arXiv math/07035192007-03-18math.NT

爬行类：具有大类数的实二次阶

Creepers: Real quadratic orders with large class number

Roger Patterson

备注：: PhD Thesis, Macquarie University, Sydney, November 2003

摘要：

二次域的Shanks序列$\Q(\sqrt{S_{n}})$其中$\Q(\sqrt{S_{n}})$是一类二次域的实例，其类数很大，因此连分数周期相对较短。实际上，周期长度随$n$线性增加，也就是说：以算术级数增加。字段具有调节器$O(n^2)$。在90年代后期，这些问题引起了欧文·卡普兰斯基（Irving Kaplansky）的兴趣，并使他计算整数$a$、$b$、$c$和$x$的序列$a^2x^{2n}+bx^{n}+c$的平方根的周期长度。简而言之，卡普毫不奇怪地发现，一般来说，三元组$(a,b,c)$是“跳跃者”：它们产生周期长度以指数速率增加的序列。但也有三元组产生具有恒定周期长度的序列，即卡普的“睡眠者”。最后，有三倍，如桑克斯序列，其期间长度在算术级数中增加。Kaplansky恰当地称这些为“爬行动物”。看起来，睡眠者和爬行者正是那些能够详细描述所有$n$的显式连分式展开的人。除其他外，本论文显着扩展已知类的爬行动物，并发现，并非所有的“kreepers”（的形状确定Kaplansky），因此不是形状的例子研究的早期作者寻找家庭的二次数域明确可计算的单位和相对较大的监管机构。本文的工作包括发现了一类新的和老的亏格为$g$的超椭圆曲线族和一类$O(g^2)$阶的挠因子。由此可见，明显的跳跃/睡眠/爬行与民间信仰相吻合，即刚刚提到的扭转是最大可能的。

英文摘要：

Shanks's sequence of quadratic fields $\Q(\sqrt{S_{n}})$ where $S_{n}=(2^n+1)^2 + 2^{n+2}$ instances a class of quadratic fields for which the class number is large and, therefore, the continued fraction period is relatively short. Indeed, that period length increases linearly with $n$, that is: in arithmetic progression. The fields have regulator $O(n^2)$. In the late nineties, these matters intrigued Irving Kaplansky, and led him to compute period length of the square root of sequences $a^2x^{2n}+bx^{n}+c$ for integers $a$, $b$, $c$, and $x$. In brief, Kap found unsurprisingly that, generically, triples $(a,b,c)$ are `leapers': they yield sequences with period length increasing at exponential rate. But there are triples yielding sequences with constant period length, Kap's `sleepers'. Finally, there are triples, as exemplified by the Shanks's sequence, for which the period lengths increase in arithmetic progression. Felicitously, Kaplansky called these `creepers'. It seems that the sleepers and creepers are precisely those for which one is able to detail the explicit continued fraction expansion for all $n$. Inter alia, this thesis noticeably extends the known classes of creepers and finds that not all are `kreepers' (of the shape identified by Kaplansky) and therefore not of the shape of examples studied by earlier authors looking for families of quadratic number fields with explicitly computable unit and of relatively large regulator. The work of this thesis includes the discovery of old and new families of hyperelliptic curves of increasing genus $g$ and torsion divisor of order $O(g^2)$. It follows that the apparent trichotomy leaper/sleeper/creeper coincides with the folk belief that the just-mentioned torsion is maximum possible.

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。

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英文摘要：

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