The structure of the space of solvable Pell-Abel equations

Given a polynomial $D$ of even order $d$ and simple roots, the Pell-Abel equation is $P^2 - D Q^2 = 1$, where $P$ and $Q$ are unknown non constant polynomials. If for a given $D$, this equation has a solution, we say that its degree is the degree of $P$. Despite having being solved by Abel and carefully studied by Chebyshev and his students, there remain many open question about this equation. In this talk I want to describe the structure of the spaces formed by polynomials of order $d$ having a solution of degree $n$. The main result is that this space is a manifold whose connected components can be classified by a simple invariant. Moreover, I will give some applications of this result and try to explain the key steps of the proof, whose main tool is the metric induced by abelian differentials on hyperelliptic curves.
The talk is based on a joint work with Andrei Bogatyrev (arXiv:2306.00884).