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Main Title: On sequences of integers of quadratic fields and computations
Translated Title: Über Folgen ganzer Elemente in quadratischen Körpern und Berechnungen
Author(s): Bircan, Nihal
Advisor(s): Pohst, Michael E.
Referee(s): Tröltzsch, Fredi
Petho, Attila
Granting Institution: Technische Universität Berlin, Fakultät II - Mathematik und Naturwissenschaften
Type: Doctoral Thesis
Language: English
Language Code: en
Abstract: In this thesis, we consider the integers $\alpha$ of the quadratic field $ \mathbb{Q} (\sqrt[]{d})$ where $d\in \Z$ is square-free and $d\equiv 1,2,3 \bmod 4$. Furthermore, let $p$ be an odd prime. Using the embedding into $ \text{GL}(2,\mathbb{R})$ we obtain bounds for the first $\nu\in\N$ such that $\alpha^\nu\equiv 1\bmod p.$ For the conductor $f$, we then study the first integer $n=n(f)$ such that $\alpha^n\in\mathcal{O}_{f}$. We obtain bounds for $n(f)$ and for $n(fp^{k})$. We allow any norm $N(\alpha)\not\eq 0$. A special case is that $\alpha$ is the fundamental unit $\varepsilon$ of the real quadratic field $ \mathbb{Q} (\sqrt[]{d})$. We consider matrices $A\in\text{GL}(2,\mathbb{C})$ and we show how the integers $\alpha$ of any quadratic field $ \mathbb{Q} ( \sqrt[]{d})$ can be embedded in GL$(2,\mathbb{R})$ where $ d=4q+r\in \mathbb{Z} $. Namely, $A$ is defined by \begin{align*} A= \text{ } \begin{pmatrix} a&b \\ bd&a \end{pmatrix}\ \text{for $ r=2,3$},\ \ A=\begin{pmatrix} \tfrac{1}{2} (a+b)&b \\ qb&\tfrac{1}{2} (a-b) \end{pmatrix}\ \text{for } r=1. \end{align*} We are now interested in the $n$ such that $ A^{n} =I$ or $A^{n} =cI$ in the residue field $ \mathbb{Z}/ p\mathbb{Z} $ where $p$ is an odd prime. As a tool we use adapted Chebyshev polynomials $t_{n} (x;s)$ and $u_{n} (x;s)$, which are monic polynomials with integer coefficients. We can write the powers of an algebraic integer $\alpha$ as \begin{align*} \alpha ^{n} =\begin{cases} \text{ } \tfrac{1}{2} t_{n} (2a)+u_{n-1} (2a)b\sqrt{d} & \text{if } r=2,3\\ \tfrac{1}{2} t_{n} (a)+\tfrac{1}{2} u_{n-1} (a)b\sqrt{d} & \text{if } r=1. \end{cases}\end{align*} Let $s=$ det $A$ and $x=$ tr $A$. We obtain results about $ A^{n}$ for all $s\neq 0$. We formulate these results in terms of $t_{n} (x;s)=$ tr $A^{n}$. Now we consider again congruences modulo $p$. In the case of the quadratic field we have $s=N(\alpha)$. The Legendre symbol $\ell\coloneqq ((x^2 -4s)/p) $ will play an important role. The values of $n$ with $ A^{n}\equiv I \bmod p$ are connected with $p-1$ if $\ell=+1$ and with $p+1$ if $\ell=-1$. In particular, we prove that if $s=1$ and $x^{2}-4\not\equiv 0\bmod p$ then $ t_{\frac{p-\ell}{2}} (x)\equiv 2((x+2)/p)\bmod p$ and to generalize this we determine the first $ n=(p-\ell)/2^{m}$ with $ t_{n} (x)\equiv 2\bmod p$ in terms of a chain of Legendre symbols. We also consider the more complicated case $s=-1$ and prove similar results. In the last chapter we consider the asymptotic behaviour of $n(p)$ by large-scale computations. Our results become easier to state if we consider the quotients $q$ defined by \begin{equation*} q= \left\{\begin{array}{lll} \frac{p\pm1}{2n(p)}& \text{if }N(\varepsilon)=+1& \text{ where } \left(\frac{d}{p}\right)=\mp1, \\ \frac {p\pm1}{n(p)} &\text{if } N(\varepsilon)=-1& \text{ where } \left(\frac{d}{p}\right)=\mp1. \end{array}\right. \end{equation*} In the computation of frequencies of $q$ we restrict ourselves to the case that $d\equiv 2,3 \bmod 4$ and we obtain numerical information about $ n(f)=n(p)=\text{min}\{ \nu\in\N : \varepsilon^{\nu}\in \mathcal{O}_{p}\}$. Our numerical results suggest that the frequencies should have a limit as the ranges of $d$ and $p$ tend simultaneously to infinity.
URI: urn:nbn:de:kobv:83-opus4-47264
Exam Date: 18-Oct-2013
Issue Date: 27-Feb-2014
Date Available: 27-Feb-2014
DDC Class: 500 Naturwissenschaften und Mathematik
Subject(s): Quadratic fields
integers of quadratic fields
Quadratische Körper
ganze Elemente in quadratischen Körpern
Appears in Collections:Inst. Mathematik » Publications

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