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The space of solvable Pell–Abel equations
- Part of
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- Journal:
- Compositio Mathematica / Volume 161 / Issue 7 / July 2025
- Published online by Cambridge University Press:
- 01 September 2025, pp. 1483-1511
- Print publication:
- July 2025
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- Article
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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.
