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

We 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) - DB2 = 1, with A, B, D is an element of C[t] and certain ramified covers P-1 -> P-1 arising from such equation and having heavy constrains on their ramification. In particular, we obtain a special case of a result of Andre, Corvaja and Zannier on the submersivity of the Betti map by studying the locus of the polynomials D that 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 when D has degree 4. In turn, this characterization gives back some precise information about the rational values of the Betti map.