No description available.

If we adjoin the cube root of a cube free rational integer <i>m</i> to the rational numbers we construct a cubic field. If we adjoin the cube roots of distinct cube free rational integers <i>m</i> and <i>n</i> to the rational numbers we construct a bicubic field. The number theoretic invariants for the cubic fields and their normal closures are well known. Some work has been done on the units, classnumbers and other invariants of the bicubic fields and their normal closures by Parry but no method is available for calculating those invariants. This dissertation provides an algorithm for calculating the number theoretic invariants of the bicubic fields and their normal closure. Among these invariants are the discriminant, an integral basis, a set of fundamental units, the class number and the rank of the 3-class group. / Ph. D.

Quintic abelian fields are characterized in terms of their conductor and a certain Galois group. From these, a generating polynomial and its roots and an integral basis are computed. A method for finding the fundamental units, regulators and class numbers is then developed. Tables listing the coefficients of a generating polynomial, the regulator, the class number, and a coefficients of a fundamental unit are given for 1527 quintic abelian fields. Of the seven cases where the class group structure is not immediate from the class number, six have their structure computed. / Ph. D.

Let K/Fq be an algebraic function field with full constant field Fq and genus g. Then the divisor class number hK of K/Fq is the order of the quotient group, D0K /P(K), degree zero divisors of K over principal divisors of K. The classification of the function fields K with hK = 1 is done by MacRea, Leitzel, Madan and Queen and the classification of the extensions with class number two is done by Le Brigand. Determination of the necessary and the sufficient conditions for a function field to have class number three is done by H&uml / ulya T&uml / ore. Let k := Fq(T) be the rational function field over the finite field Fq with q elements. For a polynomial N &isin / Fq[T], we construct the Nth cyclotomic function field KN. Cyclotomic function fields were investigated by Carlitz, studied by Hayes, M. Rosen, M. Bilhan and many other mathematicians. Classification of cyclotomic function fields and subfields of cyclotomic function fields with class number one is done by Kida, Murabayashi, Ahn and Jung. Also the classification of function fields with genus one and classification of those with class number two is done by Ahn and Jung. In this thesis, we classified all algebraic function fields and subfields of cyclotomic function fields over finite fields with class number three.

In 1883 Leopold Kronecker published a paper containing “a few explanatory remarks” to an earlier paper of his from 1866. His work loosely connected the theory of integral binary bilinear forms to the theory of integral binary quadratic forms. In this dissertation we discover the statements within Kronecker's paper and offer detailed arithmetic proofs. We begin by developing the theory of binary bilinear forms and their automorphs, providing a classification of integral binary bilinear forms up to equivalence, proper equivalence and complete equivalence. In the second chapter we introduce the class number, proper class number and complete class number as well as two refinements, which facilitate the development of a connection with binary quadratic forms. Our third chapter is devoted to deriving several class number formulas in terms of divisors of the determinant. This chapter also contains lower bounds on the class number for bilinear forms and classifies when these bounds are attained. Lastly, we use the class number formulas to rigorously develop Kronecker's connection between binary bilinear forms and binary quadratic forms. We supply purely arithmetic proofs of five results stated but not proven in the original paper. We conclude by giving an application of this material to the number of representations of an integer as a sum of three squares and show the resulting formula is equivalent to the well-known result due to Gauss.

The main part of the thesis is applications of the Strong Artin conjecture to number theory. We have two applications. One is generating number fields with extreme class numbers. The other is generating extreme positive and negative values of Euler-Kronecker constants. For a given number field $K$ of degree $n$, let $\widehat{K}$ be the normal closure of $K$ with $Gal(\widehat{K}/\Bbb Q)=G.$ Let $Gal(\widehat{K}/K)=H$ for some subgroup $H$ of $G$. Then, $$ L(s,\rho,\widehat{K}/\Bbb Q)=\frac{\zeta_K(s)}{\zeta(s)} $$ where $Ind_H^G1_H = 1_G + \rho$. When $L(s,\rho)$ is an entire function and has a zero-free region $[\alpha,1] \times [-(\log N)^2, (\log N)^2]$ where $N$ is the conductor of $L(s,\rho)$, we can estimate $\log L(1,\rho)$ and $\frac{L'}{L}(1,\rho)$ as a sum over small primes: $$ \log L(1,\rho) = \sum_{p\leq(\log N)^{k}}\lambda(p)p^{-1} + O_{l,k,\alpha}(1)$$ $$ \frac{L'}{L}(1,\rho)=-\sum_{p\leq x} \frac{\lambda(p) \log{p}}{p} +O_{l,x,\alpha}(1). $$ where $0 < k < \frac{16}{1-\alpha}$ and $(\log N)^{\frac{16}{1-\alpha}} \leq x \leq N^{\frac{1}{4}}$. With these approximations, we can study extreme values of class numbers and Euler-Kronecker constants. Let $\frak{K}$ $(n,G,r_1,r_2)$ be the set of number fields of degree $n$ with signature $(r_1,r_2)$ whose normal closures are Galois $G$ extension over $\Bbb Q$. Let $f(x,t) \in \Bbb Z[t][x]$ be a parametric polynomial whose splitting field over $\Bbb Q (t)$ is a regular $G$ extension. By Cohen's theorem, most specialization $t\in \Bbb Z$ corresponds to a number field $K_t$ in $\frak{K}$ $(n,G,r_1,r_2)$ with signature $(r_1,r_2)$ and hence we have a family of Artin L-functions $L(s,\rho,t)$. By counting zeros of L-functions over this family, we can obtain L-functions with the zero-free region above. In Chapter 1, we collect the known cases for the Strong Artin conjecture and prove it for the cases of $G=A_4$ and $S_4$. We explain how to obtain the approximations of $\log (1,\rho)$ and $\frac{L'}{L}(1,\rho)$ as a sum over small primes in detail. We review the theorem of Kowalski-Michel on counting zeros of automorphic L-functions in a family. In Chapter 2, we exhibit many parametric polynomials giving rise to regular extensions. They contain the cases when $G=C_n,$ $3\leq n \leq 6$, $D_n$, $3\leq n \leq 5$, $A_4, A_5, S_4, S_5$ and $S_n$, $n \geq 2$. In Chapter 3, we construct number fields with extreme class numbers using the parametric polynomials in Chapter 2. In Chapter 4, We construct number fields with extreme Euler-Kronecker constants also using the parametric polynomials in Chapter 2. In Chapter 5, we state the refinement of Weil's theorem on rational points of algebraic curves and prove it. The second topic in the thesis is about simple zeros of Maass L-functions. We consider a Hecke Maass form $f$ for $SL(2,\Bbb Z)$. In Chapter 6, we show that if the L-function $L(s,f)$ has a non-trivial simple zero, it has infinitely many simple zeros. This result is an extension of the result of Conrey and Ghosh.

The main part of the thesis is applications of the Strong Artin conjecture to number theory. We have two applications. One is generating number fields with extreme class numbers. The other is generating extreme positive and negative values of Euler-Kronecker constants. For a given number field $K$ of degree $n$, let $\widehat{K}$ be the normal closure of $K$ with $Gal(\widehat{K}/\Bbb Q)=G.$ Let $Gal(\widehat{K}/K)=H$ for some subgroup $H$ of $G$. Then, $$ L(s,\rho,\widehat{K}/\Bbb Q)=\frac{\zeta_K(s)}{\zeta(s)} $$ where $Ind_H^G1_H = 1_G + \rho$. When $L(s,\rho)$ is an entire function and has a zero-free region $[\alpha,1] \times [-(\log N)^2, (\log N)^2]$ where $N$ is the conductor of $L(s,\rho)$, we can estimate $\log L(1,\rho)$ and $\frac{L'}{L}(1,\rho)$ as a sum over small primes: $$ \log L(1,\rho) = \sum_{p\leq(\log N)^{k}}\lambda(p)p^{-1} + O_{l,k,\alpha}(1)$$ $$ \frac{L'}{L}(1,\rho)=-\sum_{p\leq x} \frac{\lambda(p) \log{p}}{p} +O_{l,x,\alpha}(1). $$ where $0 < k < \frac{16}{1-\alpha}$ and $(\log N)^{\frac{16}{1-\alpha}} \leq x \leq N^{\frac{1}{4}}$. With these approximations, we can study extreme values of class numbers and Euler-Kronecker constants. Let $\frak{K}$ $(n,G,r_1,r_2)$ be the set of number fields of degree $n$ with signature $(r_1,r_2)$ whose normal closures are Galois $G$ extension over $\Bbb Q$. Let $f(x,t) \in \Bbb Z[t][x]$ be a parametric polynomial whose splitting field over $\Bbb Q (t)$ is a regular $G$ extension. By Cohen's theorem, most specialization $t\in \Bbb Z$ corresponds to a number field $K_t$ in $\frak{K}$ $(n,G,r_1,r_2)$ with signature $(r_1,r_2)$ and hence we have a family of Artin L-functions $L(s,\rho,t)$. By counting zeros of L-functions over this family, we can obtain L-functions with the zero-free region above. In Chapter 1, we collect the known cases for the Strong Artin conjecture and prove it for the cases of $G=A_4$ and $S_4$. We explain how to obtain the approximations of $\log (1,\rho)$ and $\frac{L'}{L}(1,\rho)$ as a sum over small primes in detail. We review the theorem of Kowalski-Michel on counting zeros of automorphic L-functions in a family. In Chapter 2, we exhibit many parametric polynomials giving rise to regular extensions. They contain the cases when $G=C_n,$ $3\leq n \leq 6$, $D_n$, $3\leq n \leq 5$, $A_4, A_5, S_4, S_5$ and $S_n$, $n \geq 2$. In Chapter 3, we construct number fields with extreme class numbers using the parametric polynomials in Chapter 2. In Chapter 4, We construct number fields with extreme Euler-Kronecker constants also using the parametric polynomials in Chapter 2. In Chapter 5, we state the refinement of Weil's theorem on rational points of algebraic curves and prove it. The second topic in the thesis is about simple zeros of Maass L-functions. We consider a Hecke Maass form $f$ for $SL(2,\Bbb Z)$. In Chapter 6, we show that if the L-function $L(s,f)$ has a non-trivial simple zero, it has infinitely many simple zeros. This result is an extension of the result of Conrey and Ghosh.

Orientador: Sueli Irene Rodrigues Costa / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-23T07:10:07Z (GMT). No. of bitstreams: 1 Ferreira_Jamil_D.pdf: 1528200 bytes, checksum: a1ca349425c4bcf544a36d17d3157b3c (MD5) Previous issue date: 2013 / Resumo: O número de classes de divisores de grau zero, h, de corpos de funções algébricas elípticos e hiperelípticos desempenha papel importante nos esquemas criptográficos baseados em curvas elípticas e hiperelípticas. Nesse contexto, h é um número grande e é usualmente procurado por meio de algoritmos (baby step - giant step, por exemplo) em um intervalo de números reais obtido após um truncamento no produto infinito de Euler da função zeta do corpo de funções. Tendo a desigualdade de Hasse-Weil como motivação, encontramos identidades finitas para h que são também explícitas no sentido de que seus custos computacionais são diretamente deduzíveis dessas identidades. Como consequência, obtivemos também identidades finitas e explícitas para os coeficientes ai do L-polinômio da função zeta. Ferramentas fundamentais nesta pesquisa foram as L-séries de Artin e outros resultados envolvendo os símbolos polinomiais de Legendre / Abstract: The divisor class number of degree zero, h, of elliptic and hyperelliptic function fields plays an important role in cryptographic schemes based on elliptic and hyperelliptic curves. In this context, h is a large number and it is usually searched by means of algorithms (baby step - giant step, for example) in an interval of real numbers obtained after truncating the infinit Euler product coming from the zeta function of the function field. Taking the Hasse-Weil inequality as motivation, we derived finite identities for h which are also explicit in the sense that their computational costs are straightforwardly derivable from these identities. We also obtained finite and explicit identities for the coefficients ai of the L-polynomialof the zeta function. Fundamental tools for this research were the Artin L-series and other results involving the Legendre polynomial symbols / Doutorado / Matematica / Doutor em Matemática

Ce mémoire s'emploie à étudier les corps quadratiques réels ainsi qu'un élément particulier de tels corps quadratiques réels : l'unité fondamentale. Pour ce faire, le mémoire commence par présenter le plus clairement possible les connaissances sur différents sujets qui sont essentiels à la compréhension des calculs et des résultats de ma recherche. On introduit d'abord les corps quadratiques ainsi que l'anneau de ses entiers algébriques et on décrit ses unités. On parle ensuite des fractions continues puisqu'elles se retrouvent dans un algorithme de calcul de l'unité fondamentale. On traite ensuite des formes binaires quadratiques et de la formule du nombre de classes de Dirichlet, laquelle fait intervenir l'unité fondamentale en fonction d'autres variables. Une fois cette tâche accomplie, on présente nos calculs et nos résultats. Notre recherche concerne la répartition des unités fondamentales des corps quadratiques réels, la répartition des unités des corps quadratiques réels et les moments du logarithme de l'unité fondamentale. (Le logarithme de l'unité fondamentale est appelé le régulateur.) / This memoir aims to study real quadratic fields and a particular element of such real quadratic fields : the fundamental unit. To achieve this, the memoir begins by presenting as clearly as possible the state of knowledge on different subjects that are essential to understand the computations and results of my research. We first introduce quadratic fields and their rings of algebraic integers, and we describe their units. We then talk about continued fractions because they are present in an algorithm to compute the fundamental unit. Afterwards, we proceed with binary quadratic forms and Dirichlet's class number formula, which involves the fundamental unit as a function of other variables. Once the above tasks are done, we present our calculations and results. Our research concerns the distribution of fundamental units in real quadratic fields, the disbribution of units in real quadratic fields and the moments of the logarithm of the fundamental unit. (The logarithm of the fundamental unit is called the regulator.)

Ce mémoire s'emploie à étudier les corps quadratiques réels ainsi qu'un élément particulier de tels corps quadratiques réels : l'unité fondamentale. Pour ce faire, le mémoire commence par présenter le plus clairement possible les connaissances sur différents sujets qui sont essentiels à la compréhension des calculs et des résultats de ma recherche. On introduit d'abord les corps quadratiques ainsi que l'anneau de ses entiers algébriques et on décrit ses unités. On parle ensuite des fractions continues puisqu'elles se retrouvent dans un algorithme de calcul de l'unité fondamentale. On traite ensuite des formes binaires quadratiques et de la formule du nombre de classes de Dirichlet, laquelle fait intervenir l'unité fondamentale en fonction d'autres variables. Une fois cette tâche accomplie, on présente nos calculs et nos résultats. Notre recherche concerne la répartition des unités fondamentales des corps quadratiques réels, la répartition des unités des corps quadratiques réels et les moments du logarithme de l'unité fondamentale. (Le logarithme de l'unité fondamentale est appelé le régulateur.) / This memoir aims to study real quadratic fields and a particular element of such real quadratic fields : the fundamental unit. To achieve this, the memoir begins by presenting as clearly as possible the state of knowledge on different subjects that are essential to understand the computations and results of my research. We first introduce quadratic fields and their rings of algebraic integers, and we describe their units. We then talk about continued fractions because they are present in an algorithm to compute the fundamental unit. Afterwards, we proceed with binary quadratic forms and Dirichlet's class number formula, which involves the fundamental unit as a function of other variables. Once the above tasks are done, we present our calculations and results. Our research concerns the distribution of fundamental units in real quadratic fields, the disbribution of units in real quadratic fields and the moments of the logarithm of the fundamental unit. (The logarithm of the fundamental unit is called the regulator.)