×Time: 14.15 on 10 Nov, 2022
Place: SR 05.002, Spiegelgasse 5
Fabrizio Barroero (Università degli Studi Roma Tre)
Betti maps, almost Belyi maps and the polynomial Pell equation
n a joint work with Laura Capuano and Umberto Zannier we studied 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 (and for this reason we call them almost Belyi maps).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 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. In turn, this characterization gives back some precise information about the rational values of the Betti map.