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

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