2021/09/28 by Barroero, Fabrizio, Capuano, Laura, Zannier, Umberto
#11D0 #11G10 #14D22 #14E22 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2109.13552
We study the Betti map of a particular (but relevant) section of the family of Jacobians of hyperelliptic curves using the polynomial Pell equation A2-DB2=1, with A,B,D∈ \mathbb C[t] and certain ramified covers \mathbb P1→ \mathbb P1 arising from such equation and having heavy constrains on their ramification. In particular, we obtain a special case of a result of André, Covaja 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 above-mentioned 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.