Betti maps, Pell equation in polynomials and almost Belyi maps

TL;DR

This paper investigates the Betti map of Jacobians of hyperelliptic curves using polynomial Pell equations, characterizing related permutation representations and analyzing the rational values of the Betti map.

Contribution

It provides a detailed study of the Betti map in relation to polynomial Pell equations and characterizes permutation representations for specific covers, including a combinatorial description for degree 4 cases.

Findings

Characterization of permutation representations for primitive Pell solutions

Analysis of the Betti map's submersivity in polynomial Pell equations

Explicit combinatorial description for degree 4 polynomial covers

Abstract

We study the Betti map of a particular (but relevant) section of the family of Jacobians of hyperelliptic curves using the polynomial Pell equation $A^2-DB^2=1$, 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 Andr\'e, 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.