超椭圆连分数的约化与特化
Reduction and specialization of hyperelliptic continued fractions
摘要：
对于偶数阶的一元多项式$D(X)$，将$\sqrt D$表示为$X^{-1}$中的洛朗级数;这产生了连分数展开式（类似于实数的连分数）：\[\sqrt D=a_0+\dfrac{1}{a_1+\dfrac{1}{a_2+\dfrac{1}{\ddots}}},\quad a_i\text{ polynomials in }X.\]这样的连分数首先由阿贝尔在1826年考虑，后来由切比雪夫考虑。事实证明，它们很少是周期性的，除非$D$定义在有限域上。大约在2001年，van der Poorten研究了$\sqrt D$的非周期性连分数，其中$D$定义在有理数上，同时$\sqrt D$的连分数模一个合适的素数$p$ ;后者的连分数自动是周期性的。他发现，一个恢复所有的converterations（有理函数逼近$\sqrt D$通过切断连续部分）的$\sqrt D \mod{p}$适当规范化，然后减少converterations的$\sqrt D$。通过发展一个一般的专门化理论的连分数的洛朗系列，我产生了严格的证明这一结果所述的范德Poorten和进一步能够显示以下：如果$D$定义在有理数上，并且$\sqrt D$的连分数是非周期的，则对于除了100个素数之外的所有素数$p \in \mathbb Z$，这个素数$p$出现在无穷多个$a_i$的前导系数的分母中。对于$\mathrm{deg}\,D = 4$，我甚至可以给出素数出现的顺序的描述，以及$a_i$和转换的$p$ -进高斯范数。这些结果也推广到数域。此外，我推导出计算二次连分数的优化公式，以及几个例子的扩展。讨论了几个已知的关于收敛高度的结果，并解释了与超椭圆曲线和Jacobian约化的关系。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
