The space of solvable Pell–Abel equations

A Pell–Abel equation is a functional equation of the form P²-DQ² = 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.