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arXiv · 2109.13552

Betti maps, Pell equation in polynomials and almost Belyi maps

Fabrizio Barroero·Laura Capuano·Umberto Zannier
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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.

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Fabrizio Barroero, Laura Capuano, Umberto Zannier. 2021-09-28. Betti maps, Pell equation in polynomials and almost Belyi maps. https://arxiv.org/abs/2109.13552

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