| 摘要: | 在這論文中我們主要是考慮兩個質數分佈的問題：第一是給定一個交換代數群 $\A$，考慮 $\A$ 模質數後，扭點是有理點的個數大小
另一個是我們考慮秩為 1 的 Drinfeld 模上的 Erd\H{o}s-Pomerance 猜想。
首先，給定一個交換代數群 $\A$ 定義在體 $K$ 上。我們固定一個正整數 $n$。對每一個 $K$ 裡的質因子 $\wp$，令 $N_{\wp,n}$ 是 $\A$ 模 $\wp$ 後， $n$-扭點是有理點的個數。我們有興趣的是這個 $N_{\wp,n}$ 的平均值，當 $\wp$ 跑遍所有 $K$ 裡的質因子。當 $A$ 是一個一維的交換代數群，我們可以給這個平均值一個明確的公式。
第二個問題，我們考慮 $K$ 是一個包含一個一次質因子 $\infty$ 的正特徵值函數體，而且令它的常數體是 $\F_q$。 讓 $A$ 是一個環收集所有 $K$ 裡只有在 $\infty$ 有奇異點的函數。令 $\mathcal{O}$ 是 $A$ 的
Hilbert 類體裡最大的整數環，讓 $\psi$ 是一個定義在 $\mathcal{O}$ 上秩為 1 的特定 Drinfeld 模。給一個 $0 \neq \alpha \in \mathcal{O}$ 和一個 $\mathcal{O}$ 裡的理想 $\frak{M}$，令
$f_{\alpha}\left(\frak{M}\right) = \left\{f \in A : \psi_{f}\left(\alpha\right) \equiv 0 \pmod{\frak{M}} \right\}$ 是一個 $A$ 裡的理想。 $\omega\big(f_\alpha\left(\frak{M}\right)\big)$ 表示為 $f_\alpha\left(\frak{M}\right)$ 相異質理想因子的個數。我們可以証明下面這個量有常態分佈的性質：
$$
\frac{\omega\big(f_\alpha\left(\frak{M}\right)\big)-\frac{1}{2}\left(\log\deg\frak{M}\right)^2}{\frac{1}{\sqrt{3}}\left(\log\deg\frak{M}\right)^{3/2}}.
$$
In this thesis, we are concerned with two problems on the distribution of primes, the distribution according to the size of torsion of a given algebraic group modulo primes and an Erdos-Pomerance conjecture for rank one Drinfeld modules.
First of all, we consider a commutative algebraic group $\A$ which is defined over a global field $K$. Then, we fix a positive integer $n$. For a prime divisor $\wp$ of $K$, let $\F_{\wp}$ denote the residue field. If $\A$ has good reduction at $\wp$, let $\tilde \A$ be the reduction of $\A$ modulo $\wp$ and let $N_{\wp,n}$ be the number of $n$-torsion points in $\tilde\A\left(\F_\wp\right)$, the set of $\F_{\wp}$-rational points in $\tilde \A$. If $\A$ has bad reduction at $\wp$, let $N_{\wp,n} = 0$. Let $\norm\wp$ denote the norm of $\wp$, equal to the cardinality of the residue field $\F_\wp$. We are interested in the average value of $N_{\wp, n}$, where $\wp$ runs through the prime divisors in $K$, namely the limit
$$
\lim\limits_{x \rightarrow \infty } \frac{1}{\pi_{K}(x)}\sum\limits_{\norm\wp \leq x}N_{\wp,n},
$$
where $\pi_{K}(x)$ is the number of primes $\wp$ with $\norm\wp \leq x$. We denote this limit by $M(\Bbb A_{/K}, n)$. We shall derive explicit formulas for the average value $M(\Bbb A_{/K}, n)$ when $\A$ is a commutative algebraic group of dimension one defined over $K$.
Secondly, we consider a global function field $k$ of positive characteristic containing a prime divisor $\infty$ of degree one and whose field of constants is $\Bbb F_q$. Let $A$ be the ring of elements of $k$ which are regular outside $\infty$. Let $\psi$ be a sgn-normalized rank one Drinfeld $A$-module defined over $\mathcal{O}$, the integral closure of $A$ in the Hilbert class field of $A$. Given any $0 \neq \alpha \in \mathcal{O}$ and an ideal $\frak{M}$ in $\mathcal{O}$, let $f_{\alpha}\left(\frak{M \right) = \left\{f \in A : \psi_{f}\left(\alpha\right)\equiv 0 \pmod{\frak{M}} \right\}$ be the ideal in $A$. We denote by $\omega\big(f_\alpha\left(\frak{M}\right)\big)$ the number of distinct prime ideal divisors of $f_\alpha\left(\frak{M}\right)$. If $q \neq 2$, we prove that the following quantity
$$
\frac{\omega\big(f_\alpha\left(\frak{M}\right)\big)-\frac{1}{2}\left(\log\deg\frak{M}\right)^2}{\frac{1}{\sqrt{3}}\left(\log\deg\frak{M}\right)^{3/2}}
$$
distributes normally. |
