Betti maps, Pell equations in polynomials and almost-Belyi maps

Abstract

AbstractWe study the Betti map of a particular (but relevant) section of the family of Jacobians of hyperelliptic curves using the polynomial Pell equation$A^2-DB^2=1$, with$A,B,D\in \mathbb {C}[t]$and certain ramified covers$\mathbb {P}^1\to \mathbb {P}^1$arising from such equation and having heavy constrains on their ramification. In particular, we obtain a special case of a result of André, Corvaja and Zannier on the submersivity of the Betti map by studying the locus of the polynomialsDthat fit in a Pell equation inside the space of polynomials of fixed even degree. Moreover, Riemann existence theorem associates to the abovementioned covers certain permutation representations: We are able to characterize the representations corresponding to ‘primitive’ solutions of the Pell equation or to powers of solutions of lower degree and give a combinatorial description of these representations whenDhas degree 4. In turn, this characterization gives back some precise information about the rational values of the Betti map.