Matching Items (6)
Description
Diophantine arithmetic is one of the oldest branches of mathematics, the search
for integer or rational solutions of algebraic equations. Pythagorean triangles are
an early instance. Diophantus of Alexandria wrote the first related treatise in the
fourth century; it was an area extensively studied by the great mathematicians of the seventeenth century, including Euler and Fermat.
The modern approach is to treat the equations as defining geometric objects, curves, surfaces, etc. The theory of elliptic curves (or curves of genus 1, which are much used in modern cryptography) was developed extensively in the twentieth century, and has had great application to Diophantine equations. This theory is used in application to the problems studied in this thesis. This thesis studies some curves of high genus, and possible solutions in both rationals and in algebraic number fields, generalizes some old results and gives answers to some open problems in the literature. The methods involve known techniques together with some ingenious tricks. For example, the equations $y^2=x^6+k$, $k=-39,\,-47$, the two previously unsolved cases for $|k|<50$, are solved using algebraic number theory and the ‘elliptic Chabauty’ method. The thesis also studies the genus three quartic curves $F(x^2,y^2,z^2)=0$ where F is a homogeneous quadratic form, and extend old results of Cassels, and Bremner. It is a very delicate matter to find such curves that have no rational points, yet which do have points in odd-degree extension fields of the rationals.
The principal results of the thesis are related to surfaces where the theory is much less well known. In particular, the thesis studies some specific families of surfaces, and give a negative answer to a question in the literature regarding representation of integers n in the form $n=(x+y+z+w)(1/x+1/y+1/z+1/w).$ Further, an example, the first such known, of a quartic surface $x^4+7y^4=14z^4+18w^4$ is given with remarkable properties: it is everywhere locally solvable, yet has no non-zero rational point, despite having a point in (non-trivial) odd-degree extension fields of the rationals. The ideas here involve manipulation of the Hilbert symbol, together with the theory of elliptic curves.
for integer or rational solutions of algebraic equations. Pythagorean triangles are
an early instance. Diophantus of Alexandria wrote the first related treatise in the
fourth century; it was an area extensively studied by the great mathematicians of the seventeenth century, including Euler and Fermat.
The modern approach is to treat the equations as defining geometric objects, curves, surfaces, etc. The theory of elliptic curves (or curves of genus 1, which are much used in modern cryptography) was developed extensively in the twentieth century, and has had great application to Diophantine equations. This theory is used in application to the problems studied in this thesis. This thesis studies some curves of high genus, and possible solutions in both rationals and in algebraic number fields, generalizes some old results and gives answers to some open problems in the literature. The methods involve known techniques together with some ingenious tricks. For example, the equations $y^2=x^6+k$, $k=-39,\,-47$, the two previously unsolved cases for $|k|<50$, are solved using algebraic number theory and the ‘elliptic Chabauty’ method. The thesis also studies the genus three quartic curves $F(x^2,y^2,z^2)=0$ where F is a homogeneous quadratic form, and extend old results of Cassels, and Bremner. It is a very delicate matter to find such curves that have no rational points, yet which do have points in odd-degree extension fields of the rationals.
The principal results of the thesis are related to surfaces where the theory is much less well known. In particular, the thesis studies some specific families of surfaces, and give a negative answer to a question in the literature regarding representation of integers n in the form $n=(x+y+z+w)(1/x+1/y+1/z+1/w).$ Further, an example, the first such known, of a quartic surface $x^4+7y^4=14z^4+18w^4$ is given with remarkable properties: it is everywhere locally solvable, yet has no non-zero rational point, despite having a point in (non-trivial) odd-degree extension fields of the rationals. The ideas here involve manipulation of the Hilbert symbol, together with the theory of elliptic curves.
ContributorsNguyen, Xuan Tho (Author) / Bremner, Andrew (Thesis advisor) / Childress, Nancy (Committee member) / Jones, John (Committee member) / Quigg, John (Committee member) / Fishel, Susanna (Committee member) / Arizona State University (Publisher)
Created2019
Description
Iwasawa theory is a branch of number theory that studies the behavior of certain objects associated to a $\mathbb{Z}_p$-extension. We will focus our attention to the cyclotomic $\mathbb{Z}_p$-extensions of imaginary quadratic fields for varying primes p, and will give some conditions for when the corresponding lambda-invariants are greater than 1.
ContributorsStokes, Christopher Mathewson (Author) / Childress, Nancy (Thesis advisor) / Sprung, Florian (Committee member) / Montaño, Johnathan (Committee member) / Paupert, Julian (Committee member) / Kaliszewski, Steven (Committee member) / Arizona State University (Publisher)
Created2023
Description
Mark and Paupert concocted a general method for producing presentations for arithmetic non-cocompact lattices, \(\Gamma\), in isometry groups of negatively curved symmetric spaces. To get around the difficulty of constructing fundamental domains in spaces of variable curvature, their method invokes a classical theorem of Macbeath applied to a \(\Gamma\)-invariant covering by horoballs of the negatively curved symmetric space upon which \(\Gamma\) acts. This thesis aims to explore the application of their method to the Picard modular groups, PU\((2,1;\mathcal{O}_{d})\), acting on \(\mathbb{H}_{\C}^2\). This document contains the derivations for the group presentations corresponding to \(d=2,11\), which completes the list of presentations for Picard modular groups whose entries lie in Euclidean domains, namely those with \(d=1,2,3,7,11\). There are differences in the method's application when the lattice of interest has multiple cusps. \(d = 5\) is the smallest value of \(d\) for which the corresponding Picard modular group, \(\PU(2,1;\mathcal{O}_5)\), has multiple cusps, and the method variations become apparent when working in this case.
ContributorsPolletta, David Michael (Author) / Paupert, Julien H (Thesis advisor) / Kotschwar, Brett (Committee member) / Fishel, Susanna (Committee member) / Kawski, Matthias (Committee member) / Childress, Nancy (Committee member) / Arizona State University (Publisher)
Created2021
Description
The classification of $\Lambda$-modules began in 1997 with Sumida [3], when he became interested in an answer to Greenberg's conjecture. Sumida's paper introduced the following set of $\Lambda$-modules when $f(T)$ is a distinguished polynomial over $\mathfrak{O}[T],$ $$M_{f(t)}=\{M |\, \text{char(M)=f(T) and M has no non-trivial finite submodules }\}.$$ Related to the classification idea is the adjoint problem, introduced by Koike in $1999$ [4]. The adjoint of a module $M,$ (denoted $\alpha(M))$ is a mysterious object in Iwasawa Theory. As the adjoint of a module is not well understood, it is easier to classify the "self-adjoint modules" of $M_{f(T)}$ instead, i.e when $M \cong \alpha(M).$ The goal of this dissertation is to improve the classification of self-adjoint modules in the case where the characteristic polynomial has degree higher than $2.$
ContributorsCvitanov, Spencer Carey (Author) / Childress, Nancy N.C (Thesis advisor) / Sprung, Florian F.S (Committee member) / Jones, John J.J (Committee member) / Fishel, Susanna S.F (Committee member) / Spielberg, Jack J.S (Committee member) / Arizona State University (Publisher)
Created2022
Description
In this paper, we study the prime factorizations of numbers slightly larger than the factorial function. While these are closely related to the factorial prime, they have more inherent structure, which allows for explicit results as of yet not established on factorial prime. Case in point, the main result of this paper is that these numbers, which are described in concrete terms below, cannot be prime powers outside of a handful of small cases; this is a generalization of a classical result stating they cannot be primes. Minor explicit results and heuristic analysis are then given to further characterize the set.
ContributorsLawson, Liam John (Author) / Jones, John (Thesis director) / Childress, Nancy (Committee member) / School of Mathematical and Statistical Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2019-12
Description
Similar to the real numbers, the p-adic fields are completions of the rational numbers. However, distance in this space is determined based on divisibility by a prime number, p, rather than by the traditional absolute value. This gives rise to a peculiar topology which offers significant simplifications for p-adic continuous functions and p-adic integration than is present in the real numbers. These simplifications may present significant advantages to modern physics – specifically in harmonic analysis, quantum mechanics, and string theory. This project discusses the construction of the p-adic numbers, elementary p-adic topology, p-adic continuous functions, introductory p-adic measure theory, the q-Volkenborn distribution, and applications of p-adic numbers to physics. We define q-Volkenborn integration and its connection to Bernoulli numbers.
ContributorsBurgueno, Alyssa Erin (Author) / Childress, Nancy (Thesis director) / Jones, John (Committee member) / School of Mathematical and Statistical Sciences (Contributor, Contributor, Contributor) / Department of Physics (Contributor) / Barrett, The Honors College (Contributor)
Created2020-05