Hyperelliptic Continued Fractions and Generalized Jacobians

TL;DR

This paper explores continued fractions of square roots of complex polynomials, revealing that the degrees of partial quotients are always pre-periodic, and connects these properties to hyperelliptic Jacobians and divisor relations.

Contribution

It establishes a general analogue of Lagrange's theorem for hyperelliptic continued fractions, proving pre-periodicity of degrees and deriving new bounds for convergents.

Findings

Pre-periodicity of degrees of partial quotients always holds.

New bounds for degrees of convergents, often optimal.

Finiteness of rational poles occurring infinitely often.

Abstract

For a complex polynomial $D(t)$ of even degree, one may define the continued fraction of $\sqrt{D(t)}$. This was found relevant already by Abel in 1826, and later by Chebyshev, concerning integration of (hyperelliptic) differentials; they realized that, contrary to the classical case of square roots of positive integers treated by Lagrange and Galois, we do not always have pre-periodicity of the partial quotients. In this paper we shall prove that, however, a correct analogue of Lagrange's theorem still exists in full generality: pre-periodicity of the {\it degrees} of the partial quotients always holds. Apparently, this fact was never noted before. This also yields a corresponding formula for the degrees of the convergents, for which we shall prove new bounds which are generally best possible (halving the known ones). We shall further study other aspects of the continued fraction, like the growth of the heights of partial quotients. Throughout, some striking phenomena appear, related to the geometry of (generalized) Hyperelliptic Jacobians. Another conclusion central in this paper concerns the poles of the convergents: there can be only finitely many rational ones which occur infinitely many times. (This is crucial for applications to a function field version of a question of McMullen.) Our methods rely, among other things, on linking Pad\'e approximants and convergents with divisor relations in generalized Jacobians; this shall allow an application of a version for algebraic groups, proved in this paper, of the Skolem-Mahler-Lech theorem.