On the Distribution of Primes

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官彥良(Yen-Liang Kuan) 查詢紙本館藏

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論文名稱

On the Distribution of Primes
(On the Distribution of Primes)

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摘要(中)

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 ErdH{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 $0 eq alpha in mathcal{O}$ 和一個 $mathcal{O}$ 裡的理想 $frak{M}$，令
$f_{alpha}left(frak{M}
ight) = left{f in A : psi_{f}left(alpha
ight) equiv 0 pmod{frak{M}}
ight}$ 是一個 $A$ 裡的理想。 $omegaig(f_alphaleft(frak{M}
ight)ig)$ 表示為 $f_alphaleft(frak{M}
ight)$ 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：
$$
frac{omegaig(f_alphaleft(frak{M}
ight)ig)-frac{1}{2}left(logdegfrak{M}
ight)^2}{frac{1}{sqrt{3}}left(logdegfrak{M}
ight)^{3/2}}.
$$

摘要(英)

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 $ ilde A$ be the reduction of $A$ modulo $wp$ and let $N_{wp,n}$ be the number of $n$-torsion points in $ ildeAleft(F_wp
ight)$, the set of $F_{wp}$-rational points in $ ilde A$. If $A$ has bad reduction at $wp$, let $N_{wp,n} = 0$. Let $ ormwp$ 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
$$
limlimits_{x
ightarrow infty } frac{1}{pi_{K}(x)}sumlimits_{ ormwp leq x}N_{wp,n},
$$
where $pi_{K}(x)$ is the number of primes $wp$ with $ ormwp 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 $0 eq alpha in mathcal{O}$ and an ideal $frak{M}$ in $mathcal{O}$, let $f_{alpha}left(frak{M
ight) = left{f in A : psi_{f}left(alpha
ight)equiv 0 pmod{frak{M}}
ight}$ be the ideal in $A$. We denote by $omegaig(f_alphaleft(frak{M}
ight)ig)$ the number of distinct prime ideal divisors of $f_alphaleft(frak{M}
ight)$. If $q eq 2$, we prove that the following quantity
$$
frac{omegaig(f_alphaleft(frak{M}
ight)ig)-frac{1}{2}left(logdegfrak{M}
ight)^2}{frac{1}{sqrt{3}}left(logdegfrak{M}
ight)^{3/2}}
$$
distributes normally.

關鍵字(中)

★ 代數群
★ 橢圓曲線
★ Drinfeld 模
★ 函數體

關鍵字(英)

★ algebraic groups
★ ellipeic curves
★ Drinfeld modules
★ function fields

論文目次

Introduction 1
Chapter 1. The Chebotarëv Density Theorem 7
1. The Dirichlet density version for global fields 7
2. The natural density version for number fields 8
3. The natural density version for function fields 10
Chapter 2. On the distribution of torsion points modulo primes: The case of number fields 13
1. Introduction 13
2. The case of one-dimensional tori T 17
3. The case of elliptic curves with complex multiplication19
Chapter 3. On the distribution of torsion points modulo primes: The case of function fields 23
1. Introduction 23
2. The cases of Ga and Gm 24
3. The case of one-dimensional tori 26
4. The case of elliptic curves 28
Chapter 4. Drinfeld Modules 33
1. Definitions 33
2. Drinfeld A-Modules over L with Generic Characteristic 34
3. The average value $M(psi_{/L}, frak{n})$ 36
Chapter 5. On an Erdos-Pomerance conjecture for rank one Drinfeld modules 41
1. Introduction 41
2. Preliminaries 45
3. Proofs of Theorem 5.1 and Theorem 5.2 53
4. Equivalent statements of Theorem 5.3 and 5.4 62
5. Proofs of Theorem 5.3 and 5.4 73
Bibliography 81

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指導教授

陳燕美(Yen-Mei J. Chen)

審核日期

2013-6-11

推文