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The number of components of the Pell–Abel equations with primitive solutions of given degree
A. B. Bogatyrev^{a}, Q. Gendron^{b} ^{a} Marchuk Institute for Numerical Mathematics of Russian Academy of Sciences
^{b} Instituto de matemáticas, UNAM, México
Received: 15.09.2022
Abel [1] considered in 1826 Pell’s Diophantine equation over the ring of polynomials. Since then the equation
$$
\begin{equation}
P^2(x)D(x)Q^2(x)=1
\end{equation}
\tag{1}
$$
bears the names of both Pell and Abel. Here $P(x)$ and $Q(x)$ are unknown polynomials of one variable and $D(x):=\prod_{e\in{\mathsf E}}(xe)$ is a given monic complex polynomial of degree $\deg D={\mathsf E}:=2g+2$ without multiple roots. A generic Pell–Abel equation admits only the trivial solutions $(P,Q)=(\pm1,0)$. For a nontrivial solution to exist some extra conditions on the coefficient $D$ should be imposed. One form of such conditions was invented by Abel himself [1], and another is used below. Given $D(x)$, the set of solutions contains a solution of minimum degree $n:=\deg P$, which is called primitive. It generates all other solutions $P$ via compositions with classical Chebyshev polynomials and changes of sign.
The main result of this note consists in finding the number of connected components of the complex Pell–Abel equations with polynomial $D$ of fixed degree $2g+2$ which admit a primitive solution $P$ of another fixed degree $n$.
Theorem 1. Let $m:=\min(g,ng1)$ and let $[\,\cdot\,]$ denote the integer part of a number. Equation (1) has neither primitive solutions of degree $n<g$ for $g>0$, nor ones of degree $n>1$ for $g=0$. Otherwise the number of components in question is equal to $[m/2]+1$ if $n+g$ is odd and to $[(m+1)/2]$ if $n+g$ is even.
We start with the transcendental criterion [2] for the solvability of the Pell–Abel equation in terms of the associated hyperelliptic curve $C$ of genus $g$, the twopoint compactification of the affine curve
$$
\begin{equation}
(x,w)\in \mathbb{C}^2\colon\!\!\quad w^2=D(x).
\end{equation}
\tag{2}
$$
Consider the unique differential $d\eta=(x^g+\cdots)w^{1}\,dx$ on $C$ with two poles at infinity, residues $\pm1$, and purely imaginary periods. Equation (1) admits a nontrivial solution with $\deg P=n$ if and only if all the periods of $d\eta$ on $C$ lie in the same lattice $2\pi i\mathbb{Z}/n$. If a Pell–Abel equation has a nontrivial solution of degree $n$, then the distinguished differential can be represented in the form $d\eta=n^{1}d\log(P(x)+wQ(x))$, and therefore the criterion holds.
A pictorial calculus that allows one to control the periods of a distinguished differential effectively in the process of deformation of the curve (2) was designed in [3] and [2] for the investigation of socalled (real) extremal polynomials, where this problem arises too. The quadratic differential $(d\eta)^2$ descends from $C$ to the plane of the variable $x$. Now, with each curve (2) we associate a finite planar graph $\Gamma(C)$ built in three steps: (i) draw all critical vertical trajectories $(d\eta)^2<0$ (see [4]) incident to branch points $e\in{\mathsf E}$; (ii) connect all zeros of the differential $d\eta^2$ other than the branch points $e$ with the vertical leaves of the foliation we have already drawn or with other zeros of $d\eta^2$ by horizontal trajectories $(d\eta)^2>0$ (because of the normalization of the distinguished differentials, this construction is well defined: we obtain finitely many regular analytic arcs); (iii) equip each edge with its length in the metric $d\eta$ induced by the distinguished differential.
One of the associated graphs $\Gamma(C)$ that can be obtained for $g=2$ is shown in Fig. 1, (b) up to isotopy of the plane. Here black vertices denote branch points $e\in{\sf E}$ of the curve, white points denote zeros of $(d\eta)^{2}$, simple lines are horizontal leaves of the foliation, and double lines are vertical leaves. Given a genus $g$, there are only finitely many admissible topological types of graphs $\Gamma$, and they can be listed axiomatically. The graphs $\Gamma(C)$ are completely determined by their properties, so that any topological weighted planar graph meeting five certain requirements stems from a unique curve (2) (up to inessential normalization; see [3] and [2]). Two of these five are as follows: $\Gamma$ is a tree, and the total weight of its vertical edges is $\pi$.
The periods of the distinguished differential of a curve can be recovered from the graph: they are integer linear combinations of the weights of vertical edges of the graph. In particular, one can define a local isoperiodic deformations which changes the conformal structure of the curve (2) and preserves all the periods of $d\eta$. Using isoperiodic graph deformations, any curve $C$ can be transformed into another curve whose graph has a standard form. The number of those standard forms gives us an upper bound for the number of components of the space of Pell–Abel equations.
To complete the proof of Theorem 1 we need a lower bound for the number of components. We consider graphs $\Gamma$ embedded in a line. They correspond to multiband real Chebyshev polynomials $P(x)$. Even in this class there are many curves that cannot be transformed isoperiodically into one another. To prove this we use the action of the braid group $\operatorname{Br}_{2g+2}$ on binary words $(b_1,b_2,\dots)\in (\mathbb{Z}/2\mathbb{Z})^{2g+1}$ and which is defined on the generators $\beta_s$ of the group satisfying the usual braid relations [5]:
$$
\begin{equation*}
\beta_s \cdot (b_1,\dots,b_{s1},b_s,b_{s+1},\dots):=(b_1,\dots,b_{s1}+b_s,b_s,b_{s+1}b_s,\dots),
\end{equation*}
\notag
$$
$s=1,\dots,2g+1$. This representation is the reduction modulo $2$ of a certain specialization of the Burau representation of braids [5]. If two curves can be deformed isoperiodically one to the other, then the associated periods generate two binary strings in the same orbit of the braid group. It turns out that there are sufficiently many orbits of braid action in $(\mathbb{Z}/2\mathbb{Z})^{2g+1}$ for the lower bound on the number of components to be equal to the upper bound.
Theorem 1 has a geometric corollary. The moduli space of primitive $k$ differentials with a unique zero of order $2k$ on curves of genus $2$ is empty for $k=2$, connected for $k=1$ or $3$, and for even $k\geqslant4$, and has two connected components for odd $k\geqslant5$.
The authors thank J.P. Serre, who initiated their collaboration.



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Citation:
A. B. Bogatyrev, Q. Gendron, “The number of components of the Pell–Abel equations with primitive solutions of given degree”, Uspekhi Mat. Nauk, 78:1(469) (2023), 209–210; Russian Math. Surveys, 78:1 (2023), 208–210
Linking options:
https://www.mathnet.ru/eng/rm10082https://doi.org/10.4213/rm10082e https://www.mathnet.ru/eng/rm/v78/i1/p209

