Geodesic
Betti maps, Pell equations in polynomials and almost-Belyi maps
Forum of Mathematics, Sigma, Tome 10 (2022)

Voir la notice de l'article provenant de la source Cambridge University Press

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 André, 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. 
@article{10_1017_fms_2022_77,
     author = {Fabrizio Barroero and Laura Capuano and Umberto Zannier},
     title = {Betti maps, {Pell} equations in polynomials and {almost-Belyi} maps},
     journal = {Forum of Mathematics, Sigma},
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     year = {2022},
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Fabrizio Barroero; Laura Capuano; Umberto Zannier. Betti maps, Pell equations in polynomials and almost-Belyi maps. Forum of Mathematics, Sigma, Tome 10 (2022). doi: 10.1017/fms.2022.77

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