# Defining differential equations for modular forms and Jacobi forms

S. Opanasenko1, E.V. Ferapontov1,2

###### Abstract

It is well known that every modular form $f$ on a discrete subgroup
$\Gamma\leqslant\textrm{SL}(2,\mathbb{R})$ satisfies a third-order nonlinear
ODE that expresses algebraic dependence of the functions $f$, $f^{\prime}$,
$f^{\prime\prime}$ and $f^{\prime\prime\prime}$. These ODEs are automatically
invariant under the Lie group $\textrm{SL}(2,\mathbb{R})$, which acts on the
solution spaces thereof with an open orbit (and the discrete stabiliser
$\Gamma$ of a generic solution). Similarly, every modular form satisfies a
fourth-order nonlinear ODE that is invariant under the Lie group
$\textrm{GL}(2,\mathbb{R})$ acting on its solution space with an open orbit.
ODEs for modular forms can be compactly expressed in terms of the differential
invariants of these actions. The invariant forms of both ODEs define plane
algebraic curves naturally associated with every modular form; the
corresponding ODEs can be seen as modular parametrisations of the associated
curves.

After reviewing examples of nonlinear ODEs satisfied by classical modular
forms (such as Eisenstein series, modular forms on congruence subgroups of
level two and three, theta constants, and some newforms of weight two), we
generalise these results to Jacobi forms; these satisfy involutive third-order
PDE systems that are invariant under the Lie group
$\textrm{SL}(2,\mathbb{R})\ltimes H$ where $H$ is the Heisenberg group.

1 Department of Mathematical Sciences

Loughborough University

Loughborough, Leicestershire LE11 3TU, UK

2Institute of Mathematics, Ufa Federal Research Centre

Russian Academy of Sciences, 112 Chernyshevsky Street

Ufa 450008, Russian Federation

e-mails:

<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
##  1 Introduction

Modular forms on a discrete subgroup
$\Gamma\leqslant\textrm{SL}(2,\mathbb{R})$ are holomorphic functions $f(\tau)$
on the upper half-plane $\mathcal{H}$ that satisfy the modular transformation
property

$f\left({a\tau+b\over
c\tau+d}\right)=(c\tau+d)^{k}f(\tau),\qquad\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)\in\Gamma,$

where $k$ is the weight of a modular form; we refer to [6] for a general
theory. It is well known that every modular form satisfies a third-order
nonlinear ODE that expresses algebraic dependence of the functions $f$,
$f^{\prime}$, $f^{\prime\prime}$ and $f^{\prime\prime\prime}$, which is shown
in, e.g., [40, Theorem 2] based on the prior idea of [23] that two meromorphic
functions on a Riemann surface are algebraically dependent. What was not
explicitly noted however is that every such ODE is automatically
$\mathrm{SL}(2,\mathbb{R})$-invariant and can be represented by an algebraic
relation

$F(I_{k},J_{k})=0$

where $I_{k}$ and $J_{k}$ are differential invariants of a certain
$\mathrm{SL}(2,\mathbb{R})$-action,

$\displaystyle\displaystyle I_{k}={kff^{\prime\prime}-(k+1)f^{\prime 2}\over
f^{2+{4\over k}}},\quad\displaystyle
J_{k}={k^{2}f^{2}f^{\prime\prime\prime}-3k(k+2)ff^{\prime}f^{\prime\prime}+2(k+1)(k+2)f^{\prime
3}\over f^{3+{6\over k}}},$

see Section 2.1 for their expressions in terms of the Rankin–Cohen brackets.
Thus, there is a plane algebraic curve $C\colon F(I_{k},J_{k})=0$ naturally
associated with every modular form (note that relation $F$ depends on the
modular form $f$). The case of particular interest in number theory is where
$f$ is a newform of weight $k=2$ on a congruence subgroup $\Gamma_{0}(N)$. In
this case, modular functions $I_{2}$ and $J_{2}$ provide a modular
parametrisation of $C$. For special values of $N\geq 11$, one obtains elliptic
curves over $\mathbb{Q}$ with a modular parametrisation; these curves $C$ have
the same $j$-invariants as the elliptic curves associated with the modular
form $f(\tau)$ via the Taniyama-Shimura-Weil conjecture, see e.g. [3, 41, 42]
and Examples of Section 2.2 for some explicit formulae and further discussion.

Similarly, every modular form $f(\tau)$ of weight $k$ satisfies a fourth-order
ODE, compare with [40, Remark, p. 342],

${\cal F}(P_{k},Q_{k})=0,$

where $P_{k}$ and $Q_{k}$ are differential invariants of the order $3$ and $4$
of a certain $\mathrm{GL}(2,\mathbb{R})$-action,

$\begin{array}[]{c}\displaystyle
P_{k}={(k^{2}f^{2}f^{\prime\prime\prime}-3k(k+2)ff^{\prime}f^{\prime\prime}+2(k+1)(k+2)f^{\prime
3})^{2}\over(kff^{\prime\prime}-(k+1)f^{\prime 2})^{3}},\\\ \\\ \displaystyle
Q_{k}={f^{2}\left(k(k{+}1)ff^{\prime\prime\prime\prime}-4(k{+}1)(k{+}3)f^{\prime}f^{\prime\prime\prime}+3(k{+}2)(k{+}3)f^{\prime\prime
2}\right)\over(kff^{\prime\prime}-(k+1)f^{\prime 2})^{2}},\end{array}$

see Section 2.1 for their expressions in terms of the Rankin–Cohen brackets.
Thus, there is another plane algebraic curve ${\cal C}:{\cal
F}(P_{k},Q_{k})=0$ naturally associated with every modular form (the relation
$\cal F$ depends on the modular form $f$). Note that the invariants $P_{k}$
and $Q_{k}$ are modular functions for every $k$, and provide a modular
parametrisation of $\cal C$. There is a natural covering map $C\to{\cal C}$,
see Remark 2 in Section 2.1. In all examples discussed in this paper the
second curve $\cal C$ turns out to be rational (even for congruence subgroups
$\Gamma_{0}(N)$ of higher genus), although we have no general explanation of
this fact.

The origin of ODEs for classical modular forms on $\textrm{SL}(2,\mathbb{Z})$
are the Ramanujan equations for the Eisenstein series. In what follows, we use
the notation $q=\mathrm{e}^{2\pi i\tau}$ and denote by prime the operator
$q{d\over dq}={1\over 2\pi i}{d\over d\tau}$. The Eisenstein series are
defined as

$E_{k}(\tau)=1-{2k\over B_{k}}\sum_{n=1}^{\infty}\sigma_{k-1}(n)q^{n},$

where $B_{k}$ are the Bernoulli numbers and $\sigma_{k-1}(n)$ denotes the sum
of the $(k-1)$st powers of the positive divisors of $n$. Explicitly, we have

$\displaystyle
E_{2}(\tau)=1-24\sum_{n=1}^{\infty}\sigma_{1}(n)q^{n}=1-24q-72q^{2}-\dots,$
$\displaystyle
E_{4}(\tau)=1+240\sum_{n=1}^{\infty}\sigma_{3}(n)q^{n}=1+240q+2160q^{2}+\dots,$
$\displaystyle
E_{6}(\tau)=1-504\sum_{n=1}^{\infty}\sigma_{5}(n)q^{n}=1-504q-16632q^{2}-\dots,$

note that $E_{2}$ is only quasi-modular. The Eisenstein series $E_{2}$,
$E_{4}$ and $E_{6}$ satisfy the Ramanujan system of ODEs,

$E_{2}^{\prime}={E_{2}^{2}-E_{4}\over 12},\qquad
E_{4}^{\prime}={E_{2}E_{4}-E_{6}\over 3},\qquad
E_{6}^{\prime}={E_{2}E_{6}-E_{4}^{2}\over 2},$ (1.1)

which is invariant under the action of the Lie group
$\mathrm{SL}(2,\mathbb{R})$ defined as

$\tilde{\tau}={a\tau+b\over
c\tau+d},\quad\tilde{E}_{2}=(c\tau+d)^{2}E_{2}+12c(c\tau+d),\quad\tilde{E}_{4}=(c\tau+d)^{4}E_{4},\quad\tilde{E}_{6}=(c\tau+d)^{6}E_{6}.$

Every modular form $f$ on $\textrm{SL}(2,\mathbb{Z})$ is a homogeneous
polynomial in $E_{4}$ and $E_{6}$. Differentiation of $f$ with the help of
(1.1) gives four polynomial expressions for $f$, $f^{\prime}$,
$f^{\prime\prime}$, $f^{\prime\prime\prime}$ in terms of $E_{2}$, $E_{4}$ and
$E_{6}$. The elimination of $E_{2}$, $E_{4}$, $E_{6}$ leads to a third-order
nonlinear ODE for $f$, which inherits $\mathrm{SL}(2,\mathbb{R})$-symmetry
from the Ramanujan equations.

In special cases, both (third- and fourth-order) ODEs for modular forms and
their invariance properties have been discussed in the literature, see, e.g.,
[1, 6, 26, 34]. For instance, the modular discriminant

$\Delta=q\prod_{n=1}^{\infty}(1-q^{n})^{24}={1\over
1728}(E_{4}^{3}-E_{6}^{2})$

satisfies an $\mathrm{SL}(2,\mathbb{R})$-invariant third-order ODE ([40,
Proposition 3])

$\displaystyle 36\Delta^{4}\Delta^{\prime\prime\prime
2}-14(18\Delta\Delta^{\prime\prime}-13\Delta^{\prime
2})\Delta^{2}\Delta^{\prime}\Delta^{\prime\prime\prime}+48\Delta^{3}\Delta^{\prime\prime
3}$ $\displaystyle\qquad+285\Delta^{2}\Delta^{\prime 2}\Delta^{\prime\prime
2}-468\Delta\Delta^{\prime 4}\Delta^{\prime\prime}+169\Delta^{\prime
6}+48\Delta^{7}=0,$

as well as the more well-known van der Pol–Rankin equation, which is
$\mathrm{GL}(2,\mathbb{R})$-invariant fourth-order ODE,

$\displaystyle
2\Delta^{3}\Delta^{\prime\prime\prime\prime}-10\Delta^{2}\Delta^{\prime}\Delta^{\prime\prime\prime}-3\Delta^{2}\Delta^{\prime\prime
2}+24\Delta\Delta^{\prime 2}\Delta^{\prime\prime}-13\Delta^{\prime 4}=0,$

possessing an extra scaling symmetry $\Delta\to\lambda\Delta$ that is not
present in the third-order ODE; see Example 1 of Section 2.2 for the invariant
forms of both equations. Note that the fourth-order ODE is a differential
consequence of the third-order ODE.

It is precisely via the differential equations $F(I_{k},J_{k})=0$ and
$\mathcal{F}(P_{k},Q_{k})=0$ that modular forms feature in various contexts in
mathematical physics. In this paper, we discuss the invariance properties of
ODEs for modular forms from the point of view of symmetry analysis of
differential equations, as well as an extension of these results to Jacobi
forms. Below we list some examples of ODEs for modular forms originating from
the theory of dispersionless integrable PDEs in 3D.

First-order integrable Lagrangians. Paper [18] gives a characterisation of
first-order integrable Lagrangians of the form $\int
F(u_{x_{1}},u_{x_{2}},u_{x_{3}})\,\mathrm{d}x$. It was observed in [16, 12]
that the corresponding Lagrangian densities $F$ are related to Picard modular
forms. In particular, for densities of the form
$F=u_{x_{1}}u_{x_{2}}f(u_{x_{3}})$, the integrability conditions lead to a
fourth-order $\mathrm{GL}(2,\mathbb{R})$-invariant ODE for $f(\tau)$,

$ff^{\prime\prime\prime\prime}(ff^{\prime\prime}-2f^{\prime
2})-f^{2}f^{\prime\prime\prime 2}+2f^{\prime}(ff^{\prime\prime}+4f^{\prime
2})f^{\prime\prime\prime}-9f^{\prime 2}f^{\prime\prime 2}=0,$

whose general solution is the Eisenstein series $E_{1,3}(\tau)$,

$f(\tau)=E_{1,3}(\tau)=\sum_{(\alpha,\beta)\in\mathbb{Z}^{2}}q^{(\alpha^{2}-\alpha\beta+\beta^{2})}=1+6q+6q^{3}+6q^{4}+12q^{7}+.....,$

see Example 4 of Section 2.2 for further details.

Hirota type equations. Paper [17] studies integrability of dispersionless
Hirota-type equations in 3D, $F(u_{x_{i}x_{j}})=0$, where
$u(x_{1},x_{2},x_{3})$ is a function of three independent variables, and
$u_{x_{i}x_{j}}$ denote second-order partial derivatives. It was shown in [11]
that the ‘generic’ integrable Hirota master-equation is expressible via genus
three theta constants. In particular, for equations of the form

$u_{x_{3}x_{3}}-{u_{x_{1}x_{2}}\over u_{x_{1}x_{3}}}-{1\over
6}h(u_{x_{1}x_{1}})u_{x_{1}x_{3}}^{2}=0,$

the integrability conditions lead to the $\mathrm{SL}(2,\mathbb{R})$-invariant
Chazy equation for $h(t)$ [36],

$h_{ttt}+2hh_{tt}-3h_{t}^{2}=0,$

whose general solution is expressed in terms of the Eisenstein series $E_{2}$,

$h(t)=E_{2}(it/\pi)=1-24\sum_{n=1}^{\infty}\sigma_{1}(n)\mathrm{e}^{-2nt}.$

Second-order quasilinear PDEs. Paper [7] studies integrability of 3D second-
order quasilinear PDEs of the form
$\sum_{i,j}f_{ij}(u_{x_{1}},u_{x_{2}},u_{x_{3}})u_{x_{i}x_{j}}=0$ where
$u(x_{1},x_{2},x_{3})$ is a function of three independent variables. In
particular, for equations of the form

$u_{xy}+(u_{x}u_{y}r(u_{t}))_{t}=0,$

the integrability conditions result in an
$\mathrm{SL}(2,\mathbb{R})$-invariant third-order ODE for $r(s)$,

$r_{sss}(r_{s}-r^{2})-r_{ss}^{2}+4{r}^{3}{r_{ss}}+2r_{s}^{3}-6{r}^{2}r_{s}^{2}=0,$

which has appeared in the context of modular forms of level two [1]. Its
generic solution is given by the Eisenstein series

$r(s)=1-8\sum_{n=1}^{\infty}{(-1)^{n}nw^{n}\over 1-w^{n}},\quad
w=\mathrm{e}^{4s},$

which is associated with the congruence subgroup $\Gamma_{0}(2)$ of the
modular group; see Section 2.2 for further details.

Second-order integrable Lagrangians. Second-order integrable Lagrangians of
the form $\int F(u_{xx},u_{xy},u_{yy})\,\mathrm{d}x\mathrm{d}y$ were
investigated in [19]. Under the ansatz
$F=\mathrm{e}^{u_{xx}}g(u_{xy},u_{yy})$, the integrability conditions lead to
the following constraints for $g(z,c)$ (we set $z=u_{xy}$, $c=u_{yy}$):

$\begin{array}[]{c}gg_{zcc}=3g_{cc}g_{z}-2g_{zc}g_{c},\\\
gg_{zzz}=g_{z}g_{zz}+4g_{zc}g-4g_{z}g_{c},\\\
gg_{ccc}=g_{c}g_{cc}+2g_{cc}g_{zz}-2(g_{zc})^{2},\\\
gg_{zzc}=2g_{z}g_{zc}-g_{c}g_{zz}+2gg_{cc}-2(g_{c})^{2}.\end{array}$ (1.2)

This over-determined system for $g$ is in involution and its generic solution
can be represented in the form

$g(z,c)=[\Delta(ic/\pi)]^{-1/8}\theta_{1}(ic/\pi,z)$

where $\Delta$ is the modular discriminant and $\theta_{1}$ is the Jacobi
theta function,

$\displaystyle\theta_{1}(\tau,z)=2\sum_{n=0}^{\infty}(-1)^{n}\mathrm{e}^{\pi
i(n+1/2)^{2}\tau}\sin[(2n+1)z].$

The reason for the occurrence of modular forms in the above classification
results is a remarkable construction of [31] that parametrises broad classes
of dispersionless integrable systems in 3D via generalised hypergeometric
functions. For special values of the parameters (where the monodromy group of
hypergeometric system is a lattice), this parametrisation leads to integrable
PDEs whose coefficients are expressible via modular forms.

The structure of the paper is as follows. In Section 2 we discuss differential
equations for modular forms by first describing a general construction of
$\mathrm{SL}(2,\mathbb{R})$\- and $\mathrm{GL}(2,\mathbb{R})$-invariant
equations in terms of the differential invariants of the corresponding group
actions (Section 2.1), and then illustrating the general constructions by
numerous examples of modular forms on various congruence subgroups (Section
2.2). Differential systems for Jacobi forms are discussed in Section 3 by
first describing differential invariants of the Jacobi group (Section 3.1) and
then illustrating the general theory by several examples (Section 3.2).

##  2 Differential equations for modular forms

After reviewing differential invariants of the standard
$\mathrm{SL}(2,\mathbb{R})$\- and $\mathrm{GL}(2,\mathbb{R})$-actions
occurring in the theory of modular forms, we provide third-order and fourth-
order invariant differential equations for the Eisenstein series $E_{4}$ and
$E_{6}$, the modular discriminant $\triangle$, Jacobi theta constants,
Eisenstein series $E_{1,3}$, some modular forms of level two, and some
newforms of weight two on various congruence subgroups.

###  2.1 Differential invariants

Consider the group $G_{k}$, which is isomorphic to
$\mathrm{SL(2,\mathbb{R})}$, of point transformations on a space with
coordinates $(\tau,f)$, which is relevant for modular forms of weight $k$,

$\tilde{\tau}={a\tau+b\over
c\tau+d},\quad\tilde{f}=(c\tau+d)^{k}f,\quad\text{where}\
\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in\mathrm{SL}(2,\mathbb{R}).$

Its lowest-order differential invariants are

$\begin{array}[]{c}\displaystyle I_{k}={kff^{\prime\prime}-(k+1)f^{\prime
2}\over f^{2+{4\over k}}}={[f,f]_{2}\over(k+1)f^{2+{4\over k}}},\\\ \\\
\displaystyle
J_{k}={k^{2}f^{2}f^{\prime\prime\prime}-3k(k+2)ff^{\prime}f^{\prime\prime}+2(k+1)(k+2)f^{\prime
3}\over f^{3+{6\over k}}}={[f,\,[f,f]_{2}]_{1}\over(k+1)f^{3+{6\over
k}}},\end{array}$ (2.3)

where $[\cdot,\cdot]_{i}$ is the $i$th Rankin–Cohen bracket; we follow the
notation of [6, p. 53]. Any third-order $G_{k}$-invariant ODE can be written
in terms of these two invariants as

$F(I_{k},J_{k})=0.$ (2.4)

Note that the Lie algebra $\mathfrak{g}_{k}$ associated with $G_{k}$ is
spanned by the vector fields $\partial_{\tau}$,
$2\tau\partial_{\tau}-kf\partial_{f}$ and $\tau^{2}\partial_{\tau}-k\tau
f\partial_{f}$. In what follows, we will also need the easily verifiable
relation

${dI_{k}\over J_{k}}={2\pi i\over k}f^{2/k}d\tau.$ (2.5)

######  Theorem 1.

The $G_{k}$-action on the solution space of equation (2.4) is locally
transitive, that is, it possesses an open orbit.

###### Proof.

Let us write equation (2.4) in the form
$f_{\tau\tau\tau}=r(f,f_{\tau},f_{\tau\tau})$. The projections of the
evolutionary forms111Every vector field on a jet space with coordinates
$(\tau,f)$ can be prolonged uniquely to the vector field on an infinite jet
space with coordinates $(\tau,f,f_{\tau},f_{\tau\tau},\dots)$, which has an
equivalent evolutionary vector field
$\sum_{i=0}^{\infty}\mathrm{D}_{{\mathrm{d}^{i}f\over\mathrm{d}\tau^{i}}}\chi\
\partial_{{\mathrm{d}^{i}f\over\mathrm{d}\tau^{i}}}$ where $\mathrm{D}$ is the
total derivative, see [35, Chapter 5] for rigorous details. of the three
vector fields spanning $\mathfrak{g}_{k}$ to its three-dimensional solution
space,

$\displaystyle
f_{\tau}\partial_{f}+f_{\tau\tau}\partial_{f_{\tau}}+r\partial_{f_{\tau\tau}},\quad(kf+2\tau
f_{\tau})\partial_{f}+((k+2)f_{\tau}+2\tau
f_{\tau\tau})\partial_{f_{\tau}}+((k+4)f_{\tau\tau}+2\tau
r)\partial_{f_{\tau\tau}},$ $\displaystyle(\tau^{2}f_{\tau}+k\tau
f)\partial_{f}+((k+2)\tau
f_{\tau}+kf+\tau^{2}f_{\tau\tau})\partial_{f_{\tau}}+((k+4)\tau
f_{\tau\tau}+2(k+1)f+\tau^{2}r)\partial_{f_{\tau\tau}},$

are linearly independent. Indeed, the determinant

$\displaystyle\left|\begin{array}[]{ccc}f_{\tau}&f_{\tau\tau}&r\\\ kf+2\tau
f_{\tau}&(k+2)f_{\tau}+2\tau f_{\tau\tau}&(k+4)f_{\tau\tau}+2\tau r\\\
\tau^{2}f_{\tau}+k\tau f&(k+2)\tau f_{\tau}+kf+\tau^{2}f_{\tau\tau}&(k+4)\tau
f_{\tau\tau}+2(k+1)f_{\tau}+\tau^{2}r\end{array}\right|$
$\displaystyle=k^{2}ff_{\tau}f_{\tau\tau\tau}-k((k+4)f_{\tau}^{2}+2(k+1)f^{2})f_{\tau\tau}+2(k+1)(k+2)ff_{\tau}^{2}.$

is not $G_{k}$-invariant, and thus it can not be a left-hand side of the
equation satisfied by $f$. ∎

Consider also the group $\mathcal{G}_{k}$, which is isomorphic to
$\mathrm{GL}(2,\mathbb{R})$, of point transformations on a space with
coordinates $(\tau,f)$,

$\tilde{\tau}={a\tau+b\over
c\tau+d},\quad\tilde{f}=(c\tau+d)^{k}f,\quad\text{where}\
\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in\mathrm{GL}(2,\mathbb{R}).$

Its lowest-order differential invariants are

$\begin{array}[]{c}\displaystyle
P_{k}={(k^{2}f^{2}f^{\prime\prime\prime}-3k(k+2)ff^{\prime}f^{\prime\prime}+2(k+1)(k+2)f^{\prime
3})^{2}\over(kff^{\prime\prime}-(k+1)f^{\prime
2})^{3}}={(k+1)[f,\,[f,f]_{2}]_{1}^{2}\over[f,f]_{2}^{3}},\\\ \\\
\displaystyle
Q_{k}={f^{2}\left(k(k{+}1)ff^{\prime\prime\prime\prime}-4(k{+}1)(k{+}3)f^{\prime}f^{\prime\prime\prime}+3(k{+}2)(k{+}3)f^{\prime\prime
2}\right)\over(kff^{\prime\prime}-(k+1)f^{\prime
2})^{2}}={12(k+1)^{2}f^{2}[f,f]_{4}\over(k+2)(k+3)[f,f]_{2}^{2}}.\end{array}$

Any fourth-order $\mathcal{G}_{k}$-invariant ODE can be written in terms of
these two invariants as

$\mathcal{F}(P_{k},Q_{k})=0.$ (2.6)

Note that the Lie algebra associated with the group $\mathcal{G}_{k}$ is
spanned by the vector fields $\partial_{\tau}$, $\tau\partial_{\tau}$,
$\tau^{2}\partial_{\tau}-k\tau f\partial_{f}$ and $f\partial_{f}$. Similarly
to the proof of Theorem 1, one can show that the $\mathcal{G}_{k}$-action on
the solution space of equation (2.6) is locally transitive (possesses an open
orbit).

There exists a simple link between the ODEs described above. Namely, every
$G_{k}$-invariant third-order ODEs (2.4) possesses, as its differential
consequence, a $\mathcal{G}_{k}$-invariant fourth-order ODE (2.6);
furthermore, every $\mathcal{G}_{k}$-invariant fourth-order ODE (2.6) arises
in this way. These results are summarised in the two propositions below.

######  Proposition 1.

Every $G_{k}$-invariant third-order ODE possesses, as a differential
consequence, a $\mathcal{G}_{k}$-invariant fourth-order ODE.

###### Proof.

Consider a third-order $G_{k}$-invariant third-order ODE of type (2.4). Let us
differentiate it using the formulas

$I_{k}^{\prime}={\sqrt{P_{k}}\over k}I_{k}^{{3\over 2}}f^{{2\over k}},\qquad
J_{k}^{\prime}={k^{2}Q_{k}-6(k+2)^{2}\over k(k+1)}I_{k}^{2}f^{{2\over k}},$

which can be verified by direct calculation. This gives

$0=F_{I_{k}}I_{k}^{\prime}+F_{J_{k}}J_{k}^{\prime}={I_{k}^{{3\over
2}}f^{{2\over k}}\over
k}\left(F_{I_{k}}\sqrt{P_{k}}+F_{J_{k}}{k^{2}Q_{k}-6(k+2)^{2}\over
k+1}I_{k}^{{1\over 2}}\right).$

Using the relation $J_{k}=\sqrt{P_{k}}I_{k}^{{3\over 2}}$ and eliminating
$I_{k}$ from the pair of equations

$F(I_{k},J_{k})=0,\qquad
F_{I_{k}}\sqrt{P_{k}}+F_{J_{k}}{k^{2}Q_{k}-6(k+2)^{2}\over k+1}I_{k}^{{1\over
2}}=0,$ (2.7)

we obtain the required $\mathcal{G}_{k}$-invariant fourth-order ODE of type
(2.6). Note that algebraic third-order ODEs (2.4) produce algebraic fourth-
order ODEs (2.6). ∎

Proposition 1 is a direct corollary of a more general group-theoretic result.

######  Theorem 2.

Let $H$ and $G$ be groups of point transformations of the space $(\tau,f)$
with $H$ being a proper subgroup of $G$. If $H$ is a symmetry group of an ODE
$\mathcal{E}$ for $f(\tau)$, then there is a $G$-invariant differential
consequence of $\mathcal{E}$.

###### Proof.

Let the space on which the groups $H$ and $G$ are acting be coordinatised by
$(\tau,f)$, their dimensions be $m$ and $n$, respectively, $n>m$, and the
equation $\mathcal{E}$ take the form ${F(\tau,f,\dots,f_{r})=0}$ where
$f_{i}={\mathrm{d}^{i}f\over\mathrm{d}\tau^{i}}$. The order of a lowest-order
invariant $I_{m-1}$ of $H$ is $m-1$, and its higher-order invariants can be
obtained from $I_{m-1}$ with the help of the $H$-invariant differentiation
operator $\mathrm{D}$, $I_{m+k-1}=\mathrm{D}^{k}I_{m-1}$, $k\in\mathbb{N}$.
Since equation $\mathcal{E}$ is $H$-invariant it can be written as
$\bar{F}_{r}(I_{m-1},\dots,I_{r})=0$ ($r$ is necessarily greater or equal to
$m-1$). Differentiating $n-r$ times the equation $\bar{F}=0$ with the help of
the operator $\mathrm{D}$, we obtain a system of $n-r+1$ equations
$\bar{F}_{r+i}(I_{m-1},\dots,I_{m-1+i})=0$, $i=0,\dots,n-r$, on $n-r+1$
invariants $I_{k}$ (we assume that $n>r$, otherwise we would have to keep
differentiating). Using the implicit function theorem, the lower-order $n-r$
invariants can be excluded, which results in a single equation containing the
remaining invariants $I_{k}$, but they can be rewritten in terms of the
invariants $J_{l}$ of $G$. Indeed, since $H<G$, the invariants $J_{l}$ of $G$
are invariants of $H$ as well, and thus $J_{l}$ are functions of $I_{k}$,
$J_{i}=f_{i}(I_{m-1},\dots,I_{i})$, $i\geqslant n-1$. Thus we have a desired
$G$-invariant differential consequence of $\mathcal{E}$. ∎

The generalisation of this result to systems of PDEs is straightforward.

######  Proposition 2.

Every $\mathcal{G}_{k}$-invariant fourth-order ODE is a differential
consequence of some $G_{k}$-invariant third-order ODE.

###### Proof.

Given a $\mathcal{G}_{k}$-invariant fourth-order ODE, let us seek a third-
order ODE (2.4) in the form $F(I_{k},J_{k})=J_{k}-S(I_{k})=0$ where $S(I_{k})$
is a function to be determined. Using the relation
$J_{k}=\sqrt{P_{k}}I_{k}^{{3\over 2}}$, the corresponding equations (2.7) can
be written as

$P_{k}=I_{k}^{-3}S^{2}(I_{k}),\qquad Q_{k}=6{(k+2)^{2}\over k^{2}}+{k+1\over
k^{2}}I_{k}^{-2}S(I_{k}){\mathrm{d}S(I_{k})\over\mathrm{d}I_{k}}.$ (2.8)

The substitution of these relations into the fourth-order ODE,
$\mathcal{F}(P_{k},Q_{k})=0$, gives a first-order differential equation for
$S(I_{k})$ (whose general solution depends on one arbitrary constant). Thus,
there is a one-parameter family of third-order ODEs with the required
property. ∎

######  Corollary 1.

Every modular form of weight $k$ satisfies a
$\mathrm{GL}(2,\mathbb{R})$-invariant fourth-order ODE.

######  Remark 1.

For a modular form $f(\tau)$ of weight $k$, the corresponding
$G_{k}$-invariant third-order ODE (2.4) is necessarily algebraic, thus, there
is a (singular) plane algebraic curve $C:F(I_{k},J_{k})=0$ associated to every
modular form. Formulae (2.3) provide a local parametrisation of $C$. Note that
this parametrisation is not necessarily modular since the denominators in
(2.3) are not modular forms in general. In the particularly interesting case
where $f$ is a modular form of weight $k=2$ on a suitable congruence subgroup
$\Gamma_{0}(N)$, the functions $I_{2}$ and $J_{2}$ become modular,

$\displaystyle\displaystyle I_{2}={2ff^{\prime\prime}-3f^{\prime 2}\over
f^{4}},\quad\displaystyle
J_{2}={4f^{2}f^{\prime\prime\prime}-24ff^{\prime}f^{\prime\prime}+24f^{\prime
3}\over f^{6}},$

and formula (2.5) reduces to

${\mathrm{d}I_{2}\over J_{2}}=\pi if(\tau)\mathrm{d}\tau.$

It shows that, up to a constant factor, $f(\tau)\mathrm{d}\tau$ is a pull-back
of the holomorphic differential on $C$ (see [5, Section 10.4], for an
alternative derivation of a third-order ODE for modular forms of weight
$k=2$).

For several examples of modular forms of weight $k$ discussed in Section 2.2,
the curve $C$ is a nodal cubic,

$F(I_{k},J_{k})=J_{k}^{2}+aI_{k}^{3}+bI_{k}^{2}=0,$

where $a,b\in\mathbb{Q}$ are some constants (that depend on $k$). In view of
(2.7), the corresponding $\mathcal{G}_{k}$-invariant fourth-order ODE takes
the form

${\cal F}(P_{k},Q_{k})=2k^{2}Q_{k}-2(k+1)P_{k}+(k+1)a-12(k+2)^{2}=0,$

which is a linear relation between $P_{k}$ and $Q_{k}$.

######  Remark 2.

For every modular form $f(\tau)$ of weight $k$, the corresponding
${\cal{G}}_{k}$-invariant fourth-order ODE (2.6) is necessarily algebraic. In
other words, there is another (singular) plane algebraic curve ${\cal
C}\colon{{\cal F}(P_{k},Q_{k})=0}$ associated to every modular form,
furthermore, formulae (2.3) provide a modular parametrisation of $\cal C$
(note that $P_{k}$ and $Q_{k}$ are modular functions for every $k$). There is
a natural covering map $C\to{\cal C}$ defined as

$P_{k}={J_{k}^{2}\over I_{k}^{3}},\qquad Q_{k}=6{(k+2)^{2}\over
k^{2}}-{k+1\over k^{2}}{J_{k}F_{I_{k}}\over I_{k}^{2}F_{J_{k}}};$

use the equation $J_{k}=\sqrt{P_{k}}I_{k}^{{3\over 2}}$ and the second
equation (2.7). We emphasise that in all examples discussed so far, the curve
$\cal C$ has been rational. Although the rationality of $\cal C$ clearly holds
for modular forms $f(\tau)$ on genus zero congruence subgroups $\Gamma_{0}(N)$
(in which case both $P_{k}$ and $Q_{k}$ become rational functions of the
corresponding Hauptmodul), it also holds for congruence subgroups of higher
genus; see Examples 2.2–2.2 of Section 2.2 where we derived differential
equations for newforms on genus $1$ congruence subgroups
$\Gamma_{0}(11),\Gamma_{0}(14),\Gamma_{0}(15)$, $\Gamma_{0}(17)$, and
$\Gamma_{0}(37)$.

###  2.2 Examples

First of all, we would like to revamp some classical results on the forms
$E_{4}$, $E_{6}$, $\Delta$ and $\theta$.

Example 1: Eisenstein series $E_{4}$ and $E_{6}$. It has already been
mentioned that the Eisenstein series $E_{2}$, $E_{4}$ and $E_{6}$ satisfy the
Ramanujan system (1.1). Eliminating consequently $E_{4}$ and $E_{6}$ we arrive
at the Chazy equation for $E_{2}$,

$2E_{2}^{\prime\prime\prime}-2E_{2}E_{2}^{\prime\prime}+3(E_{2}^{\prime})^{2}=0.$

Analogously, we can arrive at the equation for $E_{4}$, which has appeared
earlier in [40, Proposition 4], and for $E_{6}$, a calculation which was too
‘distasteful’ for Resnikoff to perform (without the use of computer), see [40,
p. 344]. The Eisenstein series $E_{4}$ satisfies the third-order ODE

$\displaystyle 80E_{4}^{2}E_{4}^{\prime\prime\prime 2}+$ $\displaystyle
120(5E_{4}^{\prime
3}-6E_{4}E_{4}^{\prime}E_{4}^{\prime\prime})E_{4}^{\prime\prime\prime}+576E_{4}E_{4}^{\prime\prime
3}-20(27E_{4}^{\prime 2}+4E_{4}^{3})E_{4}^{\prime\prime 2}$ (2.9)
$\displaystyle+200E_{4}^{2}E_{4}^{\prime
2}E_{4}^{\prime\prime}-125E_{4}E_{4}^{\prime 4}=0,$

which takes a nice form

$5J_{4}^{2}+144I_{4}^{3}-80I_{4}^{2}=0$

in terms of the invariants of the group $G_{4}$. In its turn, the
$\mathcal{G}_{4}$-invariant equation for $E_{4}$ is

$16Q_{4}-5P_{4}-144=0.$

The explicit ODEs for $E_{6}$ are indeed quite long so that we only present
their invariant forms,

$\displaystyle 343(J_{6}^{3}-216I_{6}^{3})+2(256I_{6}^{3}+7J_{6}^{2})^{2}=0,$

and

$(6Q_{6}-32)^{2}-7(Q_{6}-4)P_{6}=0.$

Example 2: Modular discriminant $\Delta$. The modular discriminant $\Delta$ is
a cusp form of weight 12 defined by the formula

$\Delta=q\prod_{n=1}^{\infty}(1-q^{n})^{24}.$

It can be expressed as $\Delta={1\over 1728}(E_{4}^{3}-E_{6}^{2})$, and thus
we can construct a $G_{12}$-invariant third-order ODE satisfied by $\Delta$,
which is

$\begin{array}[]{c}36\Delta^{4}\Delta^{\prime\prime\prime
2}-14(18\Delta\Delta^{\prime\prime}-13\Delta^{\prime
2})\Delta^{2}\Delta^{\prime}\Delta^{\prime\prime\prime}+48\Delta^{3}\Delta^{\prime\prime
3}\\\ \qquad+285\Delta^{2}\Delta^{\prime 2}\Delta^{\prime\prime
2}-468\Delta\Delta^{\prime 4}\Delta^{\prime\prime}+169\Delta^{\prime
6}+48\Delta^{7}=0.\end{array}$ (2.10)

Its invariant form defines an elliptic curve (equianharmonic case),

$J_{12}^{2}+16I_{12}^{3}+27648=0.$

Equation (2.10) was first found in [40]. On the other hand, $\Delta$ is known
[37, 39] to satisfy the $\mathcal{G}_{12}$-invariant fourth-order ODE

$\displaystyle
2\Delta^{3}\Delta^{\prime\prime\prime\prime}-10\Delta^{2}\Delta^{\prime}\Delta^{\prime\prime\prime}-3\Delta^{2}\Delta^{\prime\prime
2}+24\Delta\Delta^{\prime 2}\Delta^{\prime\prime}-13\Delta^{\prime 4}=0,$
(2.11)

whose invariant form is

$Q_{12}=6.$

Note that (2.11) is a differential consequence of (2.10).

Example 3: Jacobi theta constants. Jacobi theta constants (thetanulls) are
defined as

$\theta_{2}=\sum_{n=-\infty}^{\infty}\mathrm{e}^{(n-1/2)^{2}\pi
i\tau},\quad\theta_{3}=\sum_{n=-\infty}^{\infty}\mathrm{e}^{n^{2}\pi
i\tau},\quad\theta_{4}=\sum_{n=-\infty}^{\infty}(-1)^{n}\mathrm{e}^{n^{2}\pi
i\tau}.$

They are known to satisfy the same third-order ODE [24]

$(\theta^{2}\theta_{\tau\tau\tau}-15\theta\theta_{\tau}\theta_{\tau\tau}+30\theta_{\tau}^{3})^{2}+32(\theta\theta_{\tau\tau}-3\theta_{\tau}^{2})^{3}+\pi^{2}\theta^{10}(\theta\theta_{\tau\tau}-3\theta_{\tau}^{2})^{2}=0.$

This equation is $G_{1/2}$-invariant, which reflects the fact that the
thetanulls are modular forms of weight $1/2$, and can be presented as

$J_{1/2}^{2}+16\,I_{1/2}^{3}-{1\over 16}I_{1/2}^{2}=0.$

Jacobi theta constants also satisfy a nonlinear fourth-order ODE with
$\mathrm{GL}(2,\mathbb{R})$-symmetry,

$\theta^{3}(\theta\theta_{\tau\tau}-3\theta_{\tau}^{2})\theta_{\tau\tau\tau\tau}-\theta^{4}\theta_{\tau\tau\tau}^{2}+2\theta^{2}\theta_{\tau}(\theta\theta_{\tau\tau}+12\theta_{\tau}^{2})\theta_{\tau\tau\tau}+\theta^{3}\theta_{\tau\tau}^{3}-24\theta^{2}\theta_{\tau}^{2}\theta_{\tau\tau}^{2}-18\theta\theta_{\tau}^{4}\theta_{\tau\tau}+18\theta_{\tau}^{6}=0,$

see [34, eq. (5.5)], whose invariant form is

$Q_{1/2}-6P_{1/2}-102=0.$

Now we would like to consider some recently discovered systems of ODEs for
modular forms on congruence subgroups of $\mathrm{SL}(2,\mathbb{Z})$. Thus,
there are Ramanujan-like systems for modular forms of level two [38], three
[22, 29, 32], five [27, 28], six [30], etc. (See many more results on various
relations between modular forms on congruence subgroups of
$\mathrm{SL}(2,\mathbb{Z})$ in [13].)

Example 4: Modular forms on $\Gamma_{0}(2)$. Ramamani [38] found the analogue

$\displaystyle\mathcal{P}^{\prime}={\mathcal{P}^{2}-\mathcal{Q}\over
4},\quad\tilde{\mathcal{P}}^{\prime}={\mathcal{P}\tilde{\mathcal{P}}-\mathcal{Q}\over
2},\quad\mathcal{Q}^{\prime}=(\mathcal{P}-\tilde{\mathcal{P}})\mathcal{Q},$
(2.12)

of the Ramanujan system for the congruence subgroup $\Gamma_{0}(2)$ of
$\mathrm{SL}(2,\mathbb{Z})$,

$\Gamma_{0}(2):=\left\\{\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\mathrm{SL}(2,\mathbb{Z})\mid c\equiv 0\bmod 2\right\\}.$

Here, $\mathcal{P}$ and $\mathcal{Q}$ are normalized Eisenstein series on
$\Gamma_{0}(2)$ of weights 2 and 4, respectively, and $\tilde{\mathcal{P}}$ is
a modular form of weight 2,

$\mathcal{P}(q)=1-8\sum\limits_{n=1}^{\infty}{(-1)^{n}nq^{n}\over
1-q^{n}},\quad\mathcal{Q}(q)=1+16\sum\limits_{n=1}^{\infty}{(-1)^{n}n^{3}q^{n}\over
1-q^{n}},\quad\tilde{\mathcal{P}}(q)=1+24\sum\limits_{n=1}^{\infty}{nq^{n}\over
1+q^{n}}.$

It was shown in [1] that $\tilde{\mathcal{P}}$ can be presented as
$\tilde{\mathcal{P}}={3\over 2}\mathcal{P}-{1\over 2}E_{2}$ where
$\mathcal{P}$ satisfies a third-order ODE,

$2(\mathcal{P}^{2}-4\mathcal{P}^{\prime})\mathcal{P}^{\prime\prime\prime}+8(\mathcal{P}^{\prime\prime})^{2}-2\mathcal{P}^{3}\mathcal{P}^{\prime\prime}+(3\mathcal{P}^{2}-4\mathcal{P}^{\prime})(\mathcal{P}^{\prime})^{2}=0,$

and the weight-4 modular form $\mathcal{D}={1\over
64}(\tilde{\mathcal{P}}^{2}-\mathcal{Q})$, which can be seen as an analogue of
$\Delta$ for the congruence group $\Gamma_{0}(2)$, satisfies a fourth-order
ODE,

$\displaystyle(8\mathcal{D}^{4}\mathcal{D}^{\prime\prime}-10\mathcal{D}^{3}\mathcal{D}^{\prime
2})\mathcal{D}^{\prime\prime\prime\prime}-8\mathcal{D}^{4}\mathcal{D}^{\prime\prime\prime
2}+(10\mathcal{D}^{2}\mathcal{D}^{\prime
3}+16\mathcal{D}^{3}\mathcal{D}^{\prime}\mathcal{D}^{\prime\prime})\mathcal{D}^{\prime\prime\prime}$
$\displaystyle-20\mathcal{D}^{3}\mathcal{D}^{\prime\prime
3}+39\mathcal{D}^{2}\mathcal{D}^{\prime 2}\mathcal{D}^{\prime\prime
2}-60\mathcal{D}\mathcal{D}^{\prime
4}\mathcal{D}^{\prime\prime}+25\mathcal{D}^{\prime 6}=0.$

We can build on these results by showing additionally that
$\tilde{\mathcal{P}}$, which is a modular form of weight two, satisfies the
third-order ODE

$\displaystyle 3\tilde{\mathcal{P}}^{2}\tilde{\mathcal{P}}^{\prime\prime\prime
2}-36(\tilde{\mathcal{P}}\tilde{\mathcal{P}}^{\prime}\tilde{\mathcal{P}}^{\prime\prime}-\tilde{\mathcal{P}}^{\prime
3})\tilde{\mathcal{P}}^{\prime\prime\prime}+32\tilde{\mathcal{P}}\tilde{\mathcal{P}}^{\prime\prime
3}-3(12\tilde{\mathcal{P}}^{\prime
2}+\tilde{\mathcal{P}}^{4})\tilde{\mathcal{P}}^{\prime\prime
2}+9\tilde{\mathcal{P}}^{3}\tilde{\mathcal{P}}^{\prime
2}\tilde{\mathcal{P}}^{\prime\prime}-{27\over
4}\tilde{\mathcal{P}}^{2}\tilde{\mathcal{P}}^{\prime 4}=0,$
$\displaystyle\qquad\text{or, in invariant terms,}\quad
3J_{2}^{2}+64I_{2}^{3}-12I_{2}^{2}=0,$

the weight-4 modular form $\mathcal{Q}$ satisfies the third-order ODE

$\displaystyle 4\mathcal{Q}^{4}\mathcal{Q}^{\prime\prime\prime
2}-6\mathcal{Q}^{2}\mathcal{Q}^{\prime}(6\mathcal{Q}\mathcal{Q}^{\prime\prime}-5\mathcal{Q}^{\prime
2})\mathcal{Q}^{\prime\prime\prime}+16\mathcal{Q}^{3}\mathcal{Q}^{\prime\prime
3}+\mathcal{Q}^{2}(21\mathcal{Q}^{\prime
2}-4\mathcal{Q}^{3})\mathcal{Q}^{\prime\prime 2}$
$\displaystyle-10\mathcal{Q}\mathcal{Q}^{\prime 2}(6\mathcal{Q}^{\prime
2}-\mathcal{Q}^{3})\mathcal{Q}^{\prime\prime}+25\mathcal{Q}^{\prime
6}-{25\over 4}\mathcal{Q}^{3}\mathcal{Q}^{\prime 4}=0,$
$\displaystyle\qquad\text{or, in invariant terms,}\quad
J_{4}^{2}+16I_{4}^{3}-16I_{4}^{2}=0,$

the weight-4 modular form $\mathcal{D}$ satisfies the third-order ODE

$\displaystyle 4\mathcal{D}^{4}\mathcal{D}^{\prime\prime\prime
2}-6\mathcal{D}^{2}\mathcal{D}^{\prime}(6\mathcal{D}\mathcal{D}^{\prime\prime}-5\mathcal{D}^{\prime
2})\mathcal{D}^{\prime\prime\prime}+16\mathcal{D}^{3}\mathcal{D}^{\prime\prime
3}-\mathcal{D}^{2}(256\mathcal{D}^{3}-21\mathcal{D}^{\prime
2})\mathcal{D}^{\prime\prime 2}$
$\displaystyle+20\mathcal{D}\mathcal{D}^{\prime
2}(32\mathcal{D}^{3}-3\mathcal{D}^{\prime
2})\mathcal{D}^{\prime\prime}+25\mathcal{D}^{\prime
6}-400\mathcal{D}^{3}\mathcal{D}^{\prime 4}=0,$ $\displaystyle\qquad\text{or,
in invariant terms,}\quad J_{4}^{2}+16I_{4}^{3}-1024I_{4}^{2}=0,$

and the weight-8 cusp form $\tilde{\mathcal{D}}=\mathcal{D}\mathcal{Q}$
satisfies the third-order ODE

$\displaystyle
16\tilde{\mathcal{D}}^{4}\tilde{\mathcal{D}}^{\prime\prime\prime
2}-30\tilde{\mathcal{D}}^{2}\tilde{\mathcal{D}}^{\prime}(4\tilde{\mathcal{D}}\tilde{\mathcal{D}}^{\prime\prime}-3\tilde{\mathcal{D}}^{\prime
2})\tilde{\mathcal{D}}^{\prime\prime\prime}+32\tilde{\mathcal{D}}^{3}\tilde{\mathcal{D}}^{\prime\prime
3}+117\tilde{\mathcal{D}}^{2}\tilde{\mathcal{D}}^{\prime
2}\tilde{\mathcal{D}}^{\prime\prime
2}-8\tilde{\mathcal{D}}(16\tilde{\mathcal{D}}^{5}+27\tilde{\mathcal{D}}^{\prime
4})\tilde{\mathcal{D}}^{\prime\prime}$
$\displaystyle+81\tilde{\mathcal{D}}^{\prime
6}+144\tilde{\mathcal{D}}^{5}\tilde{\mathcal{D}}^{\prime 2}=0,$
$\displaystyle\qquad\text{or, in invariant terms,}\quad
J_{8}^{2}+16I_{8}^{3}-496I_{8}=0.$

Note that the last invariant equation defines an elliptic curve (lemniscatic
case). A $\mathcal{G}_{8}$-invariant ODE for $\tilde{\mathcal{D}}$ is
$128Q_{8}-9P_{8}-912=0$. We refer to Remark 1 of Section 2.1 for the fourth-
order $\mathcal{G}_{k}$-invariant equations for the modular forms
$\tilde{\mathcal{P}},\mathcal{Q},\mathcal{D}$ (note that $\mathcal{P}$ is only
quasi-modular).

Example 5: Eisenstein series $E_{1,3}$. This modular form of weight $1$ and
level $3$ is defined as

$E_{1,3}(\tau)=\sum_{(\alpha,\beta)\in\mathbb{Z}^{2}}q^{(\alpha^{2}-\alpha\beta+\beta^{2})}=1+6q+6q^{3}+6q^{4}+12q^{7}+.....$

Matsuda [29] presented several systems satisfied by $E_{1,3}$, one of which
was first derived by Huber in [22],

$E_{1,3}^{\prime}={ME_{1,3}-N\over 3},\quad M^{\prime}={M^{2}-NE_{2}\over
3},\quad N^{\prime}=(M-E_{2}^{2})N,$

where $M$ and $N$ are some functions whose specific forms are irrelevant here.
Eliminating $M$ and $N$, we obtain a $G_{1}$-invariant third-order equation
for $f=E_{1,3}$,

$\displaystyle f^{2}f^{\prime\prime\prime
2}-6f^{\prime}(3ff^{\prime\prime}-4f^{\prime
2})f^{\prime\prime\prime}+18ff^{\prime\prime 3}-(f^{6}+27f^{\prime
2})f^{\prime\prime 2}+4f^{5}f^{\prime 2}f^{\prime\prime}-4f^{4}f^{\prime
4}=0,$ (2.13)

whose invariant form is

$J_{1}^{2}+18I_{1}^{3}-I_{1}^{2}=0.$

It was also shown in [16] that $f=E_{1,3}$ satisfies the fourth-order ODE

$\displaystyle ff^{\prime\prime\prime\prime}(ff^{\prime\prime}-2f^{\prime
2})-f^{2}f^{\prime\prime\prime 2}+2f^{\prime}(ff^{\prime\prime}+4f^{\prime
2})f^{\prime\prime\prime}-9f^{\prime 2}f^{\prime\prime 2}=0,$ (2.14)

whose invariant form is

$Q_{1}-2P_{1}-36=0.$

Finally, if one does not know a particular ODE or a system thereof satisfied
by a given modular form, it is possible to construct a third-order ODE
directly by choosing an appropriate (polynomial in $I_{k}$ and $J_{k}$)
ansatz. This is what we have done in the examples below.

Example 6: Newform of weight 2 on $\Gamma_{0}(11)$. There is a unique cusp
form of weight $2$ on the congruence subgroup $\Gamma_{0}(11)$,

$f(\tau)=q\prod\limits_{n=1}^{\infty}(1-q^{n})^{2}(1-q^{11n})^{2},$

labelled as case 11.2.a.a in the LMFDB database [25]. It satisfies a
$G_{11}$-invariant equation

$\displaystyle
J_{2}^{4}+32(I_{2}-8)(I_{2}^{2}+72I_{2}-944)J_{2}^{2}+256(I_{2}+8)(I_{2}^{3}+152I_{2}^{2}-704I_{2}+1168)(I_{2}-8)^{2}=0,$

which defines a singular algebraic curve $C$ of genus $1$ with the
$j$-invariant $j=-2^{12}31^{3}11^{-5}$. Remarkably, this value coincides with
the $j$-invariant of the curve $y^{2}+y=x^{3}-x^{2}-10x-20$, which can be
uniformised by the same cusp form $f$, see [3, Table 1, case $11B$]. Thus,
both curves are birationally equivalent via the map

$I_{2}=-{8x^{2}+8x-119\over(x-5)^{2}},\quad
J_{2}=-{44(4x-9)(2y+1)\over(x-5)^{3}}.$

The inverse transformation is

$\displaystyle x={-11J_{2}^{2}+16(I_{2}-8)(9I_{2}^{2}-2468I_{2}+11712)\over
64(I_{2}-83)(I_{2}^{2}-64)},$ $\displaystyle
y={(11I_{2}-264)J_{2}^{3}+176(I_{2}-8)(I_{2}^{3}+80I_{2}^{2}-5584I_{2}+43904)J_{2}\over
512(I_{2}^{2}-64)^{2}(I_{2}-83)}+{1\over 2}.$

Note that ${\mathrm{d}I_{2}}/{J_{2}}$ is the holomorphic differential on $C$.
By Remark 1 of Section 2.1, it equals $\pi if(\tau)\mathrm{d}\tau$. The form
$f(\tau)$ also satisfies the fourth-order $\mathcal{G}_{2}$-invariant ODE

$\displaystyle
5616022359375P_{2}^{4}-2^{4}3^{5}5^{3}(34618195Q_{2}-763426383)P_{2}^{3}$
$\displaystyle-2^{8}3^{3}(173368000Q_{2}^{3}-8479136175Q_{2}^{2}+183916606320Q_{2}-1561600055241)P_{2}^{2}$
$\displaystyle+2^{12}3^{2}(64349800Q_{2}^{4}{-}3828348951Q_{2}^{3}{+}88775864253Q_{2}^{2}{-}1000262056761Q_{2}{+}4759648412715)P_{2}$
$\displaystyle+131072(4Q_{2}-63)(5329Q_{2}^{3}-204861Q_{2}^{2}+2745099Q_{2}-14039703)(2Q_{2}-105)^{2}=0,$

which defines a singular rational curve $\cal C$ in the plane $(P_{2},Q_{2})$.

Example 7: Newform of weight 2 on $\Gamma_{0}(14)$. A unique cusp form of
weight $2$ on the congruence subgroup $\Gamma_{0}(14)$, labelled as 14.2.a.a
in [25],

$f(\tau)=q\prod\limits_{n=1}^{\infty}(1-q^{n})(1-q^{2n})(1-q^{7n})(1-q^{14n}),$

satisfies the following third-order ODE,

$\displaystyle
J_{2}^{4}+32(I_{2}-5)(I_{2}^{2}+49I_{2}-350)J_{2}^{2}+256(I_{2}+20)(I_{2}-4)(I_{2}^{2}+86I_{2}-199)(I_{2}-5)^{2}=0.$

The genus of this singular algebraic curve $C$ is $1$, the $j$-invariant is
$j=5^{3}11^{3}31^{3}2^{-3}7^{-6}$ and the holomorphic differential is
${\mathrm{d}I_{2}}/{J_{2}}$. This curve is birationally equivalent to the
curve [3, Table 1, case 14D], $y^{2}+(x+1)y=x^{3}-36x-70$, via the
transformation

$I_{2}={4x^{2}+18x-41\over(x+4)^{2}},\quad
J_{2}=-{28(x+11)(2y+x+1)\over(x+4)^{3}}.$

The curve $\cal C$ corresponding to the fourth-order
$\mathcal{G}_{2}$-invariant ODE for $f(\tau)$ is rational (we do not present
it explicitly due to its complexity).

Example 8: Newform of weight 2 on $\Gamma_{0}(15)$. A unique cusp form of
weight $2$ on the congruence subgroup $\Gamma_{0}(15)$, labelled as 15.2.a.a
in [25],

$f(\tau)=q\prod\limits_{n=1}^{\infty}(1-q^{n})(1-q^{3n})(1-q^{5n})(1-q^{15n}),$

satisfies the following third-order ODE,

$\displaystyle
J_{2}^{4}+32(I_{2}-5)(I_{2}^{2}+33I_{2}-406)J_{2}^{2}+256(I_{2}+76)(I_{2}+4)(I_{2}^{2}-10I_{2}+89)(I_{2}-5)^{2}=0.$

The genus of this singular algebraic curve $C$ is $1$, the $j$-invariant is
$j=23^{3}73^{3}3^{-2}5^{-8}$ and the holomorphic differential is
${\mathrm{d}I_{2}}/{J_{2}}$. This curve is birationally equivalent to the
curve [3, Table 1, case 15F], $y^{2}+(x+1)y=x^{3}+x^{2}+35x-28$, via the
transformation

$I_{2}=-{4x^{2}+74x+61\over(x-2)^{2}},\quad
J_{2}=-{180(x+3)(2y+x+1)\over(x-2)^{3}}.$

The curve $\cal C$ corresponding to the fourth-order
$\mathcal{G}_{2}$-invariant ODE for $f(\tau)$ is rational (we do not present
it explicitly due to its complexity).

Example 9: Newform of weight 2 on $\Gamma_{0}(17)$. A cusp form of weight $2$
on the congruence subgroup $\Gamma_{0}(17)$ labelled as 17.2.a.a in [25],

$f(\tau)=q-q^{2}-q^{4}-2q^{5}+4q^{7}+3q^{8}-3q^{9}+O(q^{10}),$

satisfies the third-order ODE

$\displaystyle
9J_{2}^{8}{+}96(7I_{2}^{3}{+}3I_{2}^{2}{-}1734I_{2}{+}21340)J_{2}^{6}$
$\displaystyle{+}256(73I_{2}^{6}{+}156I_{2}^{5}{-}45153I_{2}^{4}{+}734762I_{2}^{3}{-}2275710I_{2}^{2}{-}12691872I_{2}{+}175400560)J_{2}^{4}$
$\displaystyle{+}8192(28I_{2}^{5}{+}993I_{2}^{4}{-}11044I_{2}^{3}{+}13213I_{2}^{2}{+}854526I_{2}{+}6877196)(I_{2}^{4}{-}30I_{2}^{3}{+}309I_{2}^{2}{-}584I_{2}{+}5232)J_{2}^{2}$
$\displaystyle{+}65536(16I_{2}^{4}{+}1132I_{2}^{3}{+}13477I_{2}^{2}{+}91338I_{2}{+}212581)(I_{2}^{4}{-}30I_{2}^{3}{+}309I_{2}^{2}{-}584I_{2}{+}5232)^{2}=0,$

which defines a singular algebraic curve $C$ of genus $1$ with the
$j$-invariant $j=-3^{3}11^{3}17^{-4}$. Once again, this value coincides with
the $j$-invariant of the curve $y^{2}+(x+1)y=x^{3}-x^{2}-x-14$, which can be
uniformised by the same cusp form, see [3, Table 1, case 17C]. Thus, both
curves are birationally equivalent. The holomorphic differential is again
${\mathrm{d}I_{2}}/{J_{2}}$.

The curve $\cal C$ corresponding to the fourth-order
$\mathcal{G}_{2}$-invariant ODE for $f(\tau)$ is rational (we do not present
it explicitly due to its complexity).

Example 10: Newform of weight 2 on $\Gamma_{0}(37)$. A cusp form of weight $2$
on the congruence subgroup $\Gamma_{0}(37)$ labelled as 37.2.a.b in [25],

$f(\tau)=q+q^{3}-2q^{4}-q^{7}-2q^{9}+O(q^{10}),$

satisfies a third-order ODE $F(I_{37},J_{37})=0$ which defines a singular
algebraic curve $C$ of genus $1$ with the $j$-invariant
$j=2^{12}3^{3}37^{-1}$. This value coincides with the $j$-invariant of the
curve $y^{2}-y=x^{3}-x$, which can be uniformised by the same cusp form, [3,
Table 1, case 37A], compare with [41]. Thus, both curves are birationally
equivalent. The holomorphic differential is ${\mathrm{d}I_{2}}/{J_{2}}$. Note
that in this example the modular curve $\Gamma_{0}(37)\backslash{\mathcal{H}}$
has genus two and, according to [42], the map
$\Gamma_{0}(37)\backslash{\mathcal{H}}\rightarrow C$ is a two-sheeted
covering.

The curve $\cal C$ corresponding to the fourth-order
$\mathcal{G}_{2}$-invariant ODE for $f(\tau)$ is rational (we do not present
it explicitly due to its complexity).

##  3 Differential equations for Jacobi forms

######  Definition 1.

A Jacobi form of weight $k$ and index $m$ is a holomorphic function
$f\colon\mathcal{H}\times\mathbb{C}\mapsto\mathbb{C}$ with the transformation
property

$\tilde{\tau}={a\tau+b\over c\tau+d},\quad\tilde{z}={z+\lambda\tau+\mu\over
c\tau+d},\quad\tilde{f}=(c\tau+d)^{k}\mathrm{e}^{2\pi
im\left({c(z+\lambda\tau+\mu)^{2}\over c\tau+d}-\lambda^{2}\tau-2\lambda
z-\lambda\mu\right)}f,$

where $\tau\in\mathcal{H}$, $z\in\mathbb{C}$, $\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\mathrm{SL}(2,\mathbb{Z})$, $\lambda,\mu\in\mathbb{Z}$, and
such that it has a Fourier expansion of the form

$f(\tau,z)=\sum_{\begin{subarray}{c}n=0\\\ r\in\mathbb{Z},\ r^{2}\leqslant
4nm\end{subarray}}^{\infty}c(n,r)q^{n}\zeta^{r},$

where $q=\mathrm{e}^{2\pi i\tau}$ and $\zeta=\mathrm{e}^{2\pi iz}$. If $f$ has
a Fourier expansion of the same form but with $r^{2}<4nm$ then $f$ is called a
Jacobi cusp form of weight $k$ and index $m$. If we drop the holomorphicity
condition, the function $f$ is called a weak Jacobi form if $c(n,r)=0$ unless
$n\geqslant n_{0}$ for some possible negative integer $n_{0}$

Classical examples of Jacobi forms are modular forms, theta series, Fourier
coefficients of Siegel modular forms and the Weierstrass $\wp$-function. See
more details in [15].

Similarly to modular forms, there is a notion of the Rankin–Cohen bracket [8]
that assigns to Jacobi forms $f_{1}$ and $f_{2}$ of weights $k_{1}$ and
$k_{2}$ and indices $m_{1}$ and $m_{2}$, respectively, a Jacobi form of weight
$k_{1}+k_{2}+2n$ and index $m_{1}+m_{2}$,

$[[f_{1},f_{2}]]_{n}:=\sum\limits_{i=0}^{n}(-1)^{i}{k_{1}+n-3/2\choose
n-i}{k_{2}+n-3/2\choose
i}m_{1}^{n-i}m_{2}^{i}L_{m_{1}}^{i}(f_{1})L_{m_{2}}^{n-i}(f_{2}),$

where $L_{m}:=8\pi im\,\partial_{\tau}-\partial_{z}^{2}$ is the heat operator.
The theory of Jacobi forms features an additional family of Rankin–Cohen
operators [9] parametrised by an arbitrary complex number $X$ and a non-
negative integer $n$,

$\displaystyle[f_{1},f_{2}]_{X,2n}:=\sum\limits_{r+s+p=n}C_{r,s,p}(k_{1},k_{2})(1+m_{1}X)^{s}(1-m_{2}X)^{r}L_{m_{1}+m_{2}}^{p}(L_{m_{1}}^{r}(f_{1})L_{m_{2}}^{s}(f_{2})),$
$\displaystyle[f_{1},f_{2}]_{X,2n+1}=m_{1}[f_{1},\partial_{z}f_{2}]_{X,2n}-m_{2}[\partial_{z}f_{1},f_{2}]_{X,2n}$

where

$C_{r,s,p}(k_{1},k_{2}):={(k_{1}+n-3/2)_{s+p}\over
r!}{(k_{2}+n-3/2)_{r+p}\over s!}{(3/2-k_{1}-k_{2}-n)_{r+s}\over p!}$

and $(x)_{l}=\prod\limits_{0\leqslant i\leqslant l-1}(x-i)$.

###  3.1 Differential invariants of the Jacobi group

Consider the six-dimensional group $G_{k,m}$ of point transformations called
the Jacobi group and acting on a space with coordinates $(\tau,z,f)$,

$\tilde{\tau}={a\tau+b\over c\tau+d},\quad\tilde{z}={z+\lambda\tau+\mu\over
c\tau+d}\quad\tilde{f}=(c\tau+d)^{k}\mathrm{e}^{2\pi
im\left({c(z+\lambda\tau+\mu)^{2}\over c\tau+d}-\lambda^{2}\tau-2\lambda
z-\lambda\mu+\kappa\right)}f,$

where $\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in\mathrm{SL}(2,\mathbb{R})$ and
$\lambda,\mu,\kappa\in\mathbb{R}$. In the generic case

$2kff_{zz}{-}8m\pi iff_{\tau}{-}(2k{-}1)f_{z}^{2}\neq 0\quad\text{and}\quad
m\neq 0,$

its second-order differential invariants are

$\displaystyle L_{k,m}=$ $\displaystyle{1\over(2kff_{zz}{-}8m\pi
iff_{\tau}{-}(2k{-}1)f_{z}^{2})^{2}}\Big{(}64m^{2}\pi^{2}f^{3}f_{\tau\tau}{+}32m\pi
if^{2}f_{z}f_{\tau z}{-}4k(k{+}1)f^{2}f_{zz}^{2}$
$\displaystyle{+}4(8m(k{+}1)\pi
iff_{\tau}{+}(2k^{2}{+}k{-}2)f_{z}^{2})ff_{zz}{-}(16(2k{+}3)m\pi
iff_{\tau}{+}(4k^{2}{-}7)f_{z}^{2})f_{z}^{2}\Big{)},$ $\displaystyle M_{k,m}=$
$\displaystyle{4m\pi if^{2}f_{\tau z}{-}ff_{z}f_{zz}{-}(4m\pi
iff_{\tau}{-}f_{z}^{2})f_{z}\over(2kff_{zz}{-}8m\pi
iff_{\tau}{-}(2k{-}1)f_{z}^{2})^{3/2}},$

and the third-order differential invariants are

$\displaystyle N_{k,m}=$ $\displaystyle{1\over(2kff_{zz}{-}8m\pi
iff_{\tau}{-}(2k{-}1)f_{z}^{2})^{3}}\Big{(}512m^{3}\pi^{3}if^{5}f_{\tau\tau\tau}{-}384m^{2}\pi^{2}f^{4}f_{z}f_{\tau\tau
z}{-}96m\pi if^{3}f_{z}^{2}f_{\tau zz}$
$\displaystyle{+}8f^{2}f_{z}^{3}f_{zzz}{-}96m\pi
f^{2}(2(k{+}2)ff_{zz}{-}(2k{+}5)f_{z}^{2})(2m\pi ff_{\tau\tau}{+}if_{z}f_{\tau
z})$ $\displaystyle{-}4(k{+}2)f^{2}f_{zz}^{2}\left(24(k+1)m\pi
iff_{\tau}{-}2k(k+1)ff_{zz}{+}3(k-1)(2k{+}3)f_{z}^{2}\right)$
$\displaystyle{-}6(16(2k{+}3)(k{+}2)m\pi
iff_{\tau}{+}(4k^{3}{+}8k^{2}{-}11k{-}24)f_{z}^{2})ff_{z}^{2}f_{zz}$
$\displaystyle{-}24(4k^{2}{+}16k{+}17)m\pi
iff_{\tau}f_{z}^{4}{-}(8k^{3}{+}12k^{2}{-}38k{-}65)f_{z}^{6}\Big{)},$
$\displaystyle P_{k,m}=$ $\displaystyle{1\over(2kff_{zz}{-}8m\pi
iff_{\tau}{-}(2k{-}1)f_{z}^{2})^{5/2}}\Big{(}16m^{2}\pi^{2}f^{4}f_{\tau\tau
z}{+}8m\pi if^{3}f_{z}f_{\tau zz}{-}f^{2}f_{z}^{2}f_{zzz}$
$\displaystyle{-}16m^{2}\pi^{2}f^{3}f_{z}f_{\tau\tau}{+}4(2(k{+}2)m\pi
iff_{zz}{-}(2k{+}7)m\pi if_{z}^{2})f^{2}f_{\tau
z}{-}2(k{+}2)f^{2}f_{z}f_{zz}^{2}$ $\displaystyle{-}2((4m\pi
i(k+2)ff_{\tau}{-}(2k+5)f_{z}^{2})ff_{z}f_{zz}{+}(4m\pi
iff_{\tau}{-}f_{z}^{2})(2k{+}5)f_{z}^{3}\Big{)},$ $\displaystyle Q_{k,m}=$
$\displaystyle{4m\pi i(ff_{\tau zz}{-}2f_{z}f_{\tau
z})f^{2}{-}f^{2}(f_{z}f_{zzz}{+}f_{zz}^{2}){-}(4m\pi
iff_{\tau}{-}5f_{z}^{2})ff_{zz}{+}(8m\pi
iff_{\tau}{-}3f_{z}^{2})f_{z}^{2}\over(2kff_{zz}{-}8m\pi
iff_{\tau}{-}(2k{-}1)f_{z}^{2})^{2}},$ $\displaystyle R_{k,m}=$
$\displaystyle{f^{2}f_{zzz}{-}3ff_{z}f_{zz}{+}2f_{z}^{3}\over(2kff_{zz}{-}8m\pi
iff_{\tau}{-}(2k{-}1)f_{z}^{2})^{3/2}}.$

The Lie algebra of the group $G_{k,m}$ is spanned by the vector fields

$\langle\partial_{\tau},\ 2\tau\partial_{\tau}+z\partial_{z},\
\tau^{2}\partial_{\tau}+\tau z\partial_{z}-(2m\pi iz^{2}f+k\tau
f)\partial_{f},\ \tau\partial_{z}-4m\pi izf\partial_{f},\ \partial_{z},\
f\partial_{f}\rangle.$

Note that the invariance of the above expressions under $G_{k,m}$-action makes
them Jacobi functions, that is, Jacobi forms or weight 0 and order 0.
Furthermore, their numerators and denominators are Jacobi forms of the same
weight and index. Therefore, there is an analogy between invariants of the
group $G_{k,m}$ and invariants of the groups $G_{k}$ and $\mathcal{G}_{k}$.
The expression $D=2kff_{zz}-8m\pi iff_{\tau}-(2k-1)f_{z}^{2}$ is a conditional
invariant of the group $G_{k,m}$ (that is, its vanishing is an invariant
condition). Moreover, given a Jacobi form $f$ of weight $k$ and index $m$, the
function $D$ is a Jacobi form of weight $2k+2$ and index $2m$. Indeed,
$mfD=[[f,f^{2}]]_{1}$, or alternatively $D={2\over 2k-1}[f,f]_{X,2}$, with the
right-hand side being actually $X$-independent. Introducing

$\displaystyle M:={[f,f^{2}]_{{2k\over m(10k-3)},3}\over 6k-1},\quad
Q:={(10k-3)[f,M]_{X,1}\over 3m^{2}(2k-1)(4k-1)},$ $\displaystyle
L:={\left[f^{2},\left[f,f\right]_{X,2}\right]_{-{2k\over
m(8k+3)},2}+k(32k^{3}-14k+3)Q-{(4k+3)(4k+11)\over(2k-1)}[f,f]_{X,2}^{2}\over
4(2k-1)(4k-1)(4k+3)},$

we can write down all the above invariants of the $G_{k,m}$-action in terms of
Rankin–Cohen brackets,

$\displaystyle L_{k,m}={(2k-1)^{2}L\over 4[f,f]_{X,2}^{2}},\quad
M_{k,m}=-{(10k-3)M\over 3m(2k-1)(4k-1)(6k-1)\left({2\over
2k-1}[f,f]_{X,2}\right)^{3/2}},$ $\displaystyle N_{k,m}={{1\over
4(8k-7)}[f^{2},L]_{{-2k\over
m(12k+7),2}}-{2(4k^{2}+3k-1)\over(2k-1)}[f,f]_{X,2}Q-{4k+23\over
4(2k-1)}[f,f]_{X,2}L-{2(10k-3)^{2}\over
9m^{2}(2k-1)^{2}(4k-1)}M^{2}\over(4k-1){8\over(2k-1)^{3}}[f,f]_{X,2}^{3}}$
$\displaystyle
P_{k,m}={{8(k+2)(10k-3)\over(2k-1)(4k-1)}\left[f,f^{2}\right]_{{2k\over
m(10k-3)},3}[f,f]_{X,2}-{10k-3\over(4k-1)(6k+5)}\left[\left[f,f^{2}\right]_{{2k\over
m(10k-3)},3},f^{2}\right]_{{2k\over 5m(2k+1)},2}\over
3m(2k-1)(4k-1)(6k-1)({2\over 2k-1}[f,f]_{X,2})^{5/2}},$ $\displaystyle
Q_{k,m}={-(2k-1)^{2}Q\over 4[f,f]_{X,2}^{2}},\quad
R_{k,m}={8(3k-1)[f,f^{2}]_{{1\over 4m(3k-1)},3}\over
3(2k-1)(4k-1)(6k-1)({2\over 2k-1}[f,f]_{X,2})^{3/2}}.$

The expressions for the brackets with the unspecified parameter $X$ do not
involve $X$ in their expanded forms.

Any generic third-order $G_{k,m}$-invariant involutive PDE system (governing
Jacobi forms) can be obtained by expressing all third-order invariants
$N_{k,m}$, $P_{k,m}$, $Q_{k,m}$, $R_{k,m}$ as functions of the second-order
invariants $L_{k,m},M_{k,m}$, see Examples 3.2 and 3.2 where we present such
systems for the weak Jacobi forms $\varphi_{-1,2}(\tau,z)$ and
$\varphi_{-2,1}(\tau,z)$. The action of the group $G_{k,m}$ on the six-
dimensional solution space of any such system is locally transitive (possesses
an open orbit).

There also exist two different non-generic $G_{k,m}$-invariant involutive PDE
systems, first of which contain the equation $D=0$, equivalently,

$\displaystyle 8m\pi if_{\tau}=2kf_{zz}-(2k-1)f_{z}^{2}/f;$ (3.15a) note that
the substitution $f=\mathrm{e}^{\varphi}$ reduces (3.15a) to a potential
Burgers equation for $\varphi$, namely, $8m\pi
i\varphi_{\tau}=2k\varphi_{zz}+\varphi_{z}^{2}$, while the substitution
$f=\psi^{2k}$ linearises equation (3.15a) to $4m\pi i\psi_{t}=k\psi_{zz}$. In
the non-generic case the above invariants are not defined, and therefore we
consider instead $\displaystyle\mathcal{L}_{k,m}={1\over
L_{k,m}},\quad\mathcal{M}_{k,m}={L_{k,m}^{{1\over 2}}\over M_{k,m}^{{2\over
3}}},\quad\mathcal{N}_{k,m}={N_{k,m}\over L_{k,m}^{{3\over 2}}},$
$\displaystyle\mathcal{P}_{k,m}={P_{k,m}\over L_{k,m}^{{1\over
2}}M_{k,m}},\quad\mathcal{Q}_{k,m}={Q_{k,m}\over
L_{k,m}},\quad\mathcal{R}_{k,m}={R_{k,m}\over M_{k,m}}.$ In view of equation
(3.15a), the invariants $\mathcal{L}_{k,m}$, $\mathcal{Q}_{k,m}$ and
$\mathcal{R}_{k,m}$ have fixed values, $\mathcal{L}_{k,m}=0$,
$\mathcal{Q}_{k,m}=-{1\over 4k}$, $\mathcal{R}_{k,m}={1\over k}$, and thus the
essential invariants are $\mathcal{M}_{k,m}$, $\mathcal{N}_{k,m}$ and
$\mathcal{P}_{k,m}$, which involve $z$-derivatives of $f$ only and are of
order 4, 6 and 5, respectively. In particular, to obtain a non-generic
$G_{k,m}$-invariant involutive PDE system with a transitive $G_{k,m}$-action
on the solution space, one has to add to equation (3.15a) a sixth-order
equation that is a function of $\mathcal{M}_{k,m}$, $\mathcal{N}_{k,m}$ and
$\mathcal{P}_{k,m}$. One particular choice of this kind is
$\displaystyle\left({(\ln f)_{zzzzz}\over(\ln f)_{zzz}}+{6\over k}(\ln
f)_{zz}\right)_{z}=0.$ (3.15b)

The system (3.15) is in involution and admits $G_{k,m}$ as a symmetry group.
It can be represented in invariant form as

$\mathcal{L}_{k,m}=0,\quad\mathcal{M}_{k,m}^{3}(\mathcal{N}_{k,m}-2\mathcal{P}_{k,m})-8=0,$

see Example 3.2 of Section 3.2 where we obtain a system of this kind for the
Jacobi theta functions.

Another type of non-generic $G_{k,m}$-invariant involutive PDE systems is
associated with the value $m=0$. In this case, the invariants $L_{k,0}$ and
$M_{k,0}$ are functionally dependent. Moreover, the group $G_{k,0}$ is five-
dimensional since the $f$-scalings are no longer admissible. The Lie algebra
of the group $G_{k,0}$ is spanned by the vector fields

$\langle\partial_{\tau},\ \partial_{z},\ \tau\partial_{z},\
2\tau\partial_{\tau}+z\partial_{z}-kf\partial_{f},\
\tau^{2}\partial_{\tau}+\tau z\partial_{z}-k\tau f\partial_{f}\rangle.$

The invariants of the group $G_{k,0}$ that are necessary for the exposition
below are as follows

$\displaystyle\mathscr{I}_{k}={(kff_{zz}-(k+1)f_{z}^{2})f_{\tau\tau}-kff_{\tau
z}^{2}+(k+1)f_{\tau}(2f_{z}f_{\tau z}-f_{\tau}f_{zz})\over f^{1+{4\over
k}}(kff_{zz}-(k+1)f_{z}^{2})},$
$\displaystyle\mathscr{J}_{k}={(kff_{zz}-(k+1)f_{z}^{2})f_{\tau zz}-(kff_{\tau
z}-(k+1)f_{\tau}f_{z}^{2})f_{zzz}+(k+2)f_{zz}(f_{z}f_{\tau
z}-f_{\tau}f_{zz})\over f^{1+{4\over k}}(kff_{zz}-(k+1)f_{z}^{2})},$
$\displaystyle\mathscr{L}_{k}={f_{z}\over f^{1+{1\over
k}}},\quad\mathscr{M}_{k}={f_{zz}\over f^{1+{2\over
k}}},\quad\mathscr{N}_{k}={f_{zzz}\over f^{1+{3\over k}}},$

see Example 3.2 of Section 3.2 where derive differential system for the
Weierstrass $\wp$-function.

###  3.2 Examples

Here we provide examples of nonlinear involutive third-order PDE systems that
characterise Jacobi forms uniquely up to the action of the corresponding six-
dimensional symmetry group $G_{k,m}$. We refer to [2] for an alternative
construction of linear modular differential equations satisfied by Jacobi
forms.

Example 11. The weak Jacobi form $\varphi_{-1,2}(\tau,z)$ of weight $-1$ and
index $2$ is defined as
$\varphi_{-1,2}(\tau,z)=\Delta^{-1/8}(\tau)\vartheta_{1}(\tau,2z)$ where
$\Delta$ is the modular discriminant and $\vartheta_{1}$ is a Jacobi theta
function, see [14, formula (4.31)],

$\vartheta_{1}(\tau,z)=2\sum_{n=0}^{\infty}(-1)^{n+1}\mathrm{e}^{\pi
i\left(n+{1\over 2}\right)\tau}\sin((2n+1)\pi z).$

The function $f(\tau,z)=\varphi_{-1,2}(\tau,z)$ satisfies the following
$G_{-1,2}$-invariant overdetermined involutive PDE system,

$\displaystyle 2\pi iff_{\tau\tau\tau}=2\pi
if_{\tau}f_{\tau\tau}+f_{\tau\tau}f_{zz}-f_{\tau z}^{2},$ $\displaystyle
ff_{\tau\tau z}=3f_{z}f_{\tau\tau}-2f_{\tau}f_{\tau z},$ $\displaystyle
ff_{\tau zz}=8\pi i(ff_{\tau\tau}-f_{\tau}^{2})+2f_{z}f_{\tau
z}-f_{\tau}f_{zz},$ $\displaystyle ff_{zzz}=16\pi i(ff_{\tau
z}-f_{\tau}f_{z})+f_{z}f_{zz},$

which can be obtained from the system (1.2) by the change of variables

$f(\tau,z)=g(\tilde{\tau},\tilde{z}),\quad\tau={i\tilde{\tau}\over\pi},\quad
z={\tilde{z}\over 2\pi}.$

Invariant form of the above system is

$N_{-1,2}=-32M_{-1,2}^{2}+L_{-1,2},\quad P_{-1,2}=-M_{-1,2},\quad
Q_{-1,2}=-{1\over 4}(L_{-1,2}+1),\quad R_{-1,2}=2M_{-1,2}.$

Example 12. The weak Jacobi form $\varphi_{-2,1}(\tau,z)$ of weight $-2$ and
index $1$ is defined as
$\varphi_{-2,1}(\tau,z)=\Delta^{-1/4}(\tau)\vartheta_{1}^{2}(\tau,z)$, see
[14, formula (4.29)]. The function $f(\tau,z)=\varphi_{-2,1}(\tau,z)$
satisfies the following $G_{-2,1}$-invariant overdetermined involutive PDE
system,

$\displaystyle 2\pi if^{2}f_{\tau\tau\tau}=(2ff_{\tau\tau}-f_{\tau}^{2})(2\pi
if_{\tau}+f_{zz})-f_{z}^{2}f_{\tau\tau}-2ff_{\tau z}^{2}+2f_{\tau}f_{z}f_{\tau
z},$ $\displaystyle f^{2}f_{\tau\tau z}=f_{z}(2ff_{\tau\tau}-f_{\tau}^{2}),$
$\displaystyle f^{2}f_{\tau zz}=2\pi
if(ff_{\tau\tau}-f_{\tau}^{2})+f_{z}(2ff_{\tau z}-f_{\tau}f_{z}),$
$\displaystyle f^{2}f_{zzz}=4\pi if(ff_{\tau
z}-f_{\tau}f_{z})+f_{z}(2ff_{zz}-f_{z}^{2}),$

which can be obtained from the system (1.2) by the change of variables

$f(\tau,z)=g^{2}(\tilde{\tau},\tilde{z}),\quad\tau={i\tilde{\tau}\over\pi},\quad
z={\tilde{z}\over\pi}.$

Invariant form of the above system is

$N_{-2,1}=-32M_{-2,1}^{2}+2L_{-2,1}+1,\quad P_{-2,1}=0,\quad Q_{-2,1}=-{1\over
8}(L_{-2,1}+1),\quad R_{-2,1}=M_{-2,1}.$

Example 13. The Jacobi theta function $\vartheta_{1}$ is a Jacobi form of
index ${1\over 2}$ and weight ${1\over 2}$. It satisfies the heat equation

$\displaystyle 4\pi i(\vartheta_{1})_{\tau}=(\vartheta_{1})_{zz},$ (3.16a)
which coincides with equation (3.15a) for $k=m={1\over 2}$, as well as a
sixth-order equation involving $z$-derivatives of $\vartheta_{3}$ only,
$\displaystyle\left({(\ln\vartheta_{1})_{zzzzz}\over(\ln\vartheta_{1})_{zzz}}+12(\ln\vartheta_{1})_{zz}\right)_{z}=0,$
(3.16b)

which coincides with (3.15b) for $k={1\over 2}$. The system (3.16) is in
involution and admits the Jacobi group $G_{{1\over 2},{1\over 2}}$ as a
symmetry group. It also holds for Jacobi theta functions $\vartheta_{2}$,
$\vartheta_{3}$ and $\vartheta_{4}$ due to the fact that the transformations
between different theta functions belong to the group $G_{{1\over 2},{1\over
2}}$. In a somewhat different form, the system (3.16) for Jacobi theta
functions was obtained in [11, Example 4.1]. We also refer to [4, 36] for
other differential systems satisfied by the Jacobi theta functions.

The system (3.16) can be written in invariant form as

$\mathcal{L}_{{1\over 2},{1\over 2}}=0,\quad\mathcal{M}_{{1\over 2},{1\over
2}}^{3}(\mathcal{N}_{{1\over 2},{1\over 2}}-2\mathcal{P}_{{1\over 2},{1\over
2}})-8=0.$

Example 14. The Weierstrass $\wp$-function is a Jacobi form of weight $2$ and
index $0$,

$\wp(\tau,z)={1\over z^{2}}+\sum\limits_{\omega\in
L/\\{0\\}}\left({1\over(z-\omega)^{2}}-{1\over\omega^{2}}\right),$

where the lattice $L=\mathbb{Z}+\mathbb{Z}\tau$. Using differentiation rules
from [4, Formulae (37), (100) and (101)], we find that the Weierstrass
$\wp$-function satisfies the involutive $G_{2,0}$-invariant system of
differential equations

$\displaystyle\wp_{\tau\tau}={{1\over 16}\phi+3\pi^{2}(2\wp\wp_{\tau
z}^{2}-6\wp_{\tau}\wp_{z}\wp_{\tau z}+3\wp_{\tau}^{2}\wp_{zz})\over
3\pi^{2}(2\wp\wp_{zz}-3\wp_{z}^{2})},$ $\displaystyle\wp_{\tau zz}={{i\over
4}\phi+\pi(36\wp\wp_{z}(2\wp\wp_{\tau
z}-3\wp_{\tau}\wp_{z})-12\wp_{zz}(\wp_{z}\wp_{\tau
z}-\wp_{\tau}\wp_{zz}))\over 3\pi(2\wp\wp_{zz}-3\wp_{z}^{2})},$
$\displaystyle\wp_{zzz}=12\wp\wp_{z},\quad\text{where}\quad\phi:=16\wp_{zz}^{3}-72\wp^{2}\wp_{zz}^{2}-216\wp\wp_{z}^{2}\wp_{zz}+54\wp_{z}^{4}+864\wp^{3}\wp_{z}^{2}.$

Given $\wp_{z}^{2}=4\wp^{3}-g_{2}(\tau)\wp-g_{3}(\tau)$ where
$g_{2}(\tau)={4\over 3}\pi^{4}E_{4}(\tau)$ and $g_{3}(\tau)={8\over
27}\pi^{6}E_{6}(\tau)$, we obtain
$\phi=-2(g_{2}^{3}-27g_{3}^{2})=-2(2\pi)^{12}\Delta(\tau)$. Note that the
first of the above equations can be written in a symmetric Monge–Ampère form,

$2\wp(\wp_{zz}\wp_{\tau\tau}-\wp_{z\tau}^{2})=3\wp_{z}^{2}\wp_{\tau\tau}-6\wp_{\tau}\wp_{z}\wp_{\tau
z}+3\wp_{\tau}^{2}\wp_{zz}-{(2\pi)^{10}\over 6}\Delta(\tau).$

The invariant form of the above system is

$\displaystyle\mathscr{N}_{2}=12\mathscr{L}_{2},\quad\mathscr{J}_{2}=-4\pi
i\mathscr{I}_{2},$ $\displaystyle
24\pi^{2}(2\mathscr{M}_{2}-3\mathscr{L}_{2}^{2})\mathscr{I}_{2}=8\mathscr{M}_{2}^{3}-36\mathscr{M}_{2}^{2}-108\mathscr{L}_{2}^{2}\mathscr{M}_{2}+27\mathscr{L}_{2}^{2}(\mathscr{L}_{2}^{2}+16).$

##  4 Concluding remarks

  * •

The results of this paper can be extended to other types of modular forms
(such as Siegel modular forms, Picard modular forms, etc), namely, every
modular form $f$ on a discrete subgroup $\Gamma$ of a Lie group $G$ should
solve a nonlinear PDE system $\Sigma$ such that:

    * –

system $\Sigma$ is involutive (compatible);

    * –

system $\Sigma$ is of finite type (has finite-dimensional solution space);

    * –

system $\Sigma$ is $G$-invariant, furthermore, the Lie group $G$ acts on the
solution space of $\Sigma$ locally transitively and with an open orbit (thus,
the dimension of the solution space of $\Sigma$ equals ${\rm dim}\,G$);

    * –

the modular form $f$ is a generic solution of system $\Sigma$ ($f$ belongs to
the open orbit), in particular, solution $f$ has discrete stabiliser $\Gamma$;

    * –

system $\Sigma$ is expressible via algebraic relations among differential
invariants of a suitable action of $G$.

In the case of classical modular forms $f$ considered in this paper, we have:
$\Gamma=\textrm{SL}(2,\mathbb{Z})$, $G=\textrm{SL}(2,\mathbb{R})$, and system
$\Sigma$ is a third-order nonlinear $\textrm{SL}(2,\mathbb{R})$-invariant ODE
for $f$.

In particular, involutive differential systems for Siegel modular forms should
be based on differential invariants of the symplectic group $\textrm{Sp}(2g)$.
Some results in this direction are already available, thus, differential
systems for theta constants were discussed in [33] (genus $g=2$) and [43]
(general $g$).

  * •

Although modular forms provide generic solutions of the ODEs discussed in this
paper, the same ODEs possess non-generic rational solutions which may also be
of interest. Thus, as already mentioned in the introduction, the integrability
condition for the Lagrangian density $u_{x}u_{y}f(u_{t})$ is the fourth-order
ODE (2.14) for $f(\tau)$. The generic solution of this ODE is the Eisenstein
series, $f(\tau)=E_{1,3}(\tau)$, however, it also possesses a simple non-
generic solution $f(\tau)=\tau$ which corresponds to the Lagrangian density
$u_{x}u_{y}u_{t}$ (with interesting properties, see [18]).

Same applies to differential systems for Jacobi forms.

  * •

Every third-order ODE with $G_{k}$ symmetry can be linearised by a standard
procedure as discussed, e.g., in [10]: the general solution $f(\tau)$ of any
such equation can be represented parametrically as

$\tau={\tilde{w}\over w},\qquad f={w^{k}\over W^{k/2}}$ (4.17)

where $w(s)$ and $\tilde{w}(s)$ are two linearly independent solutions of a
second-order linear equation $w_{ss}+pw_{s}+qw=0$, and
$W=\tilde{w}_{s}w-w_{s}\tilde{w}$ is the Wronskian of $w$ and $\tilde{w}$.
Here the coefficients $p(s)$ and $q(s)$ depend on the third-order ODE and can
be efficiently reconstructed. The details are as follows. Differentiating the
second equation (4.17) with respect to $\tau$ using the relations ${ds\over
d\tau}={w^{2}\over W}$ and $W_{s}=-pW$, one obtains

$I_{k}=(2\pi ik)^{2}\delta\quad{\rm and}\quad J_{k}=(2\pi ik)^{3}\delta_{s}$

where $\delta={1\over 2}p_{s}-q+{1\over 4}p^{2}$ and $I_{k},J_{k}$ are the
invariants from Section 2.1. Thus, for a given third-order ODE
$F(I_{k},J_{k})=0$, one has to choose the coefficients $p(s),\,q(s)$ such that
$F((2\pi ik)^{2}\delta,\,(2\pi ik)^{3}\delta_{s})=0$.

This linearisation procedure leads to familiar parametrisations of modular
forms by hypergeometric functions.

## Acknowledgements

We thank F. Cléry, G. van der Geer, M. Pavlov, R.O. Popovych, A. Prendergast-
Smith, F. Stromberg, A. Veselov, C. Wuthrich and V. Zudilin for useful
discussions. The research of SO was supported by the NSERC Postdoctoral
Fellowship program.

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