Title: Reduction and specialization of hyperelliptic continued fractions,
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.