As shown by Zagier, singular moduli can be represented by the coefficients of a certain half integer weight modular form. Congruences for singular moduli modulo arbitrary primes have been proved by Ahlgren and Ono, Edixhoven, and Jenkins. Computation suggests that stronger congruences hold for small primes $p \in \{2, 3, 5, 7, 11\}$. Boylan proved stronger congruences hold in the case where $p=2$. We conjecture congruences for singular moduli modulo powers of $p \in \{3, 5, 7, 11\}$. In particular, we study the case where $p=3$ and reduce the conjecture to a congruence for a simpler modular form.

We give congruences modulo powers of p in {3, 5, 7} for the Fourier coefficients of certain modular functions in level p with poles only at 0, answering a question posed by Andersen and Jenkins and continuing work done by the Jenkins, the author, and Moss. The congruences involve a modulus that depends on the base p expansion of the modular form's order of vanishing at infinity.

Doctor of Philosophy / Department of Mathematics / Todd E. Cochrane / Let $m$ be a positive integer, $p$ be an odd prime, and $\mathbb{Z}_{p^m } = \mathbb{Z}/(p^m )$ be the ring of integers modulo $p^m $. Let $$Q({\mathbf{x}}) = Q(x_1 ,x_2 ,...,x_n ) = \sum\limits_{1 \leqslant i \leqslant j \leqslant n} {a_{ij} x_i x_j } ,$$ be a quadratic form with integer coefficients. Suppose that $n$ is even and $\det A_Q \not \equiv 0\;(\bmod p)$. Set $\Delta = (( - 1)^{n/2} \det A_Q /p)$, where $( \cdot /p)$ is the Legendre symbol and $\left\| {\mathbf{x}} \right\| = \max \left| {x_i } \right|$. Let $V$ be the set of solutions the congruence $ $Q({\mathbf{x}})\, \equiv \;0\quad (\bmod p^m ) \quad(1)$$, contained in $\mathbb{Z}^n $ and let $B$ be any box of points in $\mathbb{Z}^n $of the type $$B = \left\{ {{\mathbf{x}} \in \mathbb{Z}^n \left| {\,a_i \leqslant x_i < a_i + m_i ,\;\,1 \leqslant i \leqslant n} \right.} \right\},$$ where $a_i ,m_i \in \mathbb{Z},\;1 \leqslant m_i \leqslant p^m $. In this dissertation we use the method of exponential sums to investigate how large the cardinality of the box $B$ must be in order to guarantee that there exists a solution ${\mathbf{x}}$of (1) in $ B$. In particular we will focus on cubes (all $m_i $equal) centered at the origin in order to obtain primitive solutions with $\left\| {\mathbf{x}} \right\|$ small. For $m = 2$ and $n \geqslant 4$ we obtain a primitive solution with $\left\| {\mathbf{x}} \right\| \leqslant \max \left\{ {2^5 p,2^{18} } \right\}$. For $m = 3$, $n \geqslant 6$, and $\Delta = + 1$, we get $\left\| {\mathbf{x}} \right\| \leqslant \max \left\{ {2^{2/n} p^{(3/2) + (3/n)} ,2^{(2n + 4)/(n - 2)} } \right\}$. Finally for any $m \geqslant 2$, $n \geqslant m,$ and any nonsingular quadratic form we obtain $\left\| {\mathbf{x}} \right\| \leqslant \max \{ 6^{1/n} p^{m[(1/2) + (1/n)]} ,2^{2(n + 1)/(n - 2)} 3^{2/(n - 2)} \} $. Others results are obtained for boxes $B$ with sides of arbitrary lengths.

In der vorliegenden Arbeit werden Geradenkongruenzen des projektiv abgeschlossenen dreidimensionalen euklidischen Raumes differentialgeometrisch untersucht. Nach J. PLÜCKER lassen sich Geraden in gleicher Weise als Grundelemente eines Geradenraumes auffassen wie die Punkte in einem Punktraum. Unter Beachtung dieser Überlegung scheint eine &amp;quot;natürliche&amp;quot; Behandlung der Geradenkongruenzen interessant und sinnvoll. Sie bildet den Gegenstand der vorliegenden Dissertation. Ein besonderes Augenmerk richtet sich dabei auf die Frage nach &amp;quot;kleinsten&amp;quot; Geradenkongruenzen (&amp;quot;Minimalkongruenzen&amp;quot;) in der Geradenmenge des reellen projektiv abgeschlossenen dreidimensionalen euklidischen Raumes. Dahinter verbirgt sich eine gewisse Analogiebildung in der Liniengeometrie, die der klassischen Differentialgeometrie entstammt. Die Geradenkongruenzen bilden hierbei das liniengeometrische Analogon zu den Flächen des dreidimensionalen (Punkt-)Raumes. Das Wort &amp;quot;Kleinste&amp;quot; stellt im Geradenraum einen Bezug zu den Minimalflächen in der Differentialgeometrie her. Nun gestatten diese Fragestellungen in der Liniengeometrie eine anschauliche Interpretation, sobald man ein Punktmodell des Geradenraumes vorliegen hat. Einparametrige Geradenmannigfaltigkeiten (Regelflächen) lassen sich darin als Kurven und Geradenkongruenzen als zweidimensionale Flächen auffassen. Die vierparametrige Geradenmenge des reellen projektiven dreidimensionalen Raumes ist in diesem Modell eine Quadrik vom Index 2 in einem reellen projektiven fünfdimensionalen Raum, die so genannte KLEINsche Hyperquadrik. Der Modellwechsel wird durch die KLEINsche Abbildung vollzogen. / In the available work line congruences of the projectively extended three-dimensional euclidean space will be analysed. Following to J. PLÜCKER lines can be seen as basic elements of an line space like in the same way points in a point-space. Taking this fact in consideration a &amp;quot;natural&amp;quot; handling with line congruences might be interesting and reasonable. A special detail in the thesis is the question to minimal congruences in the set of lines of the projectively extended euclidean three-space. It can also be seen as an analogous problem in the geometry of lines which can be find in the differential geometry of surfaces. In this case the line congruences are similar to the surfaces of the three-dimensional (point-)space. The phrase &amp;quot;minimal&amp;quot; means in the line space the connection to the minimal surfaces in the differential geometry. These questions offer in line geometry demonstrative interpretation possibilities if a point-model in the line space exists. One-parameter manifolds of lines (rule surfaces) can be seen in this ambiance as curves and line congruences as two dimensional surfaces. The four-parametric set of lines in the projectively extended three-dimensional euclidian space is in this model a quadric of the index 2 in a real projective five-dimensional space, the so called KLEIN-quadric. The changing of the model is managed by the KLEIN-mapping.

In der vorliegenden Arbeit werden Geradenkongruenzen des projektiv abgeschlossenen dreidimensionalen euklidischen Raumes differentialgeometrisch untersucht. Nach J. PLÜCKER lassen sich Geraden in gleicher Weise als Grundelemente eines Geradenraumes auffassen wie die Punkte in einem Punktraum. Unter Beachtung dieser Überlegung scheint eine &amp;quot;natürliche&amp;quot; Behandlung der Geradenkongruenzen interessant und sinnvoll. Sie bildet den Gegenstand der vorliegenden Dissertation. Ein besonderes Augenmerk richtet sich dabei auf die Frage nach &amp;quot;kleinsten&amp;quot; Geradenkongruenzen (&amp;quot;Minimalkongruenzen&amp;quot;) in der Geradenmenge des reellen projektiv abgeschlossenen dreidimensionalen euklidischen Raumes. Dahinter verbirgt sich eine gewisse Analogiebildung in der Liniengeometrie, die der klassischen Differentialgeometrie entstammt. Die Geradenkongruenzen bilden hierbei das liniengeometrische Analogon zu den Flächen des dreidimensionalen (Punkt-)Raumes. Das Wort &amp;quot;Kleinste&amp;quot; stellt im Geradenraum einen Bezug zu den Minimalflächen in der Differentialgeometrie her. Nun gestatten diese Fragestellungen in der Liniengeometrie eine anschauliche Interpretation, sobald man ein Punktmodell des Geradenraumes vorliegen hat. Einparametrige Geradenmannigfaltigkeiten (Regelflächen) lassen sich darin als Kurven und Geradenkongruenzen als zweidimensionale Flächen auffassen. Die vierparametrige Geradenmenge des reellen projektiven dreidimensionalen Raumes ist in diesem Modell eine Quadrik vom Index 2 in einem reellen projektiven fünfdimensionalen Raum, die so genannte KLEINsche Hyperquadrik. Der Modellwechsel wird durch die KLEINsche Abbildung vollzogen. / In the available work line congruences of the projectively extended three-dimensional euclidean space will be analysed. Following to J. PLÜCKER lines can be seen as basic elements of an line space like in the same way points in a point-space. Taking this fact in consideration a &amp;quot;natural&amp;quot; handling with line congruences might be interesting and reasonable. A special detail in the thesis is the question to minimal congruences in the set of lines of the projectively extended euclidean three-space. It can also be seen as an analogous problem in the geometry of lines which can be find in the differential geometry of surfaces. In this case the line congruences are similar to the surfaces of the three-dimensional (point-)space. The phrase &amp;quot;minimal&amp;quot; means in the line space the connection to the minimal surfaces in the differential geometry. These questions offer in line geometry demonstrative interpretation possibilities if a point-model in the line space exists. One-parameter manifolds of lines (rule surfaces) can be seen in this ambiance as curves and line congruences as two dimensional surfaces. The four-parametric set of lines in the projectively extended three-dimensional euclidian space is in this model a quadric of the index 2 in a real projective five-dimensional space, the so called KLEIN-quadric. The changing of the model is managed by the KLEIN-mapping.

Using Fermat’s Little Theorem, it can be shown that Σmi=1 i m−1 ≡ −1 (mod m) if m is prime. It has been conjectured that the converse is true as well. Namely, that Σmi=1 i m−1 ≡ −1 (mod m) only if m is prime. We shall present some necessary and sufficient conditions for the conjecture to hold, and we will demonstrate that no counterexample exists for m ≤ 1012 .

Igusa's p-adic zeta function is associated to a polynomial f in several variables over the integers and to a prime p. It is a meromorphic function which encodes for every i the number of solutions M_i of f=0 modulo p^i. The intensive study of Igusa's p-adic zeta function by using an embedded resolution of f led to the introduction of the topological zeta function. This geometric invariant of the zero locus of a polynomial f in several variables over the complex numbers was introduced in the early nineties by Denef and Loeser. It is a rational function which they obtained as a limit of Igusa's p-adic zeta functions and which is defined by using an embedded resolution.<br />I have studied the smallest poles of the topological zeta function and the smallest real parts of the poles of Igusa's p-adic zeta function. For n=2 and n=3, I obtained results by using an embedded resolution of singularities. I discovered that the smallest real part of a pole of Igusa's p-adic zeta function is related with the divisibility of the M_i by powers of p. I obtained a general theorem on the divisibility of the M_i by powers of p, which I used to obtain the optimal lower bound for the real part of a pole of Igusa's p-adic zeta function in arbitrary dimension n. I obtained this lower bound also for the topological zeta function by taking the limit.

Gauss introduit la notion de congruence en 1801 dans les Disquisitiones Arithmeticae. L'historiographie classique relie le plus souvent l'histoire de cette notion au développement de la théorie des nombres algébriques, une histoire construite autour d'un groupe de mathématiciens allemands. Pourtant, d'autres auteurs ont publié des travaux en lien avec les congruences dans la première moitié du XIXe siècle, et ce dans des perspectives différentes. Dans ce travail, nous nous proposons de rendre compte de ces dernières en nous concentrant sur les travaux de la scène française publiés entre 1801 et 1850. À partir d'une première lecture globale des textes de notre corpus, nous montrons d'abord que les congruences n'y ont pas connu un développement autonome mais ont été étudiées dans un lien étroit avec les équations. Toutefois, les différentes pratiques rencontrées sont très variées, que ce soit du point de vue des méthodes, des outils en jeu ou des configurations disciplinaires en jeu. Nous étudions ensuite plusieurs travaux arithmétiques d'Euler, de Lagrange, de Legendre et de Gauss afin de comprendre certaines origines de cette activité multiforme mise en évidence dans notre première partie. Nous nous concentrons enfin sur les travaux de deux auteurs de notre corpus, Louis Poinsot et Augustin Louis Cauchy, qui ont joué un rôle important dans l'élaboration et la diffusion de résultats et de pratiques liés aux congruences, même s'ils ont pratiquement disparu des histoires de la théorie des nombres publiées au XXe siècle.

The objective of this paper is to present a context in which mathematics can be glimpsed in a more attractive and dynamic way. For this, succinct but regular knowledge will be presented to the basic understanding of cryptography. We will cite examples of gures and how some of them work, in addition to some de nitions, theorems and demonstrations on topics such as divisibility, congruence, functions and arrays. Finally, we will make suggestions of a set of activities that provide the interconnection of mathematical and daily knowledge of the student, since this combination is an attractive factor for a better assimilation of content. However, the teacher has the insight to make choices and to judge what he thinks it's most interesting. / O objetivo deste trabalho é apresentar um contexto no qual a matemática pode ser vislumbrada de forma mais atrativa e dinâmica. Para isso, serão apresentados conhecimentos sucintos, mas regulares, para a compreensão básica da criptografia. Citaremos exemplos de cifras e de como funcionam algumas delas, mostraremos algumas definições, teoremas e demonstrações sobre assuntos, tais como, divisibilidade, congruência, funções e matrizes. Por fim, faremos sugestões de um conjunto de atividades que proporcionam a interligação de conhecimento matemático e conhecimento diário do aluno, visto que essa combinação é um fator atrativo para melhor assimilação do conteúdo. No entanto, o professor tem o discernimento para realizar escolhas e julgar o que achar mais interessante.

Orientador: Jose Plinio de Oliveira Santos / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-14T08:31:50Z (GMT). No. of bitstreams: 1 Reis_JacksonMartins_M.pdf: 998765 bytes, checksum: ea7248e69be4c892e184263be7050375 (MD5) Previous issue date: 2009 / Resumo: Neste trabalho foram abordados tópicos de Teoria dos Números e alguns testes de primalidade. Mostramos propriedades dos números inteiros, bem como alguns critérios de divisibilidade. Apresentamos também, além das propriedades do Máximo Divisor Comum e Mínimo Múltiplo Comum, interpretações geométricas dos mesmos. Foram estudados Tópicos da Teoria de Congruências e por fim trabalhamos alguns Testes de Primalidade, com respectivos exemplos. / Abstract: In this work were discussed topics of the theory of numbers and some primality tests. We show properties of whole numbers, and some criteria for divisibility. We also present, beyond the properties of the Common Dividing Maximum and Minimum Common Multiple, geometric interpretations of the same ones. They had been study topics of theory of congruences and finally we work some of primality tests, whith respective applications. / Mestrado / Teoria dos Numeros / Mestre em Matemática