Elliptic functions, differential equations and dynamical systems

Denote by $\mathcal E_0$ the space of the universal bundle of elliptic curves with the space of parameters $g_2$, $g_3$ as base, and as fiber over the point $(g_2,g_3)$ the corresponding elliptic curve in the standard Weierstrass form with $t$ as coordinate. The field of Abelian functions of $t$ on $\mathcal E_0$ is determined by the Weierstrass function $\sigma(t;g_2,g_3)$, which is a section of the linear complex bundle over $\mathcal E_0$. The Weierstrass function $\wp=\wp(t;g_2,g_3)=-(\ln\sigma(t;g_2,g_3))''$ determines a birational equivalence $\mathcal E_0\to\mathbb C^3\colon(t,g_2,g_3)\to(\wp,\wp',\wp'')$, by which the differentiation along the fiber of the bundle $\mathcal E_0$ (the differentiation of functions along $t$) induces a classical algebraic dynamical system on $\mathbb C^3$. The algebra of differential operators along $t$, $g_2$, and $g_3$, which annihilate the $\sigma$-function, is extracted from classical works and leads to a solution of the well-known problem of differentiation of elliptic functions along parameters and, correspondingly, the problem of differentiation of a dynamical system solution along initial data. Using the generators of this algebra, we get dynamics in the space of parameters $g_2$, $g_3$, and on this basis the solution of the heat equation in terms of the $\sigma$-function. The dynamics are determined by a solution of the Shazy equation.
Let $\mathcal E_1$ be the space of the bundle with the space of parameters $g_2$, $g_3$ as base, and the fiber over the point $(g_2,g_3)$ the corresponding elliptic curve with coordinate $t$ and a marked point $\tau$. We obtain the bundle $\mathcal E_1\to\mathcal E_0$ with the universal bundle of elliptic curves with parameter $\tau$ as base, and as fiber the elliptic curve with $t$ as parameter. The field of Abelian functions of $t$ and $\tau$ on $\mathcal E_1$ is determined by the function $\sigma(\tau;g_2,g_3)$ and the Baker-Akhiezer function $\Phi(t,\tau,g_2,g_3)$, which is a section of the linear complex bundle over $\mathcal E_1$. The function $\Phi(t,\tau,g_2,g_3)$ gives a solution of the Lame equation. It is a common eigenfunction of the Sturm-Liouville operator $\mathcal L_2$ with the potential $2\wp(t;g_2,g_3)$ and a third-order differential operator $\mathcal L_3$, which commutes with $\mathcal L_2$. The commutativity condition for the operators $\mathcal L_2$ and $\mathcal L_3$ is equivalent to the condition that the function $\wp$ is a solution of the stationary KdV equation.
We give differential equations on $\Phi(t,\tau,g_2,g_3)$, describing its dependence on parameters $g_2$, $g_3$. These equations completely determine the operators of differentiation of elliptic functions along the parameters. The function $P=-(\ln\Phi(t,\tau,g_2,g_3))'$ is elliptic along $t$ and $\tau$ and symmetric with respect to these variables. Using a differential equation on this function, we describe the algebraic surface $\mathcal W$ in $\mathbb C^5$ and a birational equivalence $\mathcal E_1\to\mathcal W$, which is fiberwise with respect to a projection $\mathbb C^5\to\mathbb C^3$. As a corollary, we obtain an algebraic dynamical system in $\mathbb C^5$ integrable in elliptic functions. We obtain three integrals of this system. We give differential equations that describe the dependence of a solution of the dynamical system on the initial data.
New results presented in the talk were obtained in recent joint works with E. Yu. Bunkova. The talk is addressed to a wide audience. Main definitions will be introduced during the talk.