Some remarkable autonomous systems

TL;DR

This paper explores the mathematical structures connecting Darboux-Halphen-Ramanujan systems, contact geometry, Frobenius manifolds, and projective connections, emphasizing the role of contact geometry and autonomous systems derived from group determinants.

Contribution

It establishes new links between complex geometric structures and autonomous systems, highlighting the significance of contact geometry in this context.

Findings

Connections between Darboux-Halphen-Ramanujan systems and contact geometry

Identification of canonical coordinates in Frobenius manifolds

Analysis of autonomous systems from group determinants

Abstract

We study the links of the Darboux-Halphen-Ramanujan system, with contact geometry, canonical coordinates of some $3$-dimensional Frobenius manifolds and projective connections on Riemann surfaces. One of our important goals is to highlight the role of contact geometry in this setting. We also study autonomous systems "derived" from the potential given by the Group-determinant, of any cyclic group $\Z/n\Z$, $n\geqslant3$, and the Klein group $\Z/2\Z\times\Z/2\Z$.

Equations

$$\begin{array}{rl}{E}_4(\tau )& =1+240\sum _{n=1}^{\infty }\frac{n^3{q}^n}{1-q^n}=1+240\sum _{n=1}^{\infty }{\sigma }_3(n)q^n\\ E_6(\tau )& =1-504\sum _{n=1}^{\infty }\frac{n^5{q}^n}{1-q^n}=1-504\sum _{n=1}^{\infty }{\sigma }_5(n)q^n\\ E_2(\tau )& =1-24\sum _{n=1}^{\infty }\frac{n{q}^n}{1-q^n}=1-24\sum _{n⩾1}{\sigma }_1(n)q^n,\end{array}$$

$$\wp (u)=\wp (u;{\omega }_1,{\omega }_2)=\frac{1}{u^2}+\sum _{m^2+n^2\ne 0}\left(\frac{1}{(u+2m{\omega }_1+2n{\omega }_2{)}^2}-\frac{1}{(2m{\omega }_1+2n{\omega }_2{)}^2}\right).$$

$$\begin{array}{r}{e}_1:=\wp ({\omega }_1;{\omega }_1,{\omega }_2)\\ e_2:=\wp ({\omega }_2;{\omega }_1,{\omega }_2)\\ e_3:=\wp ({\omega }_3;{\omega }_1,{\omega }_2)\end{array}$$

$$\begin{array}{r}{e}_k=:\frac{1}{(2{\omega }_1{)}^2}{e}_k(\tau )\\ k=1,2,3.\end{array}$$

$$\begin{array}{rl}{g}_2& =60\sum _{m^2+n^2\ne 0}\frac{1}{(2m{\omega }_1+2n{\omega }_2{)}^4},\\ g_3& =140\sum _{m^2+n^2\ne 0}\frac{1}{(2m{\omega }_1+2n{\omega }_2{)}^6}.\end{array}$$

$$\begin{array}{rl}{E}_4(\tau )=& 12{\left(\frac{{\omega }_1}{\pi }\right)}^4{g}_2=1+240\sum _{n=1}^{\infty }\frac{n^3{q}^n}{1-q^n}\\ E_6(\tau )=& 216{\left(\frac{{\omega }_1}{\pi }\right)}^6{g}_3=1-504\sum _{n=1}^{\infty }\frac{n^5{q}^n}{1-q^n}\end{array}$$

$${\wp }^{\prime 2}=4{\wp }^3-g_2\wp -g_3$$

$${\wp }^{\prime \prime }=6{\wp }^2-\frac{1}{2}{g}_2.$$

$$\begin{split}&f(\tau)=(c\tau+d)^{-2k}f\left(\dfrac{a\tau+b}{c\tau+d}\right)\\ &\forall\,\left(\begin{array}[]{cc}a&b\\ c&d\end{array}\right)\in PSL(2,\mathbb{Z})=SL(2,\mathbb{Z})/\{\pm I_{2}\}.\end{split}$$

$$\begin{array}{r}f(\tau +1)=f(\tau )\end{array}$$

$$\begin{array}{r}f(-1/\tau )={\tau }^{2k}f(\tau ).\end{array}$$

$$\tilde{f}(q)=\sum _{n\in \mathbb{Z}}{a}_n{q}^n$$

$$\begin{array}{rl}\frac{1}{2i\pi }\frac{d}{d\tau }{E}_4& =\frac{1}{3}(E_2{E}_4-E_6)\\ \frac{1}{2i\pi }\frac{d}{d\tau }{E}_6& =\frac{1}{2}(E_2{E}_6-E_4^2)\\ \frac{1}{2i\pi }\frac{d}{d\tau }{E}_2& =\frac{1}{12}(E_2^2-E_4).\end{array}$$

$$y^{\prime}=\frac{i}{\pi}\Bigl{(}-y^{2}+\frac{\pi^{2}}{3}E_{2}(\tau)y+\frac{2}{9}\pi^{4}E_{4}(\tau)\Bigr{)}.$$

$$\frac{1}{2i\pi }\frac{d}{d\tau }\left(e_k\right)=\frac{1}{2{\pi }^2}\left(-e_k^2+\frac{{\pi }^2}{3}{E}_2(\tau )e_k+\frac{2}{9}{\pi }^4{E}_4(\tau )\right),1\le k\le 3.$$

$$x=\frac{{\pi }^2}{12}{E}_2,y=\frac{{\pi }^4}{12}{E}_4,z=\frac{{\pi }^6}{216}{E}_6,$$

$$\begin{array}{rl}\frac{dx}{dt}& =\frac{1}{2}{x}^2-\frac{1}{24}y\\ \frac{dy}{dt}& =2xy-3z\\ \frac{dz}{dt}& =3xz-\frac{1}{6}{y}^2.\end{array}$$

$$\begin{array}{rl}\frac{d}{dt}(X_1+X_2)& =X_1{X}_2\\ \frac{d}{dt}(X_2+X_3)& =X_2{X}_3\\ \frac{d}{dt}(X_3+X_1)& =X_3{X}_1,\end{array}$$

$$\begin{array}{rl}x& =\frac{1}{3}\left(X_1+X_2+X_3\right)\\ y& =\frac{4}{3}\left(X_1^2+X_2^2+X_3^2-X_1{X}_2-X_2{X}_3-X_3{X}_1\right)\\ z& =\frac{4}{27}\left(2X_1-X_2-X_3\right)\left(2X_2-X_3-X_1\right)\left(2X_3-X_1-X_2\right).\end{array}$$

$$\begin{array}{rl}\mathcal{X}& =1-24\sum _{n=1}^{\infty }{\sigma }_1(n)q^n\\ \mathcal{Y}& =1+240\sum _{n=1}^{\infty }{\sigma }_3(n)q^n\\ \mathcal{Z}& =1-504\sum _{n=1}^{\infty }{\sigma }_5(n)q^n,\end{array}$$

$$\begin{array}{rl}\frac{d\mathcal{X}}{dt}& =\frac{1}{12}(\mathcal{Y}-{\mathcal{X}}^2)\\ \frac{d\mathcal{Y}}{dt}& =\frac{1}{3}(\mathcal{Z}-\mathcal{XY})\\ \frac{d\mathcal{Z}}{dt}& =\frac{1}{2}({\mathcal{Y}}^2-\mathcal{XZ}),\end{array}$$

$$\omega \wedge (d\omega {)}^m\text{-}\ne 0.$$

$${\omega }_{\alpha }\wedge (d{\omega }_{\alpha }{)}^m\ne \text{-}0.$$

$$\Omega :TM\stackrel{\sim }{\to }{T}^{\star }M.$$

$$i_{X_{\mathcal{F}}}\Omega =-d\mathcal{F}.$$

$${\mathcal{L}}_Y\Omega =\Omega .$$

$$\begin{array}{rl}{\dot{\mathcal{X}}}_1& =\frac{1}{2}{\mathcal{X}}_1^2-\frac{1}{24}{\mathcal{Y}}_1\\ {\dot{\mathcal{Y}}}_1& =2{\mathcal{X}}_1{\mathcal{Y}}_1-3{\mathcal{Z}}_1\\ {\dot{\mathcal{Z}}}_1& =3{\mathcal{X}}_1{\mathcal{Z}}_1-\frac{1}{6}{\mathcal{Y}}_1^2,\dot{}=\frac{d}{dt}.\end{array}$$

$$\mathcal{F}=\frac{1}{2}{q}_1^2{p}_1-\frac{1}{48}{p}_1^2{p}_2^8+3q_1{q}_2{p}_2.$$

$$\begin{array}{rl}{\dot{\mathcal{X}}}_1& =\frac{1}{2}{\mathcal{X}}_1^2-\frac{{\lambda }^9}{24}{\mathcal{Y}}_1\\ \lambda {\dot{\mathcal{Y}}}_1& =2\lambda {\mathcal{X}}_1{\mathcal{Y}}_1-3\lambda {\mathcal{Z}}_1\\ {\dot{\mathcal{Z}}}_1& =-\frac{1}{6}{\lambda }^9{\mathcal{Y}}_1^2+3{\mathcal{X}}_1{\mathcal{Z}}_1\\ \dot{\lambda }& =-3\lambda {\mathcal{X}}_1.\end{array}$$

$$Y_{\mathcal{F}}:=(1/2x_1^2-{\lambda }^9{y}_1)\frac{\partial }{\partial x_1}+(2x_1{y}_1-3z_1)\frac{\partial }{\partial y_1}+(-1/6{\lambda }^9{y}_1^2+3x_1{z}_1)\frac{\partial }{\partial z_1}-3\lambda x_1\frac{\partial }{\partial \lambda }.$$

Full text

1. Introduction

The system of Darboux-Halphen was first exhibited by Darboux [11] in his study of triply orthogonal systems, that is a system of three one-parameter family of surfaces in $\mathbb{R}^{3}$ such that at any common point to three representatives of each family, the tangent planes of the surfaces are $2$ by $2$ orthogonal. Their solutions were given in terms of (logarithmic) derivatives of theta functions by [6, 20, 21]. Later on they appeared in the mathematical physics literature as some reduction, of the selfdual Yang-Mills equations [5, 2, 3, 4, 1, 35]. One of the remarquable works in the field of Darboux-like autonomous polynomials differential systems were made in [28, 29, 30, 31]. These works in particular associate to any second order linear ODE with regular or irregular singular points on the projective line $\mathbb{C}P_{1}$, a generalized Darboux-Halphen system. Darboux-Halphen systems, see Section 2 below, are related to the Chazy III equation which was first introduced in [8], in his study of third order ODEs solved in the highest derivative, which are autonomous and rational in the other derivatives, and which possess a Painlevé like property, of having a movable natural boundary. Chazy [8] found twelve such equations in total. We refer to [2, 3, 4, 1, 9] for important contributions regarding the study of Chazy equations.

On the other hand the differential system of Ramanujan was invented by Ramanujan [32] in his study of differential properties of (quasi) modular forms for the modular group $\rm{SL}(2,\mathbb{Z})$. It implies that the three normalized Eisenstein series $E_{2}$, $E_{4}$, $E_{6}$, see Section 2 below, generate a differential ring over $\mathbb{C}$. The Ramanujan system has important arithmetical applications, see [37] for instance. It was also used in [24, 13, 15] to derive transcendence type results for the field generated by $q=e^{2i\pi\tau}$, $E_{2}(q)$, $E_{4}(q)$ and $E_{6}(q)$, $\tau\in\mathfrak{H}=\{\rm{Im}(\tau)>0\}$. Besides there has been intensive investigations of systems of generalized Darboux-Ramanujan-Halphen type in the context of Gauss-Manin connections on certain moduli stacks, classifying Abelian schemes with a preferred choice of frame on their first de Rham cohomology group, that is in the theory of the so-called Gauss-Manin connection "in disguise", we refer to [16, 25, 26, 27].

In this article we begin in Section 2 with introducing the notations, then in Subsection 2.1 we give a contact geometric interpretation of the Ramanujan system, see Theorem 2.7. Further in Subsection 2.3 we give a link between autonomous systems of ODEs and Hamiltonians systems and deduce from the Poisson theorem a way to build more first integrals of the given autonomous systems starting from a known one, see Proposition 2.10. After that we give in Theorem 3.1 the canonical coordinates of the three dimensional Frobenius manifold associated to the solution $\frac{\pi i}{3}E_{2}$ to the Chazy equation (3.6), in terms of its flat coordinates and some solutions of the Darboux-Halphen system (2.20). We also give the formula of change of basis between the flat coordinate vector fields and the canonical coordinate vector fields in Proposition 3.2. Finally in Section 4 after introducing affine and projective connections on Riemann surfaces, as well as the Klein bidifferential and Wirtinger connection associated to an even $\theta$-characteristics with non-vanishing $\theta$-constant, we prove in Theorem 4.2 that $-6e_{1}$, $-6e_{2}$, $-6e_{3}$, with $e_{1}$, $e_{2}$, $e_{3}$ defined in (2.2), (2.3), can be interpreted as the Wirtinger connections associated respectively to the even $\theta$-characteristics $\begin{bmatrix}0\\ 0\end{bmatrix},\begin{bmatrix}1\\ 0\end{bmatrix},\begin{bmatrix}0\\ 1\end{bmatrix}$, (genus one case).

2. Darboux-Halphen-Ramanujan systems and autonomous systems

Let us first recall some notions in order to fix notations and as a way of introduction. We remind that the normalized Eisenstein series are given by

[TABLE]

$\,q=e^{2i\pi\tau},\,\tau\in\mathfrak{H}:=\{\operatorname{Im}(\tau)>0\}$ and $\sigma_{k}(n)=\sum_{d|n}d^{k}$. We set $\displaystyle{\omega_{3}=\omega_{1}+\omega_{2}}$ and adopt the classical notations for Weierstrass elliptic functions

[TABLE]

Associated to the Weierstrass $\wp$ function we have the following three expressions

[TABLE]

with $\omega_{3}=\omega_{1}+\omega_{2}$. With this definition of the $e_{k}$ we further introduce another set of functions which depend on the modular variable $\tau:=\omega_{2}/\omega_{1}$

[TABLE]

We will use the functions $e_{k}(\tau)$ in the Theorem 2.2 below.

Let us introduce the Eisenstein series

[TABLE]

They are related to the normalized Eisenstein series $E_{4}$ and $E_{6}$ by the following relations [33, 7, 10]

[TABLE]

Finally we have (see for ex. [33, p.84]) that the Weierstrass function $\wp$ satisfies the fundamental differential equation

[TABLE]

so that

[TABLE]

For the following definition we refer to [7, 33, 10].

Definition 2.1**.**

Let $k$ be an integer. We say that $f$ is weakly modular of weight $2k$, if $f$ is meromorphic on the upper half plane $\mathfrak{H}$ and verifies the relation

[TABLE]

This is equivalent to

[TABLE]

[TABLE]

Condition (2.7) implies that $f$ can be expressed as a function of $q=e^{2i\pi\tau}$, function which we will denote by $\tilde{f}$; it is meromorphic in the disc $|q|<1$ with the origin removed. If $\tilde{f}$ extends to a meromorphic (resp. holomorphic) function at the origin, we say, that $f$ is meromorphic (resp. holomorphic) at infinity. This means that $\tilde{f}$ admits a Laurent expansion in a neighborhood of the origin

[TABLE]

where $a_{n}=0$, for $n$ small enough (resp. for $n<0$).

A weakly modular function $f$ is called modular if it is meromorphic at infinity. In case it is holomorphic at infinity, we define its value at infinity by $f(\infty)=\tilde{f}(0)$.

A modular function which is holomorphic everywhere (including infinity) is called a modular form; if such a function is zero at infinity, it is called a cusp form.

The Eisenstein series $E_{4}$ and $E_{6}$ above, are modular forms of weight $4$ and $6$ respectively for the group $SL(2,\mathbb{Z})$. It is a fundamental result, that every modular form for $SL(2,\mathbb{Z})$ is uniquely expressible as a polynomial in $E_{4}$ and $E_{6}$ and the extension ring ${\mathbb{C}}[E_{2},E_{4},E_{6}]$ of ${\mathbb{C}}[E_{4},E_{6}]$ is a differential ring. More precisely the following basic relations of Ramanujan hold [7, 32]:

[TABLE]

In other words, the field $\left(\mathbb{C}\left(E_{2},E_{4},E_{6}\right),\dfrac{1}{2i\pi}\dfrac{d}{d\tau}\right)$ is a differential field. The subfield of constants is the field of complex numbers $\mathbb{C}$ (as it is embedded in the field of meromorphic functions on $\mathfrak{H}$).

Theorem 2.2** ([34]).**

The functions $e_{k}(\tau),$ $k=1,2,3$, $\tau\in\mathfrak{H}$, solve the nonlinear differential equation of Riccati type, with coefficients in $\mathbb{C}\left(E_{2},E_{4},E_{6}\right)$

[TABLE]

Thus by the previous Theorem 2.2 one has that:

[TABLE]

The system of differential equations (2.9) is now necessary to show that the system (2.10) leads to a Darboux-Halphen system. Setting $\displaystyle t=\frac{4i}{\pi}\tau$ and

[TABLE]

the system (2.9) becomes

[TABLE]

With $X_{k}=\frac{\pi^{2}}{12}E_{2}+\frac{1}{4}e_{k},k=1,2,3$, the equations (2.10) take the form

[TABLE]

which is a Darboux-Halphen system. Moreover it is transformable into (2.11) by means of the substitutions:

[TABLE]

2.1. Contact geometric interpretation of the Darboux-Ramanujan system

Let $\mathfrak{H}$ be the upper-half plane and $\tau\in\mathfrak{H}$. We set as before $q=e^{2i\pi\tau}$ and consider the Eisenstein series

[TABLE]

where $\sigma_{k}(n)=\displaystyle\sum_{d|n}d^{k}$, with a change of notation from the previous subsection. We recall that $\mathscr{X}$, $\mathscr{Y}$, $\mathscr{Z}$ satisfy the following system of differential equations, hereafter called the Darboux-Ramanujan system (see equation (2.9))

[TABLE]

with $t=-2i\pi\tau$. In this section we give a contact geometric interpretation of the system (2.15). We start with some preliminaries.

Definition 2.3**.**

Let $M$ be a complex manifold of dimension $2m+1$ and $T^{\star}M$ its holomorphic cotangent bundle. A contact structure on $M$ is a holomorphic line-subbundle $L\subset T^{\star}M$ such that if $\omega$ is a local generator of $\mathcal{O}(L)\subset\Omega_{M}^{1}$, then

[TABLE]

In other words there is an open covering of $M$ by open sets $\mathcal{O}_{\alpha}$ such that:

1. (1)

On each $\mathcal{O}_{\alpha}$ there is a holomorphic one form $\omega_{\alpha}$ satisfying

[TABLE] 2. (2)

On $\mathcal{O}_{\alpha}\cap\mathcal{O}_{\beta}\-\neq\emptyset$ there is a non-vanishing holomorphic function $f_{\alpha\beta}$ such that $\omega_{\alpha}=f_{\alpha\beta}\omega_{\beta}$

Thus the condition $(1)$ is clearly independent of the chosen generator $\omega$.

Definition 2.4**.**

A holomorphic symplectic manifold is a complex manifold $M$ of dimension $2n$ with a closed non-degenerate holomorphic two-form $\Omega$(of type $(2,0)$): $\Omega$ establishes an isomorphism between the holomorphic tangent bundle $TM$ of $M$ and the holomorphic cotangent bundle $T^{\star}M$ of $M$:

[TABLE]

Definition 2.5**.**

Let $\mathscr{F}:M\to\mathbb{C}$ a holomorphic function on $M$. Then the Hamiltonian vector field associated to $\mathscr{F}$ is the holomorphic vector field defined by:

[TABLE]

Definition 2.6** ([18]).**

A Liouville vector field $Y$ on a complex symplectic manifold $M$ is a holomorphic vector field satisfying the equation

[TABLE]

In this case, the one form $\alpha:=i_{Y}\Omega:=\Omega(Y,.)$ is a contact form on any hypersurface $W$ of $M$ transverse to $Y$, i.e. with $Y$ nowhere tangent to $W$. Such hypersurfaces are called hypersurfaces of contact type.

Let us consider now $M=\mathbb{C}^{3}\times\mathbb{C}^{\times}$ with coordinates $(q_{1},q_{2},p_{1},p_{2})$ and the symplectic form $\Omega=dp_{1}\wedge dq_{1}+dp_{2}\wedge dq_{2}$ and associated Liouville form $\alpha=p_{1}dq_{1}+p_{2}dq_{2}$. We introduce the following modification of the system (2.15) by making the change of variables $\mathscr{X}_{1}=\dfrac{1}{6}\mathscr{X}$, $\mathscr{Y}_{1}=\dfrac{1}{3}\mathscr{Y}$, $\mathscr{Z}_{1}=\dfrac{1}{27}\mathscr{Z}$. The new system, which is the same as (2.11), is

[TABLE]

Theorem 2.7**.**

Consider the following Hamiltonian on $(M=\mathbb{C}^{3}\times\mathbb{C}^{\times},\Omega)$ given by

[TABLE]

Set $q_{1}=\mathscr{X}_{1}$, $q_{2}=\mathscr{Z}_{1}$, $p_{1}=\lambda\mathscr{Y}_{1}$ and $p_{2}=\lambda$. Then the Hamiltonian equations associated to $\mathscr{F}$, derived from Definition 2.5 are

[TABLE]

Moreover the projection of the system (2.18) to the hypersurface of contact-type $\lambda=1$ transverse to the Liouville vector field $Y=p_{1}\partial_{p_{1}}+p_{2}\partial_{p_{2}}$ gives the system (2.17).

Proof.

The system (2.18) is easily deduced from the definition of the hamiltonian equations. The fact that $Y=p_{1}\partial_{p_{1}}+p_{2}\partial_{p_{2}}$ is a Liouville vector field is also immediate. To prove the last assertion we remark that giving the Hamiltonian system (2.18) is equivalent to requiring that $\mathscr{X}_{1}$, $\mathscr{Y}_{1}$, $\mathscr{Z}_{1}$ and $\lambda$ be a flow of the vector field

[TABLE]

The projection of the vector field $Y_{\mathscr{F}}$ onto the hypersurface of contact type $p_{2}\equiv\lambda=1$ gives the vector field

[TABLE]

whose flow on $\mathfrak{H}$ gives the system (2.17). ∎

2.2. From the Chazy equation to the Darboux-Halphen system

Lemma 2.8**.**

Let $\gamma$ be a holomorphic function on the upper half plane such that

[TABLE]

with $R$ a polynomial with constant coefficients in $\mathbb{C}$. Let $P(\mathscr{N})=\mathscr{N}^{n}+a_{1}\gamma(\tau)\mathscr{N}^{n-1}+a_{2}\gamma^{\prime}(\tau)\mathscr{N}^{n-2}+\ldots+a_{n}\gamma^{(n-1)}(\tau)$ be a polynomial in $\mathscr{N}$ with $(a_{i})_{1\leqslant i\leqslant n}\in\mathbb{C}$ and such that the discriminant of $P(\mathscr{N})$ does not vanish on $\mathfrak{H}$. Then the roots of $P(\mathscr{N})$ satisfy an autonomous polynomial differential system.

Proof.

The elementary symmetric functions of the roots of $P(\mathscr{N})$ are expressed in terms of $a_{1}\gamma(\tau)$, $a_{2}\gamma^{\prime}(\tau)$, $\ldots$, $a_{n}\gamma^{(n-1)}(\tau)$. Taking the derivation with respect to $\tau$ of the gotten equations one obtains a linear system for the $\left(\dfrac{d\mathscr{N}_{i}}{d\tau}\right)_{1\leqslant i\leqslant n}$ with determinant (a Vandermonde determinant) given up to sign by the discriminant of the polynomial $P(\mathscr{N})$. By hypothesis this discriminant does not vanish of $\mathfrak{H}$. This enables us to apply Cramer’s rule. Finally the condition $\gamma^{(n)}=R(\gamma,\gamma^{\prime},\ldots,\gamma^{(n-1)})$ forces the $\mathscr{N}_{i},\,1\leqslant i\leqslant n$ to satisfy an autonomous polynomial differential system. ∎

Corollary 2.9** ([12, 31]).**

If $\gamma$ is the solution of the Chazy equation,

[TABLE]

given by $\gamma=\dfrac{\pi i}{3}E_{2}$. Then the three solutions $\mathscr{N}_{1}(\tau)$, $\mathscr{N}_{2}(\tau)$, $\mathscr{N}_{3}(\tau)$ of the cubic equation

[TABLE]

are solutions of the Darboux-Halphen system:

[TABLE]

which gives the equation (2.12) by setting $X_{i}=-2\mathscr{N}_{i},\,i=1,2,3$ .

Proof.

One has

[TABLE]

[TABLE]

and

[TABLE]

Derivation of the three relations with respect to $\tau$ gives

[TABLE]

[TABLE]

and

[TABLE]

This gives the system

[TABLE]

Set $\mathscr{D}=(\mathscr{N}_{1}-\mathscr{N}_{2})(\mathscr{N}_{1}-\mathscr{N}_{3})(\mathscr{N}_{2}-\mathscr{N}_{3})$. The three roots are all distinct because for $\gamma=\frac{\pi i}{3}E_{2}$ we have the algebraic independence of $\gamma,\gamma^{\prime},\gamma^{\prime\prime}$ over $\mathbb{C}$ [13, 24]; applying the Cramer formula one gets for the value of $\frac{d\mathscr{N}_{2}}{d\tau}$ the following

[TABLE]

Using the fact that $\gamma$ satisfies the Chazy equation

[TABLE]

and the expressions of $\gamma^{\prime}$ and $\gamma^{\prime\prime}$ in terms of $\mathscr{N}_{1}$, $\mathscr{N}_{2}$ and $\mathscr{N}_{3}$, one gets after a short computation the value of $\frac{d}{d\tau}\mathscr{N}_{2}$

[TABLE]

The determination of the derivatives $\frac{d}{d\tau}\mathscr{N}_{1}$ and $\frac{d}{d\tau}\mathscr{N}_{3}$ follows in the same manner. ∎

2.3. Autonomous systems of ODES and Hamiltonian systems

Consider differential equations of the form

[TABLE]

where the $(X_{i})_{1\leqslant i\leqslant n}$ are germs of holomorphic functions near a point $P$ of $\mathbb{C}^{n}$, $P=(0,0,\dots,0)$, say. Let us introduce the additional variables $y_{1}$, $y_{2}$, $\ldots$, $y_{n}$ and set

[TABLE]

then we can replace the system (2.21) by the Hamiltonian system

[TABLE]

where the last $n$ equations serve to define the $y_{i}$.

For a function $\varphi(x_{1},x_{2},\ldots,x_{n},y_{1},y_{2},\ldots,y_{n})$ to be a first integral of the Hamiltonian system (2.23), it is necessary and sufficient that the Poisson bracket

[TABLE]

Let us look in particular for a first integral of the Hamiltonian system (2.23) in the form

[TABLE]

where the $Y_{i}$ depend only on $x_{1}$, $x_{2}$, $\ldots$, $x_{n}$. Taking into account that the condition (2.24) holds for arbitrary values of $y_{1}$, $y_{2}$, $\ldots$, $y_{n}$, we obtain the $n$ equations

[TABLE]

which define the functions $Y_{i}$. For a method of integration of equations of he type given in (2.26) we refer to [23, p. 198]. Let us we recall that a function $\phi(x_{1},x_{2},\ldots,x_{n},y_{1},y_{2},\ldots,y_{n})$ is a first integral of the Hamiltonian system (2.23) if and only if $X_{H}(\phi)=0$ where

[TABLE]

Furthermore a function (possibly multivalued) $G(x_{1},x_{2},\ldots,x_{n})$ is a first integral of the system (2.21) if and only if $X(G)=0$ where

[TABLE]

Thus if $G(x_{1},x_{2},\ldots,x_{n})$ is a first integral of the system (2.21) then it is also a first integral of the Hamiltonian system (2.23). Hence by the Poisson theorem (which results from the fact that the Poisson bracket defines a Lie bracket of the set of holomorphic functions), the Poisson bracket

[TABLE]

is also a first integral of the system (2.23); since $\sum_{i=1}^{n}Y_{i}\frac{\partial G}{\partial x_{i}}$ only depends on $(x_{1},x_{2},\ldots,x_{n})$, one sees that it is in fact a first integral of the system (2.21). Thus

Proposition 2.10**.**

Given an autonomous system of differential equations of the form (2.21), there are functions $Y_{1}$, $Y_{2}$, $\ldots$, $Y_{n}$, depending on $(x_{1},x_{2},\ldots,x_{n})$, such that if $F$ is a first integral of (2.21), one can via the Hamiltonian formalism produce new first integrals of (2.21), of the form

[TABLE]

starting from $F$.

3. WDVV equations and Chazy equation

In the following we adopt the notations of [12, chap. 1, appen. C]. If one looks for a holomorphic, function $F(t_{1},t_{2},t_{3})$ on a domain of $\mathbb{C}^{3}$ (not fixed for the moment), of the form

[TABLE]

such that its third derivatives

[TABLE]

satisfy the following equations

1. (1)

Normalization

[TABLE]

is a constant non-degenerate matrix. We set $\eta^{ij}:=(\eta_{ij})^{-1}$. 2. (2)

Associativity

[TABLE]

(summation convention assumed) for any $t=(t_{1},t_{2},t_{3})$, define in the three dimensional space with basis $\mathfrak{e}_{1}$, $\mathfrak{e}_{2}$, $\mathfrak{e}_{3}$ a structure of an (commutative) associative algebra $A_{t}$ (with unity $e_{1}$)

Then one finds the table

[TABLE]

and that the two associativity conditions

[TABLE]

are equivalent to

[TABLE]

If furthermore one requires that

$(3)$ $F$ be quasihomogeneous

[TABLE]

for $c\in\mathbb{C}^{\times}$ (modulo quadratic terms) and furthermore that $F$ be periodic of period $1$ in its third variable $t_{3}$ (modulo quadratic terms), and analytic in the point $t_{1}=t_{2}=0,\;\;t_{3}=i\infty$, then one finds that $F$ must have the form

[TABLE]

for some unknown function $\gamma(\tau)$ analytic at $\tau=i\infty$

[TABLE]

The coefficients $b_{m}$ are defined up to shift

[TABLE]

and one has more precisely

[TABLE]

Moreover the function $\gamma$ must satisfy the Chazy equation

[TABLE]

In this situation the degrees of the flat (by definition) variables are

[TABLE]

the Euler vector field is

[TABLE]

and $d=1$, cf. [12, chap. 1]: $E(F)=2F=(3-d)F$. The associativity condition is a particular case of what is called the WDVV equations. Any solution of the WDVV equations endows its domain of definition with a so-called structure of Frobenius manifold [12, chap. 1]. So $F$ given in (3.3) endows $\mathbb{C}^{2}\times\mathfrak{H}$ with a structure of Frobenius manifold. We have

Theorem 3.1**.**

The canonical coordinates of the Frobenius manifold $\mathbb{C}^{2}\times\mathfrak{H}$ with flat coordinates $t_{1}$, $t_{2}$, $t_{3}$, of respective degrees given by (3.7), with Euler vector field given by (3.8) and with potential $F$ given by (3.3) can be expressed in the form

[TABLE]

Here $\mathscr{N}_{1}(t_{3})$, $\mathscr{N}_{2}(t_{3})$, $\mathscr{N}_{1}(t_{3})$ are the three roots of the cubic equation (2.19). The metric $\eta$ in flat coordinates is given by

[TABLE]

Furthermore the Poisson system on $\mathbb{C}^{3}$ with the standard Poisson structure and for the Poisson Hamiltonian

[TABLE]

given by

[TABLE]

admits the solution

[TABLE]

Proof.

By definition (cf. [12, chap. 1]) we have

[TABLE]

Hence we find

[TABLE]

and this gives immediately the metric $\eta=dt_{2}^{2}+2dt_{1}dt_{3}$. Also from [12, lect. 1] the structure constants of the deformed Frobenius algebra associated to the solution $F$ of the WDVV equation is given by

[TABLE]

where $C_{ijk}=\partial_{t_{i}}\partial_{t_{j}}\partial_{t_{k}}F$, $\eta^{ij}=(\eta_{ij})^{-1}$ and where we adopt Einstein’s summation convention. Hence for the $C_{ij}^{k}$ we obtain

[TABLE]

In order to go further we need to compute the intersection form given by [12, lect. 3] (Einstein’s summation convention assumed in the following whenever there are repeated indices)

[TABLE]

where

[TABLE]

and $E=(E^{1},E^{2},E^{3})$ is the Euler vector field written in components. The $C_{k}^{ij}$ are given by the following

[TABLE]

Using this we get for the intersection form

[TABLE]

According to [12, prop. 3.3] if $(t_{1},t_{2},t_{3})\in\mathbb{C}^{2}\times\mathfrak{H}$ is a point such that the roots of the following degree $3$ polynomial in the variable $u$ are distinct then its three roots $u_{1}(t_{1},t_{2},t_{3})$, $u_{2}(t_{1},t_{2},t_{3})$, $u_{3}(t_{1},t_{2},t_{3})$ constitute the canonical coordinates in a neighborhood of $(t_{1},t_{2},t_{3})$. The polynomial in question is

[TABLE]

where

[TABLE]

Using (3.18) we find after developing the determinant that

[TABLE]

One easily verifies that its three roots are given by

[TABLE]

where $\mathscr{N}_{1}(t_{3})$, $\mathscr{N}_{2}(t_{3})$, $\mathscr{N}_{3}(t_{3})$ are the three roots of the cubic (2.19). When $(t_{1},t_{2},t_{3})\in\mathbb{C}\times\mathbb{C}^{\times}\times\mathfrak{H}$ then $u_{1}(t_{1},t_{2},t_{3})$, $u_{2}(t_{1},t_{2},t_{3})$ and $u_{3}(t_{1},t_{2},t_{3})$ are distinct, hence they are the canonical coordinates of the underlying Frobenius manifold in that neighborhood.

One easily finds the Poisson Hamiltonian equations for the standard Poisson structure and the given Hamiltonian $H$. To see that $\Omega_{1}$, $\Omega_{2}$, $\Omega_{3}$ given in (3.11) satisfy (3.10) one uses (2.20), the chain rule and the identity

[TABLE]

which follows from

[TABLE]

and (2.19). ∎

Proposition 3.2** (Change of basis vector fields).**

For $(t_{1},t_{2},t_{3})\in\mathbb{C}\times\mathbb{C}^{\times}\times\mathfrak{H}$ and if $u_{1}(t_{1},t_{2},t_{3})$, $u_{2}(t_{1},t_{2},t_{3})$, $u_{3}(t_{1},t_{2},t_{3})$ constitute the corresponding canonical coordinates we have

[TABLE]

where

[TABLE]

Proof.

This follows straightforwardly from the chain rule

[TABLE]

and (2.19). Here we have set $\partial_{t_{i}}=\frac{\partial}{\partial t_{i}}$ for $i=1,2,3$, $\partial_{u_{i}}=\frac{\partial}{\partial u_{i}}$ for $i=1,2,3$, ${}^{.}=\frac{\partial}{\partial t_{3}}$ and we have written $\mathscr{N}_{i}$ for $i=1,2,3$, to mean $\mathscr{N}_{i}(t_{3})$. ∎

4. Wirtinger connections

In this section we study various connections. We start by recalling some classical definitions on connections [19], [36].

1. (1)

For a non-vanishing function $f=f(t)$, we note

[TABLE]

$\{f,t\}_{2}$ is the Schwarzian derivative of $f$. 2. (2)

If $X$ is a Riemann surface with a system of coordinate charts $\{(U,z)\}$, a system of functions $\{(q,U,z)\}$ (one function for each coordinate chart) defines a $k$-form or a tensor of order $k$ if, under change of coordinates it transforms according to

[TABLE]

It defines a projective connection on $X$ if it transforms as:

[TABLE]

and defines an affine connection if

[TABLE] 3. (3)

If $r$ is an affine connection, the curvature $q=\frac{\partial r}{\partial z}-\frac{1}{2}r^{2}$ is a projective connection. We thus have a way to obtain projective connections from affine connections. 4. (4)

An affine connection $r$ defines a covariant derivative $\nabla=\nabla_{\alpha}$ mapping $\alpha$-forms into $\alpha+1$-forms by

[TABLE]

An example is given by the Eisenstein series $E_{2}$ which verifies the functional equation

[TABLE]

hence $\displaystyle\frac{\pi i}{3}E_{2}$ defines an affine connection on $\mathfrak{H}/SL(2,\mathbb{Z})$ minus the $SL(2,\mathbb{Z})$-orbits of the "cusps" $i\infty$, $i$ and $e^{2i\pi/3}$, see [12, appen. C]. From the third relation in (2.9) we find that the curvature is $\frac{1}{18}\pi^{2}E_{4}$, which is a projective connection on $\mathfrak{H}/SL(2,\mathbb{Z})$ minus the $SL(2,\mathbb{Z})$-orbits of the "cusps" $i\infty$, $i$ and $e^{2i\pi/3}$. The covariant differentiation defined by this affine connection corresponds to the fact that the Serre derivative $\frac{1}{2i\pi}\frac{d}{d\tau}-\frac{2k}{12}E_{2}=\frac{1}{2i\pi}\frac{d}{d\tau}-\frac{k}{6}E_{2}$ sends the space of modular forms of weight $2k$ for ${SL(2,\mathbb{Z})}$ into the space of modular forms of weight $2k+2$.

Suppose now that $X$ is a compact Riemann surface of genus $g>0$. We fix a homology canonical basis $a_{1},\cdots,a_{g},~{}b_{1},\cdots,b_{g}$ and a basis $w_{1},\ldots,w_{g}$ of holomorphic differential such that for $i,j=1,\cdots,g$

[TABLE]

where $(\tau_{ij})_{1\leq i,j\leq g}$ is a Riemann matrix and $\delta_{i}^{j}$ is the Kronecker symbol.

In this situation, there is a unique symmetric bidifferential of the second kind $\omega$, called the Bergman Kernel of $X$, such that

1. (1)

$\rm{Bires}(\omega)=1$, 2. (2)

$\omega=\frac{1}{(u-v)^{2}}dudv+H(u,v)dudv$, with $H(u,v)$ holomorphic on $X\times X$, 3. (3)

$\int_{a_{i}}\omega(u,v)=0$ for any fixed $u\in X$ and $i=1,\cdots,g$, 4. (4)

$\int_{b_{i}}\omega(u,v)=w_{i}(u),\,i=1,\cdots,g$.

It is a general fact [36, prop. 1.3.6], that a symmetric bidifferential of the second kind $\omega$, given locally by $\displaystyle\omega=\frac{1}{(u-v)^{2}}dudv+H(u,v)dudv$, with a holomorphic $H(u,v)$ gives rise to a projective connection $h_{\omega}(Z)=6H(u,v)_{|u=v=Z}$.

For $\epsilon,\,\delta\in{\mathbb{R}}^{g}$ we define the genus $g\geqslant 1$ theta function with characteristics by

[TABLE]

where $\cdot$ is the scalar product, $z\in\mathbb{C}^{g}$ and $\tau$ is a symmetric $g\times g$ complex matrix, with positive definite imaginary part $\rm{Im}(\tau_{ij})$, a so-called Riemann matrix. We have

[TABLE]

For instance, in terms of characteristics, the four Jacobi theta functions are given as follows

[TABLE]

Every even $\theta$-characteristic $\displaystyle\begin{bmatrix}\epsilon\\ \delta\end{bmatrix}$ on $X$ (for example the first $3$ cases in (4.1)) with non-zero $\theta$-constant $\theta\begin{bmatrix}\epsilon\\ \delta\end{bmatrix}(0,\tau)\neq 0$, determines a bidifferential $\omega_{\begin{bmatrix}\epsilon\\ \delta\end{bmatrix}}$, in the following way [14, p. 20, 22] or [36]. We first observe that

[TABLE]

is a holomorphic symmetric bidifferential on $X\times X$, invariant under a change of homology basis preserving the $\theta$-characteristic $\begin{bmatrix}\epsilon\\ \delta\end{bmatrix}$, and one sets

[TABLE]

Definition 4.1**.**

The symmetric invariant bidifferential of the second kind $\omega_{\begin{bmatrix}\epsilon\\ \delta\end{bmatrix}}$ is called a Klein bidifferential. The associated projective connection is called the Wirtinger connection. Finally writing for short $\beta=\begin{bmatrix}\epsilon\\ \delta\end{bmatrix}$ and $K(u,v)$ for $\omega(u,v)$, the invariant Klein bidifferential $K_{X}(u,v)$ on the Riemann surface $X$ is defined by

[TABLE]

in case $X$ is a Riemann surface all whose even theta constants are non-zero. The associated projective connection $W_{X}$ is called the invariant Wirtinger connection. Hence $K_{X}$ is an averaged bidifferential over all the $2^{g-1}(2^{g}+1)$ even characteristics.

Let us now give the Wirtinger connections in the genus one case. We recall that classically

[TABLE]

As is well known, if we identify the opposite sides of the parallelogram generated by $2\omega_{1},2\omega_{2},\frac{\omega_{3}}{\omega_{1}}\in\mathfrak{H}$ in the complex $u$-plane, we obtain a genus one Riemann surface. The effect of a closed $a$-cycle is the parallel displacement by the vector $2\omega_{1}$ and the effect of a closed $b$-cycle is the displacement by the vector $2\omega_{2}$. With the notations of (4.2), the normalized integral of the first kind and the Riemann matrix are given respectively by

[TABLE]

Further [22, p. 428]

[TABLE]

The function

[TABLE]

is a normalized integral of the second kind, $u$ is a coordinate (global) near $P$ and $v$ is a coordinate (global) near $Q$. It has a simple pole at $P=Q$ and zero period over the $a$-cycle. The Bergman kernel in the genus one case is

[TABLE]

It is a bidifferential in $P,Q$ and has a double pole at $P=Q$ with zero residue and biresidue $1$ and with the property that

[TABLE]

According for example to [22, p. 415, p. 455], we have

[TABLE]

Using also [22, p. 428, eq. 56, 58], we find that the connection associated to the Bergman kernel is

[TABLE]

For $2\omega_{1}=1,2\omega_{2}=\tau$, the Bergman kernel is $\displaystyle K(u,v)=\wp(u-v;1,\tau)+\frac{\pi^{2}}{3}E_{2}$.

Theorem 4.2**.**

The Wirtinger connections associated to the even characteristics $\begin{bmatrix}0\\ 0\end{bmatrix},\begin{bmatrix}1\\ 0\end{bmatrix},\begin{bmatrix}0\\ 1\end{bmatrix}$ are respectively $-6e_{1},-6e_{2},-6e_{3}$.

Proof.

We use the classical formulas for the Weierstrass function $\wp(u)=\wp(u,\omega_{1},\omega_{2})$, with $u=2\omega_{1}w$

[TABLE]

and the local expansions

[TABLE]

[TABLE]

which give

[TABLE]

∎

The Klein invariant [14, p. 35] is the averaged bidifferential

[TABLE]

by virtue of [22, p. 416]. The invariant Wirtinger connection, associated with the Klein invariant, which is the projective connection, associated to this bidifferential, is equal to zero; because $\wp(u-v)=\frac{1}{(u-v)^{2}}+O(u-v)^{2}$.