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.
Barroero, F., Capuano, L., Zannier, U. (2022). Betti maps, Pell equations in polynomials and almost-Belyi maps. FORUM OF MATHEMATICS. SIGMA, 10 [10.1017/fms.2022.77].