Continued fractions in function fields: polynomial analogues of McMullen's and Zaremba's conjectures

Abstract

We examine the polynomial analogues of McMullen's and Zaremba's conjectures on continued fractions with bounded partial quotients. It has already been proved by Blackburn that if the base field is infinite, then the polynomial analogue of Zaremba's conjecture holds; we will prove this again with a different method and examine some known results for finite base fields. Translating to the polynomial setting a result of Mercat, we will prove that the polynomial analogue of McMullen's conjecture holds over infinite algebraic extensions of finite fields and that, over finite fields, it would be a consequence of the polynomial analogue of Zaremba's conjecture. We will then prove that the polynomial analogue of McMullen's conjecture holds over uncountable base fields, over $\overline{\mathbb Q}$ (thanks to the theory of reduction of a formal Laurent series modulo a prime) and over number fields. For this purpose, we will examine the connection between the continued fractions of polynomial multiples of $\sqrt D$ and pullbacks of generalized Jacobians of the hyperelliptic curve $U^2=D(T)$.