Reduction and specialization of hyperelliptic continued fractions

TL;DR

This paper studies the properties of hyperelliptic continued fractions, especially their reduction modulo primes, providing new theoretical insights, explicit formulas, and connections to algebraic geometry.

Contribution

It develops a general specialization theory for Laurent series continued fractions, proving key properties about prime divisors in the coefficients and extending results to number fields.

Findings

Non-periodic continued fractions have primes in denominators of coefficients for almost all primes p.

Explicit descriptions of prime appearances and p-adic norms for degree 4 cases.

Connections established between continued fractions, hyperelliptic curves, and Jacobians.

Abstract

For a monic polynomial $D(X)$ of even degree, express $\sqrt D$ as a Laurent series in $X^{-1}$; this yields a continued fraction expansion (similar to continued fractions of real numbers): \[\sqrt D=a_0+\dfrac{1}{a_1+\dfrac{1}{a_2+\dfrac{1}{\ddots}}},\quad a_i\text{ polynomials in }X.\] Such continued fractions were first considered by Abel in 1826, and later by Chebyshev. It turns out they are rarely periodic unless $D$ is defined over a finite field. Around 2001 van der Poorten studied non-periodic continued fractions of $\sqrt D$, with $D$ defined over the rationals, and simultaneously the continued fraction of $\sqrt D$ modulo a suitable prime $p$; the latter continued fraction is automatically periodic. He found that one recovers all the convergents (rational function approximations to $\sqrt D$ obtained by cutting off the continued fraction) of $\sqrt D \mod{p}$ by appropriately normalising and then reducing the convergents of $\sqrt D$. By developing a general specialization theory for continued fractions of Laurent series, I produced a rigorous proof of this result stated by van der Poorten and further was able to show the following: If $D$ is defined over the rationals and the continued fraction of $\sqrt D$ is non-periodic, then for all but finitely many primes $p \in \mathbb Z$, this prime $p$ occurs in the denominator of the leading coefficient of infinitely many $a_i$. For $\mathrm{deg}\,D = 4$, I can even give a description of the orders in which the prime appears, and the $p$-adic Gauss norms of the $a_i$ and the convergents. These results also generalise to number fields. Moreover, I derive optimised formulae for computing quadratic continued fractions, along with several example expansions. I discuss a few known results on the heights of the convergents, and explain some relations with the reduction of hyperelliptic curves and Jacobians.