On Criteria for Large Cyclotomic $\lambda$-invariants for Imaginary Quadratic Fields

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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…

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.
Date Created
2023
Agent

Generalizations of the Signed Selmer Groups for Cyclotomic Extensions

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Description
Let $E$ be an elliptic curve defined over a number field $K$, $p$ a rational prime, and $\Lambda(\Gamma)$ the Iwasawa module of the cyclotomic extension of $K$. A famous conjecture by Mazur states that the $p$-primary component of the Selmer…

Let $E$ be an elliptic curve defined over a number field $K$, $p$ a rational prime, and $\Lambda(\Gamma)$ the Iwasawa module of the cyclotomic extension of $K$. A famous conjecture by Mazur states that the $p$-primary component of the Selmer group of $E$ is $\Lambda(\Gamma)$-cotorsion when $E$ has good ordinary reduction at all primes of $K$ lying over $p$. The conjecture was proven in the case that $K$ is the field of rationals by Kato, but is known to be false when $E$ has supersingular reduction type. To salvage this result, Kobayashi introduced the signed Selmer groups, which impose stronger local conditions than their classical counterparts. Part of the construction of the signed Selmer groups involves using Honda's theory of commutative formal groups to define a canonical system of points. In this paper I offer an alternate construction that appeals to the Functional Equation Lemma, and explore a possible way of generalizing this method to elliptic curves defined over $p$-adic fields by passing from formal group laws to formal modules.
Date Created
2023
Agent

Presentations for Singly-Cusped Bianchi Groups

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Description

Applying a classical theorem due to Macbeath applied to a suitably sized horoball, we calculate novel group presentations for singly-cusped Bianchi groups. We find new presentations for Bianchi groups with d = -43, -67, -163. With previously known presentations for…

Applying a classical theorem due to Macbeath applied to a suitably sized horoball, we calculate novel group presentations for singly-cusped Bianchi groups. We find new presentations for Bianchi groups with d = -43, -67, -163. With previously known presentations for d = -1, -2, -3, -7, -11, -19, this constitutes a complete set of presentations for singly-cusped Bianchi groups.

Date Created
2022-05
Agent

Graphs of Sets of Reduced Words

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Description
Any permutation in the finite symmetric group can be written as a product of simple transpositions $s_i = (i~i+1)$. For a fixed permutation $\sigma \in \mathfrak{S}_n$ the products of minimal length are called reduced decompositions or reduced words, and the…

Any permutation in the finite symmetric group can be written as a product of simple transpositions $s_i = (i~i+1)$. For a fixed permutation $\sigma \in \mathfrak{S}_n$ the products of minimal length are called reduced decompositions or reduced words, and the collection of all such reduced words is denoted $R(\sigma)$. Any reduced word of $\sigma$ can be transformed into any other by a sequence of commutation moves or long braid moves. One area of interest in these sets are the congruence classes defined by using only braid moves or only commutation moves. This document will present work towards a conjectured relationship between the number of reduced words and the number of braid classes. The set $R(\sigma)$ can be drawn as a graph, $G(\sigma)$, where the vertices are the reduced words, and the edges denote the presence of a commutation or braid move between the words. This paper will present brand new work on subgraph structures in $G(\sigma)$, as well as new formulas to count the number of braid edges and commutation edges in $G(\sigma)$. The permutation $\sigma$ covers $\tau$ in the weak order poset if the length of $\tau$ is one less than the length of $\sigma$, and there exists a simple transposition $s_i$ such that $\sigma = \tau s_i$. This paper will cover new work on the relationships between the size of $R(\sigma)$ and $R(\tau)$, and how this creates a new method of writing reduced decompositions of $\sigma$ as products of permutations $\alpha$ and $\beta$, where both $\alpha$ and $\beta$ have a length greater than one. Finally, this thesis will also discuss how these results help relate the number of reduced words and the number of braid classes in certain cases.
Date Created
2021
Agent

The Geometry of 1-cusped and 2-cusped Picard Modular Groups

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…

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.
Date Created
2021
Agent

An Investigation of Supersingular Elliptic Curves in Quantum-Resistant Cryptography

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Description

Many current cryptographic algorithms will eventually become easily broken by Shor's Algorithm once quantum computers become more powerful. A number of new algorithms have been proposed which are not compromised by quantum computers, one of which is the Supersingular Isogeny…

Many current cryptographic algorithms will eventually become easily broken by Shor's Algorithm once quantum computers become more powerful. A number of new algorithms have been proposed which are not compromised by quantum computers, one of which is the Supersingular Isogeny Diffie-Hellman Key Exchange Protocol (SIDH). SIDH works by having both parties perform random walks between supersingular elliptic curves on isogeny graphs of prime degree and eventually end at the same location, a shared secret.<br/><br/>This thesis seeks to explore some of the theory and concepts underlying the security of SIDH, especially as it relates to finding supersingular elliptic curves, generating isogeny graphs, and implementing SIDH. As elliptic curves and SIDH may be an unfamiliar topic to many readers, the paper begins by providing a brief introduction to elliptic curves, isogenies, and the SIDH Protocol. Next, the paper investigates more efficient methods of generating supersingular elliptic curves, which are important for visualizing the isogeny graphs in the algorithm and the setup of the protocol. Afterwards, the paper focuses on isogeny maps of various degrees, attempting to visualize isogeny maps similar to those used in SIDH. Finally, the paper looks at an implementation of SIDH in PARI/GP and work is done to see the effects of using isogenies of degree greater than 2 and 3 on the security, runtime, and practicality of the algorithm.

Date Created
2021-05
Agent

Battleship: An Application of Multiparty Computation Encryption

Description

Cheating in Battleship is effortless. Battleship is a popular two-player board game where each player strategically places five ships on his or her concealed board. During this game, one can easily move their ships during a play, falsify an attack,…

Cheating in Battleship is effortless. Battleship is a popular two-player board game where each player strategically places five ships on his or her concealed board. During this game, one can easily move their ships during a play, falsify an attack, or not even place their ships. A solution to this concern is implementing multiparty computation (MPC) encryption to ensure that the location of both players’ ships and the result of attacking a ship is true. This document details the creation and security of a Battleship program that implements an MPC encryption method known as Poker Over the Telephone.

Date Created
2021-05
Agent

p-Adic Numbers with an Emphasis on q-Volkenborn Integration

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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…

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.
Date Created
2020-05
Agent

The Number Field Sieve

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Description
This thesis project is focused on studying the number field sieve. The number field sieve is a factoring algorithm which uses algebraic number theory and is one of the fastest known factoring algorithms today. Factoring large integers into prime factors…

This thesis project is focused on studying the number field sieve. The number field sieve is a factoring algorithm which uses algebraic number theory and is one of the fastest known factoring algorithms today. Factoring large integers into prime factors is an extremely difficult problem, yet also extremely important in cryptography. The security of the cryptosystem RSA is entirely based on the difficulty of factoring certain large integers into a product of two distinct large primes. While the number field sieve is one of the fastest factoring algorithms known, it is still not efficient enough to factor cryptographic sized integers.

In this thesis we will examine the algorithm of the number field sieve and discuss some important advancements. In particular, we will focus on the advancements that have been done in the polynomial selection step, the first main step of the number field sieve. The polynomial selected determines the number field by which computations are carried out in the remainder of the algorithm. Selection of a good polynomial allows for better time efficiency and a higher probability that the algorithm will be successful in factoring.
Date Created
2020-05
Agent

Almost-Primes Near Factorials

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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…

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.
Date Created
2019-12
Agent
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