

Figure 3. Graph of $\mathrm{LHS}(t)$, $\mathrm{RHS}(t)$, and
$g(t)=\mathrm{LHS}(t)-\mathrm{RHS}(t)$ of (76).

######  Proposition 6.8.

The function

$g(t):=\frac{432}{\pi^{2}}-\frac{F(it)}{G(it)}-\frac{725760\Delta(it)}{G(it)}\left(\frac{1}{\pi
t^{3}}-\frac{10}{3\pi^{2}t^{2}}\right)$

is monotone increasing in $t$ for $0<t<\frac{3\pi}{10}$ and $\lim_{t\to
0^{+}}g(t)=0$. Especially, we have $g(t)>0$ for all $0<t<\frac{3\pi}{10}$.

###### Proof.

After writing the limit as $\lim_{t\to 0^{+}}g(t)=\lim_{t\to\infty}g(1/t)$, we
can compute limit from Proposition 6.4 and the fact that
$\Delta|_{12}S=\Delta$ is a cusp form but
$G|_{14}S=-H_{4}^{5}(7H_{2}^{2}+7H_{2}H_{4}+2H_{4}^{2})$ is not, so the third
term in $g(t)$ vanishes as $t\to 0^{+}$. We will omit the details and focus on
the monotonicity part. We have

$\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{F(it)}{G(it)}\right)=-2\pi\frac{\mathcal{L}_{1,0}(it)}{G(it)^{2}}$

and by $\Delta^{\prime}=E_{2}\Delta$,

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\left[\frac{\Delta(it)}{G(it)}\left(\frac{1}{\pi
t^{3}}-\frac{10}{3\pi^{2}t^{2}}\right)\right]$
$\displaystyle=(-2\pi)\frac{\Delta(it)(E_{2}(it)G(it)-G^{\prime}(it))}{G(it)^{2}}\left(\frac{1}{\pi
t^{3}}-\frac{10}{3\pi^{2}t^{2}}\right)+\frac{\Delta(it)}{G(it)}\left(-\frac{3}{\pi
t^{4}}+\frac{20}{3\pi^{2}t^{3}}\right)$
$\displaystyle=(2\pi)\frac{\Delta(it)}{G(it)^{2}}\left[(\partial_{12}G)(it)\left(\frac{1}{\pi
t^{3}}-\frac{10}{3\pi^{2}t^{2}}\right)-G(it)\left(\frac{3}{2\pi^{2}t^{4}}-\frac{10}{3\pi^{3}t^{3}}\right)\right],$

so $\mathrm{d}g/\mathrm{d}t>0$ if and only if (after factoring out $1/G^{2}$)

$\widetilde{\mathcal{L}}_{1,0}(it):=\mathcal{L}_{1,0}(it)-725760\Delta(it)\left[(\partial_{12}G)(it)\left(\frac{1}{\pi
t^{3}}-\frac{10}{3\pi^{2}t^{2}}\right)-G(it)\left(\frac{3}{2\pi^{2}t^{4}}-\frac{10}{3\pi^{3}t^{3}}\right)\right]>0.$

We have $\widetilde{\mathcal{L}}_{1,0}(\frac{3\pi i}{10})>0$, since
Proposition 6.5 gives $\mathcal{L}_{1,0}(\frac{3\pi i}{10})>0$ and when
$t=\frac{3\pi}{10}$,

$(\partial_{12}G)(it)\left(\frac{1}{\pi
t^{3}}-\frac{10}{3\pi^{2}t^{2}}\right)-G(it)\left(\frac{3}{2\pi^{2}t^{4}}-\frac{10}{3\pi^{3}t^{3}}\right)=-G\left(\frac{3\pi
i}{10}\right)\cdot\left(\frac{5000}{81\pi^{6}}\right)<0.$

From Proposition 3.6 (see also Remark 3.8), it is enough to show that its
Serre derivative

$\partial_{30}\widetilde{\mathcal{L}}_{1,0}(it)=\widetilde{\mathcal{L}}_{1,0}^{\prime}(it)-\frac{5}{2}E_{2}(it)\widetilde{\mathcal{L}}_{1,0}(it)=-\frac{1}{2\pi}\frac{\mathrm{d}\widetilde{\mathcal{L}}_{1,0}(it)}{\mathrm{d}t}-\frac{5}{2}E_{2}(it)\widetilde{\mathcal{L}}_{1,0}(it)$

is positive (i.e. $t\mapsto\widetilde{\mathcal{L}}_{1,0}(it)/\eta(it)^{60}$ is
a monotone decreasing function in $t$) on $0<t<\frac{3\pi}{10}$. Recall
$\partial_{30}\mathcal{L}_{1,0}=c\Delta X_{8,2}G$ (74). Using (32),
$\partial_{12}\Delta=0$, and (73), one can check that the Serre derivative of
the second term of $\widetilde{\mathcal{L}}_{1,0}$ is $725760\Delta$ times

$\displaystyle\partial_{18}\left[(\partial_{12}G)(it)\left(\frac{1}{\pi
t^{3}}-\frac{10}{3\pi^{2}t^{2}}\right)-G(it)\left(\frac{3}{2\pi^{2}t^{4}}-\frac{10}{3\pi^{3}t^{3}}\right)\right]$
$\displaystyle=\left[\frac{37E_{4}(it)-E_{2}(it)^{2}}{24}\left(\frac{1}{\pi
t^{3}}-\frac{10}{3\pi^{2}t^{2}}\right)+E_{2}(it)\left(\frac{3}{4\pi^{2}t^{4}}-\frac{5}{3\pi^{3}t^{3}}\right)-\left(\frac{3}{\pi^{3}t^{5}}-\frac{5}{\pi^{4}t^{4}}\right)\right]G(it),$
(77)

so $\Delta G$ factors out from
$\partial_{30}\widetilde{\mathcal{L}}_{1,0}(it)>0$ and it reduces to the
positivity of

$7560X_{8,2}(it)-\frac{37E_{4}(it)-E_{2}(it)^{2}}{24}\left(\frac{1}{\pi
t^{3}}-\frac{10}{3\pi^{2}t^{2}}\right)-E_{2}(it)\left(\frac{3}{4\pi^{2}t^{4}}-\frac{5}{3\pi^{3}t^{3}}\right)+\left(\frac{3}{\pi^{3}t^{5}}-\frac{5}{\pi^{4}t^{4}}\right).$
(78)

Let $h(t)$ be the above function in (78). Then

$\displaystyle t^{-8}h\left(\frac{1}{t}\right)$
$\displaystyle=7560\left(X_{8,2}(it)+\frac{7E_{2}(it)E_{4}(it)-E_{6}(it)}{30240\pi
t}-\frac{E_{4}(it)}{1440\pi^{2}t^{2}}\right)$
$\displaystyle\quad-\frac{1}{24}\left(37E_{4}(it)-E_{2}(it)^{2}+\frac{12E_{2}(it)}{\pi
t}-\frac{36}{\pi^{2}t^{2}}\right)\left(\frac{1}{\pi
t}-\frac{10}{3\pi^{2}t^{2}}\right)$
$\displaystyle\quad-\left(-E_{2}(it)+\frac{6}{\pi
t}\right)\left(\frac{3}{4\pi^{2}t^{2}}-\frac{5}{3\pi^{3}t^{3}}\right)+\left(\frac{3}{\pi^{3}t^{3}}-\frac{5}{\pi^{4}t^{4}}\right)$
$\displaystyle=7560X_{8,2}(it)$ $\displaystyle\quad+\frac{1}{\pi
t}\left(\frac{7E_{2}(it)E_{4}(it)-E_{6}(it)}{4}-\frac{37E_{4}(it)-E_{2}(it)^{2}}{24}\right)+\frac{1}{\pi^{2}t^{2}}\left(-\frac{4E_{4}(it)+5E_{2}(it)^{2}}{36}+\frac{E_{2}(it)}{4}\right)$
(79)

and since $X_{8,2}$ is (completely) positive, it is enough to show that (79)
is positive, i.e. show the following _nonhomogeneous_ inequality (after
factoring out $1/\pi t$)

$\frac{7E_{2}(it)E_{4}(it)-E_{6}(it)}{4}-\frac{37E_{4}(it)-E_{2}(it)^{2}}{24}-\frac{1}{\pi
t}\left(\frac{5E_{2}(it)^{2}+4E_{4}(it)}{36}-\frac{1}{4}E_{2}(it)\right)>0$
(80)

for $t\geq\frac{10}{3\pi}$. We can further reduce it to an inequality with
only quasimodular terms (i.e. no rational terms) with the following lemma.

######  Lemma 6.9.

The following (nonhomogeneous) quasimodular forms are completely positive:

$\displaystyle J_{1}$
$\displaystyle=\frac{5}{36}E_{2}^{2}+\frac{1}{9}E_{4}-\frac{1}{4}E_{2},$ (81)
$\displaystyle J_{2}$ $\displaystyle=E_{2}-E_{6}.$ (82)

###### Proof.

By using the Fourier expansions of $E_{2}$, $E_{4}$, $E_{6}$, and (14), we can
compute the Fourier expansions of the above forms explicitly as

$\displaystyle J_{1}$
$\displaystyle=\frac{5}{3}E_{2}^{\prime}-\frac{1}{4}E_{2}+\frac{1}{4}E_{4}=\sum_{n\geq
1}(60\sigma_{3}(n)-40n\sigma_{1}(n)+6\sigma_{1}(n))q^{n},$ $\displaystyle
J_{2}$ $\displaystyle=\sum_{n\geq 1}(504\sigma_{5}(n)-24\sigma_{1}(n))q^{n}.$

The complete positivity of $J_{1}$ follows from the trivial estimates
$\sigma_{3}(n)>n^{3}$ and $\sigma_{1}(n)\leq 1+2+\cdots+n=\frac{n(n+1)}{2}\leq
n^{2}$, and that of $J_{2}$ follows from
$504\sigma_{5}(n)-24\sigma_{1}(n)=\sum_{d|n}(504d^{5}-24d)>0$. ∎

By Lemma 6.9, for $t\geq\frac{10}{3\pi}$ we have

$\displaystyle\frac{7E_{2}(it)E_{4}(it)-E_{6}(it)}{4}-\frac{37E_{4}(it)-E_{2}(it)^{2}}{24}-\frac{1}{\pi
t}\left(\frac{5E_{2}(it)^{2}+4E_{4}(it)}{36}-\frac{E_{2}(it)}{4}\right)$
$\displaystyle>\frac{7E_{2}(it)E_{4}(it)-E_{6}(it)}{4}-\frac{37E_{4}(it)-E_{2}(it)^{2}}{24}-\frac{3}{10}\left(\frac{5E_{2}(it)^{2}+4E_{4}(it)}{36}-\frac{E_{2}(it)}{4}\right)$
(83)
$\displaystyle=\frac{7E_{2}(it)E_{4}(it)-E_{6}(it)}{4}-\frac{63}{40}E_{4}(it)+\frac{3}{40}E_{2}(it)$
$\displaystyle>\frac{7E_{2}(it)E_{4}(it)-E_{6}(it)}{4}-\frac{63}{40}E_{4}(it)+\frac{3}{40}E_{6}(it)$
(84)
$\displaystyle=\frac{7}{4}\left(E_{2}(it)E_{4}(it)-\frac{1}{10}E_{6}(it)-\frac{9}{10}E_{4}(it)\right)=:\frac{7}{4}J_{3}$
(85)

where the positivity of $J_{1}$ and $J_{2}$ are used in (83) and (84),
respectively. Now, we can prove the positivity of (85) (i.e. $J_{3}$) as
follows. As in Lemma 6.9, we can compute the Fourier expansion of $J_{3}$ as

$\displaystyle J_{3}$
$\displaystyle=E_{2}E_{4}-\frac{1}{10}E_{6}-\frac{9}{10}E_{4}$
$\displaystyle=3E_{4}^{\prime}+\frac{9}{10}E_{6}-\frac{9}{10}E_{4}$
$\displaystyle=\sum_{n\geq
1}\left(720n\sigma_{3}(n)-\frac{2268}{5}\sigma_{5}(n)-216\sigma_{3}(n)\right)q^{n}$
$\displaystyle=:\sum_{n\geq 1}a_{n}q^{n}.$

We have $a_{1}=\frac{252}{5}>0$. For $n\geq 2$,

$n\sigma_{3}(n)\leq
n(1^{3}+2^{3}+\cdots+n^{3})=\frac{n^{3}(n+1)^{2}}{4}\leq\frac{9}{16}n^{5}<\frac{9}{16}\sigma_{5}(n)<\frac{2268}{720\cdot
5}\sigma_{5}(n)$

and we get $a_{n}<0$. From this observation, the function

$t\mapsto e^{2\pi t}J_{3}(it)=a_{1}+\sum_{n\geq 1}a_{2}e^{-2\pi(n-1)t}$

is monotone increasing, so

$e^{2\pi t}J_{3}(it)\geq
e^{2\pi}J_{3}(i)=e^{2\pi}\left(\frac{3}{\pi}-\frac{9}{10}\right)E_{4}(i)>0\Rightarrow
J_{3}(it)>0$

for $t\geq 1$, hence for $t>\frac{10}{3\pi}$. ∎

######  Remark 6.10.

The estimate $\pi<\frac{10}{3}$ is used in the proof (e.g. (83)), and this can
be verified _geometrically_ (without calculators) by considering the area of a
regular octagon circumscribed to a unit circle:

$\pi<8\tan\left(\frac{\pi}{8}\right)=8(\sqrt{2}-1)<\frac{10}{3}.$

######  Remark 6.11.

The inequalities (76) and (78) are “homogeneous” if one regard
$\frac{1}{t}=\frac{i}{z}$ and $\frac{1}{\pi}$ as “weight 1” objects, which
makes sense if we consider the transformation law of $E_{2}$ (10). However, we
had to flip it under $t\leftrightarrow\frac{1}{t}$ and prove the
nonhomogeneous inequalities (80)-(85) instead. It would be interesting if one
can prove the inequality (70) in a purely homogeneous way.

## References

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## Appendix

##  Appendix A Implementation details

All the codes are available in the GitHub repository
https://github.com/seewoo5/posqmf. Our SageMath codes are heavily based on the
current implementation of quasimodular forms, thanks to David Ayotte. In
SageMath, the ring of quasimodular forms of level $\Gamma_{0}(N)$ or
$\Gamma_{1}(N)$ are essentially implemented as polynomial rings of one
variable ($E_{2}$) with the ring of (genuine) modular forms as a coefficient
ring, based on [11]. The implementation of extremal quasimodular forms simply
follow the recurrence relations (44)-(47) and (48)-(50).

For the quasimodular forms of level $\mathrm{SL}_{2}(\mathbb{Z})$, we can
simply define the ring as QM = QuasiModularForms(1). However, for the ring of
quasimodular forms of level $\Gamma(2)$, we had to implement it ourselves
since the current implementation of the modular forms does not support the
level. The ring of quasimodular forms of level $\Gamma(2)$ is isomorphic to a
polynomial ring with three generators, namely $H_{2}=\Theta_{2}^{4}$,
$H_{4}=\Theta_{4}^{4}$, and $E_{2}$. So we simply define it as a polynomial
ring QM2.<H2,H4,E2_> = QQ[‘H2,H4,E2’], and implement functions that compute
the $q$-series, (Serre) derivatives, and plotting a graph of a given form in
$t$ for $z=it$. We use (31) to implement (Serre) derivatives as follows.

⬇

def qm2_weight(qm):

w = None

for (a, b, e) in qm.dict().keys():

if w is None:

w = 2 * a + 2 * b + 2 * e

else:

assert w == 2 * a + 2 * b + 2 * e, ”Not homogeneous”

return w

\pardef qm2_depth(qm):

dp = 0

for (_, _, e) in qm.dict().keys():

dp = max(e, dp)

return dp

\pardef qm2_derivative(qm):

r = QM2(0)

for (a, b, e), coeff in qm.dict().items():

r += (coeff/6) * H2^a * H4^b * ((a-2*b)*H2 + (2*a-b)*H4 + (a+b)*E2_) * E2_^e

if e >= 1:

r += coeff * H2^a * H4^b * e * E2_^(e-1) * (E2_^2 - E4_) / 12

return r

\pardef qm2_serre_derivative(qm, k=None):

s = qm2_depth(qm)

if k is None:

# Serre derivative that preserves depth

k = qm2_weight(qm) - qm2_depth(qm)

return qm2_derivative(qm) - (k / 12) * E2_ * qm

For $q$-series, we expressed them in qh $=q^{1/2}$ instead of q $=q$, since
the power series ring does not support non-integer powers. We also need a
function that embeds the ring QM ($\mathcal{QM}(\mathrm{SL}_{2}(\mathbb{Z}))$)
into the ring QM2 ($\mathcal{QM}(\Gamma(2))$), using (22) and (23) (l1_to_l2).

⬇

E4_ = H2^2 + H2 * H4 + H4^2

E6_ = (H2 + 2 * H4) * (2 * H2 + H4) * (H4 - H2) / 2

\pardef l1_to_l2(qm):

r = QM2(0)

for (d2, d4, d6), coeff in qm.polynomial.dict().items():

r += coeff * E2_^d2 * E4_^d4 * E6_^d6

return r

For the computations in the proof of “harder” inequality (9), we define
auxilary rings RQM and RQM2 corresponds to

$\mathcal{RQM}(\Gamma)=\mathcal{QM}(\Gamma)\left[\frac{1}{\pi},\frac{i}{z}\right]$

for $\Gamma=\mathrm{SL}_{2}(\mathbb{Z})$ and $\Gamma(2)$, by adding two formal
variables ip and ioz correspond to $1/\pi$ and $i/z=1/t$, respectively. As we
mentioned before, we regard these new elements as “weight 1” objects. Then we
extend the derivative $D$ on these rings using

$D\left(\frac{1}{\pi}\right)=0,\quad D\left(\frac{i}{z}\right)=\frac{1}{2\pi
i}\frac{\mathrm{d}}{\mathrm{d}z}\left(\frac{i}{z}\right)=\frac{1}{2}\frac{1}{\pi}\left(\frac{i}{z}\right)^{2}$

and the product rule.

⬇

# Level SL_2(Z)

RQM.<ip, ioz> = QM[’ip’,’ioz’] # ‘ip‘ = 1 / pi, ‘ioz‘ = i / z = 1 / t

\par# Weight

def rqm_weight(rqm):

w = 0

for (dip, dioz), qm in rqm.dict().items():

w = max(w, qm.weight() + dip + dioz)

return w

\pardef is_rqm_homogeneous(rqm):

w = None

for (dip, dioz), qm in rqm.dict().items():

w_ = qm.weight() + dip + dioz

if w is None:

w = w_

else:

if w != w_:

return False

return True

\par# Depth

def rqm_depth(rqm):

dp = 0

for qm in rqm.dict().values():

dp = max(dp, qm_depth(qm))

return dp

\pardef rqm_derivative(rqm):

r = 0

for (dip, dioz), qm in rqm.dict().items():

r += qm.derivative() * ip^dip * ioz^dioz

if dioz >= 1:

r += qm * ip^dip * dioz * ioz^(dioz - 1) * ((1/2) * ip * ioz^2)

return r

\par# Level \Gamma(2)

RQM2.<ip_, ioz_> = QM2[’ip’,’ioz’] # ‘ip‘ = 1 / pi, ‘ioz‘ = i / z

\par# Weight

def rqm2_weight(rqm):

w = 0

for (dip, dioz), qm in rqm.dict().items():

w = max(w, qm2_weight(qm) + dip + dioz)

return w

\pardef is_rqm2_homogeneous(rqm):

w = None

for (dip, dioz), qm in rqm.dict().items():

w_ = qm2_weight(qm) + dip + dioz

if w is None:

w = w_

else:

if w != w_:

return False

return True

\par# Depth

def rqm2_depth(rqm):

dp = 0

for qm in rqm.dict().values():

dp = max(dp, qm2_depth(qm))

return dp

\pardef rqm2_derivative(rqm):

r = 0

for (dip, dioz), qm in rqm.dict().items():

r += qm2_derivative(qm) * ip_^dip * ioz_^dioz

if dioz >= 1:

r += qm * ip_^dip * dioz * ioz_^(dioz - 1) * ((1/2) * ip_ * ioz_^2)

return r

We can also define $S$-actions on these rings, using the transformation laws
of Eisenstein series and thetanull functions. Note that, if input is a
quasimodular form without any rational terms, then the output is homogeneous.
Otherwise, the output may not be homogeneous in general. Also, we always
assume that the input is homogeneous. On each monomial, the action $|_{w}S$ on
$F\cdot(1/\pi)^{a}\cdot(i/z)^{b}$ where $F\in\mathcal{QM}_{w-a-b}(\Gamma)$
equals

$(F|_{w-a-b}S)\cdot(1/\pi)^{a}\cdot(i/(-1/z))^{b}\cdot
z^{-a-b}=(-1)^{(a+b)/2}\cdot(F|_{w-a-b}S)\cdot(1/\pi)^{a}\cdot(i/z)^{a}$

(we only deal with inputs of even weights, so $a+b$ is always even). We can
extract homogeneous components using rqm_homogeneous_comps and
rqm2_homogeneous_comps.

⬇

# For homogeneous inputs with rational terms

def rqm_S_action(rqm):

r = 0

assert is_rqm_homogeneous(rqm), ”Input is not homogeneous.”

for (dip, dioz), qm in rqm.dict().items():

r += (-1)^((dip + dioz)/2) * qm_S_action(qm) * ip^dip * ioz^dip

return r

\par# For homogeneous inputs with rational terms

def rqm2_S_action(rqm):

r = 0

assert is_rqm2_homogeneous(rqm), ”Input is not homogeneous.”

for (dip, dioz), qm in rqm.dict().items():

r += (-1)^((dip + dioz)/2) * qm2_S_action(qm) * ip_^dip * ioz_^dip

return r

\par# Extract each of homogeneous components

def rqm_homogeneous_comps(rqm):

r = dict()

for (dip, dioz), qm in rqm.dict().items():

qm_comps = qm.homogeneous_components()

for w_, qm_ in qm_comps.items():

w = w_ + dip + dioz

if w not in r:

r[w] = qm_ * ip^dip * ioz^dioz

else:

r[w] += qm_ * ip^dip * ioz^dioz

return r

\pardef qm2_homogeneous_comps(qm):

r = dict()

for (dh2, dh4, de2), coeff in qm.dict().items():

w = 2 * (dh2 + dh4 + de2)

if w not in r:

r[w] = coeff * H2^dh2 * H4^dh4 * E2_^de2

else:

r[w] += coeff * H2^dh2 * H4^dh4 * E2_^de2

return r

\pardef rqm2_homogeneous_comps(rqm):

r = dict()

for (dip, dioz), qm in rqm.dict().items():

qm_comps = qm2_homogeneous_comps(qm)

for w_, qm_ in qm_comps.items():

w = w_ + dip + dioz

if w not in r:

r[w] = qm_ * ip_^dip * ioz_^dioz

else:

r[w] += qm_ * ip_^dip * ioz_^dioz

return r

Note that these functions also help us for the limit computations such as
(60), (71), and the one in Proposition 6.8, since we essentially use
$S$-action to change the limits from $\lim_{t\to 0^{+}}$ to
$\lim_{t\to\infty}$. For example, the following code snippet recovers the
proof of Proposition 5.1, computing $F|_{12}S$ and $G|_{10}S$ and extract
their modular form components.

⬇

>>> F_8dS = qm_S_action(F_8d)

>>> G_8dS = qm2_S_action(G_8d)

>>> print_rqm(F_8dS, ”F_8d|S”)

F_8d|S

polynomial (E2^2*E4^2 - 2*E2*E4*E6 + E6^2) + (-12*E2*E4^2 +
12*E4*E6)*(1/π)*(i/z) + (36*E4^2)*(1/π)^2*(i/z)^2

weight 12

depth 2

>>> print_rqm2(G_8dS, ”G_8d|S”)

G_8d|S

polynomial (-5*H2^2*H4^3 - 5*H2*H4^4 - 2*H4^5)

weight 10

depth 0

>>> print(”F_8dS, ip^0 * ioz^0:”, F_8dS.coefficient([0, 0]))

F_8dS, ip^0 * ioz^0: 518400*q^2 + 18662400*q^3 + 255052800*q^4 +
1870387200*q^5 + O(q^6)

>>> print(”F_8dS, ip^1 * ioz^1:”, F_8dS.coefficient([1, 1]))

F_8dS, ip^1 * ioz^1: -8640*q - 2229120*q^2 - 56712960*q^3 - 570689280*q^4 -
3375043200*q^5 + O(q^6)

>>> print(”F_8dS, ip^2 * ioz^2:”, F_8dS.coefficient([2, 2]))

F_8dS, ip^2 * ioz^2: 36 + 17280*q + 2229120*q^2 + 37808640*q^3 + 285344640*q^4
+ 1350017280*q^5 + O(q^6)

>>> print(”G_8dS, ip^0 * ioz^0:”, qm2_q_series(QM2(G_8dS.coefficient([0, 0])),
10))

G_8dS, ip^0 * ioz^0: (-2) + (-240)*qh^2 + 10240*qh^3 + (-134640)*qh^4 +
1007616*qh^5 + (-5215680)*qh^6 + 20828160*qh^7 + (-69131760)*qh^8 +
199966720*qh^9 + Order(qh^10)

##  Appendix B Table of extremal forms

Table 1 gives first few extremal quasimodular forms of Kaneko and Koike [10].

$s$ | $w$ | $X_{w,s}$
---|---|---
$1$ | $6$ | $\dfrac{E_{2}E_{4}-E_{6}}{720}=q+18q^{2}+84q^{3}+292q^{4}+630q^{5}+\cdots$
$8$ | $\dfrac{-E_{2}E_{6}+E_{4}^{2}}{1008}=q+66q^{2}+732q^{3}+4228q^{4}+15630q^{5}+\cdots$
$10$ | $\dfrac{E_{2}E_{4}^{2}-E_{4}E_{6}}{720}=q+258q^{2}+6564q^{3}+66052q^{4}+390630q^{5}+\cdots$
$12$ | $\dfrac{-12E_{2}E_{4}E_{6}+5E_{4}^{3}+7E_{6}^{2}}{3991680}=q^{2}+56q^{3}+1002q^{4}+9296q^{5}+57708q^{6}+\cdots$
$14$ | $\dfrac{7E_{2}E_{4}^{3}+5E_{2}E_{6}^{2}-12E_{4}^{2}E_{6}}{4717440}=q^{2}+128q^{3}+4050q^{4}+58880q^{5}+525300q^{6}+\cdots$
$2$ | $4$ | $\dfrac{-E_{2}^{2}+E_{4}}{288}=q+6q^{2}+12q^{3}+28q^{4}+30q^{5}+\cdots$
$8$ | $\dfrac{-7E_{2}^{2}E_{4}+2E_{2}E_{6}+5E_{4}^{2}}{362880}=q^{2}+16q^{3}+102q^{4}+416q^{5}+1308q^{6}+\cdots$
$10$ | $\dfrac{5E_{2}^{2}E_{6}+2E_{2}E_{4}^{2}-7E_{4}E_{6}}{1088640}=q^{2}+\dfrac{104}{3}q^{3}+390q^{4}+2480q^{5}+11140q^{6}+\cdots$
$12$ | $\dfrac{-77E_{2}^{2}E_{4}^{2}+34E_{2}E_{4}E_{6}+50E_{4}^{3}-7E_{6}^{2}}{798336000}=q^{3}+\dfrac{51}{2}q^{4}+\dfrac{1422}{5}q^{5}+1944q^{6}+9714q^{7}+\cdots$
$14$ | $\dfrac{13E_{2}^{2}E_{4}E_{6}+E_{2}E_{4}^{3}-3E_{2}E_{6}^{2}-11E_{4}^{2}E_{6}}{415134720}=q^{3}+\dfrac{93}{2}q^{4}+810q^{5}+8004q^{6}+54474q^{7}+\cdots$
Table 1. Extremal forms of depth $\leq 2$ and weight $\leq 14$.


# Explaining Nonlinearities in Black Hole Ringdowns from Symmetries

Alex Kehagias<EMAIL_ADDRESS>Physics department, National Tecnical
University of Athens, 15780 Athens, Greece  CERN, Theoretical Physics
Department, Geneva, Switzerland  Davide Perrone<EMAIL_ADDRESS>Département de Physique Théorique, Université de Genève, 24 quai E. Ansermet,
CH-1211 Geneva, Switzerland  Antonio Riotto<EMAIL_ADDRESS>Département de Physique Théorique, Université de Genève, 24 quai E. Ansermet,
CH-1211 Geneva, Switzerland  Gravitational Wave Science Center (GWSC),
Université de Genève, 24 quai E. Ansermet, CH-1211 Geneva, Switzerland
Francesco Riva<EMAIL_ADDRESS>Département de Physique Théorique,
Université de Genève, 24 quai E. Ansermet, CH-1211 Geneva, Switzerland

###### Abstract

It has been recently pointed out that nonlinear effects are necessary to model
the ringdown stage of the gravitational waveform produced by the merger of two
black holes giving rise to a remnant Kerr black hole. We show that this
nonlinear behaviour is explained, both on the qualitative and quantitative
level, by near-horizon symmetries of the Kerr black hole within the Kerr/CFT
correspondence.

Introduction. Quasi Normal Modes (QNMs) provide a unique tool to investigate
the properties of Black Holes (BHs) whose understanding is one of the major
goals of gravitational wave astronomy Berti  _et al._ (2009). In the merger of
two BHs, during the final stage called ringdown, they dominate the BH response
to any kind of disturbance and their frequencies are uniquely determined by
the BH mass, spin and charge.

Gravitational Waves (GWs) during ringdown are well described by a
superposition of exponentially damped QNMs, labeled by two angular harmonic
numbers $(\ell,m)$ and an overtone number $n$. Their amplitude is denoted
$A_{(\ell,m,n)}$, while their oscillation frequency and decay timescale are
given by the real and imaginary parts of $\omega_{(\ell,m,n)}$. The GW strain
far from the BH source can be decomposed as

$\displaystyle h(u,\theta,\phi)$ $\displaystyle=$ $\displaystyle\sum_{\ell\geq
2}\sum_{|m|\leq\ell}h_{(\ell,m)}(u){}_{-2}Y_{(\ell,m)}(\theta,\phi),$
$\displaystyle h_{(\ell,m)}(u)$ $\displaystyle=$ $\displaystyle\sum_{n\geq
0}A_{(\ell,m,n)}e^{-i\omega_{(\ell,m,n)}(u-u_{\text{\tiny pk}})},$ (1)

where $u=(t-r)$ is the Bondi time, $u_{\text{\tiny pk}}$ is the time at which
the strain achieves its maximum value, and ${}_{-2}Y_{(\ell,m)}$ are the spin-
weight $s=-2$ spherical harmonics (the $s=+2$ mode with outgoing boundary
conditions is subleading at infinity Teukolsky (1973)). The GW strain produced
is generically modeled using first-order BH perturbation theory. However,
nonlinearities are an intrinsic property of general relativity and indeed it
has been recently pointed out that second-order effects are relevant to
describe ringdowns from BH merger simulations Cheung  _et al._ (2022); Mitman
_et al._ (2022) (see also Refs. London  _et al._ (2014); Ma  _et al._ (2022);
Lagos and Hui (2022)).

In particular, the second-order mode amplitude
$A^{(2,2,0)\times(2,2,0)}_{(4,4)}$ obtained from the square of the first-order
fundamental mode $(\ell,m)=(2,2)$ is comparable to or even larger than that of
the fundamental linear mode $(4,4)$. For the numerical simulations of
quasicircular mergers giving rise to a Kerr BH with spin $0.7$ (in units of
the BH mass and we set from now on $G_{N}=1$) Ref. Cheung  _et al._ (2022)
found

$\frac{\left|A^{(2,2,0)\times(2,2,0)}_{(4,4)}\right|}{\left|A_{(2,2,0)}\right|^{2}}=0.1637\pm
0.0018,$ (2)

where we have neglected the milder dependence on the BH mass ratio of the two
BH mergers. This result is consistent with what found in Ref. Mitman  _et al._
(2022) which quotes values in the interval $(0.15-0.2)$.

Restricting ourselves to the fundamental modes and noting that the second-
order QNM $(4,4)$ is sourced at second-order from the square of the $(2,2)$
mode with frequency $2\omega_{(2,2,0)}$, Eq. (2) can be written in the
suggestive form

$\frac{\langle h_{(2,2)}h_{(2,2)}h_{(4,4)}\rangle}{\langle
h^{2}_{(2,2)}\rangle^{2}}\simeq 0.1637\pm 0.0018.$ (3)

Similarly, Ref. Cheung  _et al._ (2022) found

$\frac{\left|A^{(2,2,0)\times(3,3,0)}_{(5,5)}\right|}{\left|A_{(2,2,0)}\right|\left|A_{(3,3,0)}\right|}=0.4735\pm
0.0062,$ (4)

again for a Kerr BH remnant of spin $\sim 0.7$. In order to correctly model
the BH ringdown one needs therefore to include nonlinear effects.

The goal of this paper is to explain the nonlinearities of the Kerr BH remnant
from symmetry arguments. Our starting point is the realization that the QNMs
are produced in the proximity of the BH horizon Maggiore (2018). In the region
very close to the horizon of an extreme Kerr BH one can set consistent
boundary conditions such that the asymptotic symmetry generators identify one
copy of the Virasoro algebra. This implies that the near-horizon quantum
states can be identified with those of (a chiral half of) a two-dimensional
conformal field theory (CFT) with finite temperature $T=1/2\pi$. This goes
under the name of the Kerr/CFT correspondence Guica  _et al._ (2009). Although
the CFT details are not exactly known, several nontrivial checks have been
studied in the past (see, for instance, Refs. Bredberg  _et al._ (2010);
Hartman  _et al._ (2010); Porfyriadis and Strominger (2014)), consolidating
the idea that there is a relation between universal properties of BHs and
CFTs, and trying also to extend the results to non-extremal cases Castro and
Larsen (2009); Matsuo  _et al._ (2010); Rasmussen (2010); Becker  _et al._
(2011).

If the correspondence is correct, the correlators of bulk fields can be
computed through the correlators of the corresponding boundary operators. In
turn, the latter are dictated by the CFT. In the case of the spin-2 strain,
the associated boundary operator is the two-dimensional energy momentum
tensor, whose correlator amplitudes are fixed by the central charge. This
simple reasoning will allow us to calculate the three-point correlators to
estimate the level of nonlinearities in the ringdowns from symmetry arguments.

Before launching ourselves in the technicalities, we briefly summarise the
basics of the Kerr/CFT correspondence. The expert reader on the subject can
skip this part.

Kerr/CFT. The Kerr BH with mass $M$ and angular momentum $J$ is described in
Boyer-Lindquist coordinates by the metric

$\displaystyle\differential s^{2}$ $\displaystyle=$
$\displaystyle-\frac{\Delta}{\rho^{2}}\left(\differential\hat{t}-a\sin^{2}\theta\right)^{2}+\frac{\rho^{2}}{\Delta}\differential\hat{r}^{2}+$
(5) $\displaystyle+$
$\displaystyle\frac{\sin^{2}\theta}{\rho^{2}}\left[(\hat{r}^{2}+a^{2})\differential\hat{\phi}-a\differential\hat{t}\right]^{2}+\rho^{2}\differential\theta^{2},$

where

$\Delta=\hat{r}^{2}-2M\hat{r}+a^{2},\quad\rho^{2}=\hat{r}^{2}+a^{2}\cos^{2}\theta,\quad
a=\frac{J}{M}.$ (6)

The two horizons are defined as the solutions to $\Delta(r)=0$:

$\hat{r}_{\pm}=M\pm\sqrt{M^{2}-a^{2}},$ (7)

and the Hawking temperature can be written as

$T_{H}=\frac{\hat{r}_{+}-\hat{r}_{-}}{8\pi M\hat{r}_{+}}\,.$ (8)

We consider the extremal case $a=M$ and the change of coordinates Bardeen and
Horowitz (1999)

$t=\frac{\lambda\hat{t}}{2M},\quad r=\frac{\hat{r}-M}{\lambda
M},\quad\phi=\hat{\phi}-\frac{\hat{t}}{2M},$ (9)

such that, in the limit $\lambda\to 0$ and keeping fixed the coordinates
$(t,r,\phi,\theta)$, one can zoom into a small region around the BH event
horizon $\hat{r}=M$. The resulting metric is the Near Horizon Extreme Kerr
(NHEK)

$\displaystyle\differential s^{2}$ $\displaystyle=$ $\displaystyle
2J\Gamma(\theta)\bigg{[}-r^{2}\differential t^{2}+\frac{\differential
r^{2}}{r^{2}}+\differential\theta^{2}+\Lambda^{2}(\theta)(\differential\phi+r\,\differential
t)^{2}\bigg{]},$ (10)
$\displaystyle\Gamma(\theta)=\frac{1+\cos^{2}\theta}{2},\quad\Lambda(\theta)=\frac{2\sin\theta}{1+\cos^{2}\theta},$

with $\phi\sim\phi+2\pi$ and $0\leq\theta\leq\pi$. For a generic value of
$\theta$ one deals with the geometry of a warped $\rm AdS_{3}$ with a ${\rm
SL}(2,\mathbb{R})\otimes U(1)$ symmetry group. At the specific value
$\theta_{0}$ such that $\Lambda(\theta_{0})=1$, the metric is that of gravity
on ${\rm AdS}_{3}$ Anninos  _et al._ (2009); Bengtsson and Sandin (2006).

At extremality, rotational perturbations along the angular azimuthal direction
correspond to excitations of the left sector of the dual CFT. The right modes
are not excited (unless one goes above extremality). On the other hand, the
left modes cannot be associated with the Hawking temperature $T_{H}$, which
vanishes for extremal Kerr BHs. Indeed, the Hartle–Hawking vacuum for quantum
fields in the region outside the Schwarzschild BH horizon (which gives rise to
a density matrix $e^{-\omega/T_{H}}$) cannot be used for the Kerr spacetime.
This is because in spacetimes lacking a globally defined timelike Killing
vector, such as the Kerr geometry, the Hartle–Hawking vacuum does not exist.
One can however use the Frolov-Thorne vacuum Frolov and Thorne (1989) which is
appropriately defined in the vicinity of a spinning BH horizon. Quantum fields
can be expanded in asymptotic energy and angular momentum eigenstates of the
operators $\partial_{\hat{t}}$ and $\partial_{\hat{\phi}}$ as

$\displaystyle\Phi_{s}(\hat{t},\hat{r},\theta,\hat{\phi})=\sum_{\omega,\ell,m}\Phi_{\omega\ell
m}e^{-i\omega\hat{t}+im\hat{\phi}}R_{\ell m}(\hat{r},\theta).$ (11)

In the near horizon coordinates ($t,r,\theta,\phi$), the asymptotic energy-
angular momentum eigenstates are expressed as

$\displaystyle e^{-i\omega\hat{t}+im\hat{\phi}}=e^{-in_{R}t+in_{L}\phi},$ (12)

where

$\displaystyle n_{L}=m,\qquad n_{R}=\frac{1}{\lambda}\left(2\omega
M-m\right).$ (13)

Then the Frolov-Thorne vacuum gives rise to a density matrix with a Boltzmann
weighting factor

$\displaystyle e^{-\frac{n_{L}}{T_{L}}-\frac{n_{R}}{T_{R}}},$ (14)

where the left- and right-temperatures are Guica  _et al._ (2009)

$\displaystyle T_{L}=\frac{r_{+}-M}{2\pi(r_{+}-a)},\qquad
T_{R}=\frac{r_{+}-M}{2\pi r_{+}\lambda}.$ (15)

In the extremal case (keeping $\lambda$ small, but finite), we end up with the
left-moving sector in the Frolov-Thorne temperature

$\displaystyle T_{L}=\frac{1}{2\pi},$ (16)

and a Boltzmann suppression factor $e^{-2\pi n_{L}}$, while the right sector
has $T_{R}=0$. At the same time, the geometry (10) has a boundary at
$r\to\infty$ (where boundary fields are a function of $t$ and $\phi$) and an
asymptotic symmetry group (with specific boundary conditions for the
perturbations of the metric in AdS) which extends the symmetry to half of a
Virasoro algebra Guica  _et al._ (2009). This result led to the Kerr/CFT
conjecture, where the NHEK BH can be described by a dual chiral CFT. According
to the correspondence, for every bulk field $\Phi$ there is a corresponding
boundary local operator ${\cal O}$ coupled to the boundary field $\Phi_{b}$
such that

$Z_{\rm AdS,\,eff}[\Phi]=e^{iS_{\rm eff}[\Phi]}=\left\langle{\rm
T}\,e^{\int_{\partial\rm AdS}\Phi_{b}\mathcal{O}}\right\rangle_{\rm CFT},$
(17)

where T is the time-ordering operator. In particular, for the helicity-2
gravitational strain the associated boundary local operator is the stress
energy-momentum tensor. The correlators of the QNMs can be therefore inferred
from the correlators of the boundary energy-momentum tensor, as we will show
in the following.

nonlinearities of the QNMs from the Kerr/CFT. The energy-momentum tensor
exists in any local CFT. In two-dimensions, its components are denoted as $T$
and $\overline{T}$, and they have weights $(h,\bar{h})=(2,0)$ and $(0,2)$,
respectively. The central charge $c=12J$ completely fixes their correlators
Guica  _et al._ (2009). By defining the energy-momentum tensor as $T_{\mu\nu}$
as the response of the Hamiltonian $\mathcal{H}$ to the transformation
$x^{\mu}\to x^{\mu}+\xi^{\mu}$, as

$\delta\mathcal{H}=-\int{\rm d}^{2}x\,\partial_{\mu}\xi_{\nu}\,T^{\mu\nu},$
(18)

the two- and three-point correlators on the complex plane turn out to be
Belavin  _et al._ (1984)

$\displaystyle\langle
T(z_{1})T(z_{2})\rangle=\frac{1}{(2\pi)^{2}}\frac{c/2}{z_{21}^{4}},$ (19)
$\displaystyle\langle
T(z_{1})T(z_{2})T(z_{3})\rangle=\frac{1}{(2\pi)^{3}}\frac{c}{z_{21}^{2}\,z_{32}^{2}\,z_{13}^{2}},$
(20)

where $z_{ij}=(z_{i}-z_{j})$ and $T=T_{zz}$. To take into account that the CFT
is at finite temperature $T_{L}$ for the left-movers, we map the complex plane
to the cylinder with complex coordinate $w=x^{1}+i\tau$ by Di Francesco  _et
al._ (1997)

$\displaystyle z=e^{2\pi T_{L}w},\qquad\bar{z}=e^{2\pi T_{L}\bar{w}}$ (21)

and identify $\tau$ with $\tau+1/T_{L}$. Then, using the transformation
property of the energy-momentum tensor under conformal transformations $z\to
w(z)$

$\displaystyle T(z)\to
T(w)=\left(\partial_{z}w\right)^{2}\left(T(z)-\frac{c}{12}\\{w;z\\}\right),$
(22)

where $\\{w;z\\}$ is the Schwarzian derivative, we find that the two- and
three-point functions at finite temperature $T_{L}$, after Wick rotating
$\tau\to ix^{0}$, are

$\displaystyle\langle
T(x_{1}^{-})T(x_{2}^{-})\rangle=\frac{c/2}{(2\pi)^{2}}\left(\frac{\pi
T_{L}}{\sinh(\pi T_{L}x_{21}^{-})}\right)^{4},$ (23) $\displaystyle\langle
T(x_{1}^{-})T(x_{2}^{-})T(x_{3}^{-})\rangle=\frac{c}{(2\pi)^{3}}\left(\frac{\pi
T_{L}}{\sinh(\pi T_{L}x_{12}^{-})}\right)^{2}\left(\frac{\pi T_{L}}{\sinh(\pi
T_{L}x_{23}^{-})}\right)^{2}\left(\frac{\pi T_{L}}{\sinh(\pi
T_{L}x_{31}^{-})}\right)^{2},$ (24)

where $x^{-}=(x^{1}-x^{0})$. The bulk isometries $\partial_{\phi}$ and
$\partial_{t}$ are identified, up to a scale, with the left and right
translations in the CFT, implying $x^{-}=\phi$ and $x^{+}=t$. Going to
momentum space and from Eqs. (12) and (13), one identifies therefore the
frequency of the left-movers with the azimuthal number $m$ Bredberg  _et al._
(2010).

We can write the above connected correlators in momentum space as

$\displaystyle\langle T_{m_{1}}T_{m_{2}}\rangle=$
$\displaystyle\frac{1}{2}\int\prod_{i=1}^{2}{\rm
d}x^{-}_{i}e^{im_{i}x^{-}_{i}}\langle T(x^{-}_{1})T(x^{-}_{2})\rangle,$

and

$\displaystyle\langle T_{m_{1}}T_{m_{2}}T_{m_{3}}\rangle=$
$\displaystyle\int\prod_{i=1}^{3}\frac{{\rm
d}x^{-}_{i}}{2^{3/2}}e^{im_{i}x^{-}_{i}}\langle
T(x^{-}_{1})T(x^{-}_{2})T(x^{-}_{3})\rangle,$

where the $T_{m}$’s are the boundary duals to the modes of the gravitational
strain with angular momentum $m$. We obtain Gubser (1997); Bredberg  _et al._
(2010); Becker  _et al._ (2014)

$\displaystyle\langle
T_{m_{1}}T_{m_{2}}\rangle^{\prime}=\frac{c}{24}\frac{(2\pi
T_{L})^{3}}{(2\pi)^{2}}\,e^{m_{1}/(2T_{L})}\left|\Gamma\left(2+i\frac{m_{1}}{2\pi
T_{L}}\right)\right|^{2}$

and

$\displaystyle\langle
T_{m_{1}}T_{m_{2}}T_{m_{3}}\rangle^{\prime}=-\frac{c}{2\sqrt{2}}\,\frac{(2\pi
T_{L})^{4}}{(2\pi)^{3}}e^{-(m_{1}+m_{2})/(2T_{L})}G^{3,3}_{3,3}\left(\begin{array}[]{ccc}-i\frac{m_{1}}{2\pi
T_{L}},&0,&i\frac{m_{2}}{2\pi T_{L}}\\\ 1-i\frac{m_{1}}{2\pi
T_{L}},&1,&1+i\frac{m_{2}}{2\pi T_{L}}\end{array}\Bigg{|}\,\,e^{i\pi}\right),$
(27)

where the primes indicate we have removed the Dirac delta function
$(2\pi)\delta(\sum_{i}m_{i}$) and $G^{3,3}_{3,3}$ is a Meijer-$G$ function.
The two- and three-point correlators of the graviton are then inferred from
the expressions Maldacena (2003)

$\langle h_{m}h_{-m}\rangle^{\prime}=-\frac{1}{2\,{\rm Re}\langle
T_{m}T_{-m}\rangle^{\prime}}\,,\quad\quad\langle
h_{m_{1}}h_{m_{2}}h_{m_{3}}\rangle^{\prime}=\dfrac{2\,{\rm Re}\langle
T_{m_{1}}T_{m_{2}}T_{m_{3}}\rangle^{\prime}}{\prod_{i=1}^{3}(-2\,{\rm
Re}\langle T_{m_{i}}T_{-m_{i}}\rangle^{\prime})},$ (28)

from where it follows that (setting finally $T_{L}=1/2\pi$)

$\displaystyle\frac{\langle
h_{m_{1}}h_{m_{2}}h_{m_{3}}\rangle^{\prime}}{\langle
h_{m_{1}}h_{-m_{1}}\rangle^{\prime}\langle
h_{m_{2}}h_{-m_{2}}\rangle^{\prime}}$ $\displaystyle=-\frac{{\rm Re}\langle
T_{m_{1}}T_{m_{2}}T_{m_{3}}\rangle^{\prime}}{{\rm Re}\langle
T_{m_{3}}T_{-m_{3}}\rangle^{\prime}}=\frac{6\sqrt{2}}{2\pi}\frac{G^{3,3}_{3,3}\left(\begin{array}[]{ccc}-im_{1},&0,&im_{2}\\\
1-im_{1},&1,&1+im_{2}\end{array}\Bigg{|}\,\,e^{i\pi}\right)}{\left|\Gamma\left(2+im_{3}\right)\right|^{2}},$
(31)

which critically does not depend upon the central charge $c$. The last passage
is to integrate over the remaining part of the spin-weighted spherical
harmonics in the polar angle $\theta$. We obtain the general expression

$\displaystyle\frac{\langle
h_{(\ell_{1},m_{1})}h_{(\ell_{2},m_{2})}h_{(\ell_{1}+\ell_{2},m_{1}+m_{2})}\rangle}{\langle
h^{2}_{(\ell_{1},m_{1})}\rangle\langle h^{2}_{(\ell_{2},m_{2})}\rangle}$
$\displaystyle=$
$\displaystyle\frac{6\sqrt{2}}{2\pi}\,{}_{-2}C_{\ell_{1},\ell_{2},\ell_{1}+\ell_{2}}^{m_{1},m_{2},m_{1}+m_{2}}\,\frac{G^{3,3}_{3,3}\left(\begin{array}[]{ccc}-im_{1},&0,&im_{2}\\\
1-im_{1},&1,&1+im_{2}\end{array}\Bigg{|}\,\,e^{i\pi}\right)}{\left|\Gamma\left(2-i(m_{1}+m_{2})\right)\right|^{2}},$
(34)

where

$\displaystyle{}_{-2}C_{\ell_{1},\ell_{2},\ell_{3}}^{m_{1},m_{2},m_{3}}$
$\displaystyle=$ $\displaystyle 2\pi\int_{0}^{\pi}{\rm
d}\theta\,\sin\theta\,{}_{-2}Y_{(\ell_{1},m_{1})}{}_{-2}Y_{(\ell_{2},m_{2})}{}_{2}\overline{Y}_{(\ell_{3},m_{3})}$
(35) $\displaystyle=$
$\displaystyle\frac{\Gamma\bigg{(}\\!\\!-2+\sum\limits_{i=1}^{3}\displaystyle\frac{|m_{i}|}{2}\bigg{)}\Gamma\bigg{(}4+\sum\limits_{i=1}^{3}\displaystyle\frac{|m_{i}|}{2}\bigg{)}}{2\sqrt{\pi}\,\Gamma\bigg{(}2+\sum\limits_{i=1}^{3}|m_{i}|\bigg{)}}\left(\prod\limits_{i=1}^{3}\frac{(2|m_{i}|+1)!}{(|m_{i}|+2)!\,\,(|m_{i}|-2)!}\right)^{1/2},$

with ${}_{s}\overline{Y}_{(\ell,m)}=(-1)^{s-m}{}_{-s}{Y}_{(\ell,-m)}$, valid
for $\ell_{1}=m_{1}$, $\ell_{2}=m_{2}$ and $\ell_{3}=-m_{3}=(m_{1}+m_{2})$.
Taking $m_{1}=m_{2}=2$, we get111The fits in Ref. Cheung  _et al._ (2022) are
obtained for positive azimuthal numbers. We thank E. Berti and M. Cheung for
exchanges about this technical point. However, notice that our results are
symmetric under the change of sign of the azimuthal numbers.

$\frac{\langle h_{(2,2)}h_{(2,2)}h_{(4,4)}\rangle}{\langle
h^{2}_{(2,2)}\rangle^{2}}\simeq
0.62\cdot\frac{5}{24}\sqrt{\frac{7}{\pi}}\simeq 0.19,$ (36)

which is quite in good agreement with the numerical results of Refs. Mitman
_et al._ (2022); Cheung  _et al._ (2022). To get closer to the spin value of
$0.7$ considered in those simulations, we can partially approximate departure
from extremality using the corresponding temperature $T_{L}$ from Eq. (15) in
our expressions. We obtain in Eq. (36) the numerical value of 0.17, which is
even astonishingly closer to the numerical result in Ref. Cheung  _et al._
(2022). For the modes $m_{1}=2$ and $m_{2}=3$, we find (summing up the two
terms from the permutation of the modes $m=2$ and 3)

$\frac{\langle h_{(2,2)}h_{(3,3)}h_{(5,5)}\rangle}{\langle
h^{2}_{(2,2)}\rangle\langle h^{2}_{(3,3)}\rangle}\simeq
1.57\cdot\frac{2}{3}\sqrt{\frac{7}{11\pi}}\simeq 0.47,$ (37)

matching again the value found in Ref. Cheung  _et al._ (2022) and giving
confidence on the validity of Eq. (34) (we obtain 0.45 taking into account the
corresponding temperature $T_{L}$ from Eq. (15) for spins equal to 0.7). It
would be nice to check the expression (34) against numerical quasicircular
simulations giving rise to fast spinning remnants from which the various
multipoles correlators may be extracted.

Conclusions. In this paper we have offered an argument based on the Kerr/CFT
correspondence to evaluate the nonlinearities of the Kerr BH ringdown. We have
shown that the Kerr/CFT correspondence provides a simple way to both explain,
at the qualitative level, the quadratic scaling of the second-order mode
amplitude with the product of the amplitudes of the fundamental modes, and
also to quantitatively predict the size of this effect in striking agreement
with numerical results found in recent literature.

Our findings hold only in the extremal case; our next step will be to study
departures from this condition. Numerical results are indeed found close to,
but not exactly at, extremality, i.e. for remnant dimensionless spins $\simeq
0.7$ Mitman  _et al._ (2022); Cheung  _et al._ (2022). Some corrections might
therefore intervene. For small deviations from extremality, a different set of
boundary conditions lead to a second copy of the Virasoro algebra Castro and
Larsen (2009) with indications of a hidden dual CFT even far from extremality
Castro  _et al._ (2010); Hui  _et al._ (2022); Kehagias  _et al._ (2022),
where the new excitations correspond to right-movers. However, the fact that
our findings are so close to the numerical fits of Refs. Mitman  _et al._
(2022); Cheung  _et al._ (2022) might indicate that the value of the remnant
spin is not so relevant (as it also appears from numerical results) and that
the right-mover sector is decoupled from the dynamics. We have indeed reasons
to believe that the right sector is associated to the $s=+2$ degree of freedom
whose contribution to the GW strain is subleading at infinity with respect to
the $s=-2$ mode Kehagias  _et al._ . Large nonlinearities have been observed
also in the case of head-on collisions, giving rise to non-spinning BHs Cheung
_et al._ (2022). This fact as well might be explained using similar symmetry
arguments, but in the context of Schwarzschild BHs Hui  _et al._ (2022). The
CFT approach may also provide new consistency relations among the QNM
amplitudes and useful predictions for the nonlinearities involved in the QNM
dynamics. Finally, it would also be interesting to go beyond the three-point
correlator, to see if sizeable effects persist at higher-orders.

We will investigate these issues in order to provide generic predictions in
terms of the multipole numbers and spins of the remnants in future work
Kehagias  _et al._ .

Acknowledgments. We thank E. Berti for many useful discussions about the
results of Ref. Cheung  _et al._ (2022). We thank E. Berti, M. Biagetti, and
V. De Luca for useful feedback on the draft. The work of D. P. and F. R. is
supported by the Swiss National Science Foundation under grants no.
200021-205016 and PP00P2-206149. A.K. is supported by the PEVE-2020 NTUA
programme for basic research with project number 65228100. A. K. thanks the
University of Geneva where part of this work was performed. A.R. is funded by
the Boninchi Foundation.

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