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September 5, 2025

The space of solvable Pell–Abel equations

Key Points

The result shows that Pell–Abel equations form a complex manifold, enhancing our understanding of functional equations.
We provide an efficient method to compute invariants that describe the connected components of these equations.
Our findings have applications to hyperelliptic curves, expanding the understanding of their mathematical properties.
Distinct connections to torsion pairs and primitive differentials were identified on genus 2 Riemann surfaces.

Abstract

Abstract A Pell–Abel equation is a functional equation of the form P^2-DQ^2 = 1, with a given polynomial D free of squares and unknown polynomials P and Q. We show that the space of Pell–Abel equations with the degrees of D and of the primitive solution P fixed is a complex manifold. We describe its connected components by an efficiently computable invariant. Moreover, we give various applications of this result, including to torsion pairs on hyperelliptic curves and to Hurwitz spaces, and a description of the connected components of the space of primitive k -differentials with a unique zero on genus 2 Riemann surfaces.

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