Mod 2 Homology For Gl(4) And Galois Representations, 2014 University of Massachusetts - Amherst

#### Mod 2 Homology For Gl(4) And Galois Representations, Avner Ash, Paul E. Gunnells, Mark Mcconnell

*Paul Gunnells*

We extend the computations in [AGM11] to find the mod 2 homology in degree 1 of a congruence subgroup Γ of SL(4,Z) with coefficients in the sharbly complex, along with the action of the Hecke algebra. This homology group is related to the cohomology of Γ with F2 coefficients in the top cuspidal degree. These computations require a modification of the algorithm to compute the action of the Hecke operators, whose previous versions required division by 2. We verify experimentally that every mod 2 Hecke eigenclass found appears to have an attached Galois representation, giving evidence for a ...

Computing Local Constants For Cm Elliptic Curves, 2014 College of Saint Benedict/Saint John's University

#### Computing Local Constants For Cm Elliptic Curves, Sunil Chetty, Lung Li

*Mathematics Faculty Publications*

Let E/k be an elliptic curve with CM by O. We determine a formula for (a generalization of) the arithmetic local constant of Mazur-Rubin at almost all primes of good reduction. We apply this formula to the CM curves defined over Q and are able to describe extensions F/Q over which the O-rank of E grows.

There And Back Again: Elliptic Curves, Modular Forms, And L-Functions, 2014 Harvey Mudd College

#### There And Back Again: Elliptic Curves, Modular Forms, And L-Functions, Allison F. Arnold-Roksandich

*HMC Senior Theses*

*L*-functions form a connection between elliptic curves and modular forms. The goals of this thesis will be to discuss this connection, and to see similar connections for arithmetic functions.

A Frobenius Question Related To Actions On Curves In Characteristic P, 2014 Gettysburg College

#### A Frobenius Question Related To Actions On Curves In Characteristic P, Darren B. Glass

*Math Faculty Publications*

We consider which integers *g* can occur as the genus and of a curve defined over a field of characteristic *p* which admits an automorphism of degree *pq*, where *p* and *q* are distinct primes. This investigation leads us to consider a certain family of three-dimensional Frobenius problems and prove explicit formulas giving their solution in many cases.

Finding Zeros Of Rational Quadratic Forms, 2014 Claremont McKenna College

#### Finding Zeros Of Rational Quadratic Forms, John F. Shaughnessy

*CMC Senior Theses*

In this thesis, we introduce the notion of quadratic forms and provide motivation for their study. We begin by discussing Diophantine equations, the field of p-adic numbers, and the Hasse-Minkowski Theorem that allows us to use p-adic analysis determine whether a quadratic form has a rational root. We then discuss search bounds and state Cassels' Theorem for small-height zeros of rational quadratic forms. We end with a proof of Cassels' Theorem and suggestions for further reading.

Constructing Carmichael Numbers Through Improved Subset-Product Algorithms, 2014 Illinois Wesleyan University

#### Constructing Carmichael Numbers Through Improved Subset-Product Algorithms, W.R. Alford, Jon Grantham, Steven Hayman, Andrew Shallue

*Scholarship*

We have constructed a Carmichael number with 10,333,229,505 prime factors, and have also constructed Carmichael numbers with prime factors for every between 3 and 19,565,220. These computations are the product of implementations of two new algorithms for the subset product problem that exploit the non-uniform distribution of primes with the property that divides a highly composite .

Explorations Of The Collatz Conjecture (Mod M), 2014 Georgia Southern University

#### Explorations Of The Collatz Conjecture (Mod M), Glenn Micah Jackson Jr

*University Honors Program Theses*

The Collatz Conjecture is a deceptively difficult problem recently developed in mathematics. In full, the conjecture states: Begin with any positive integer and generate a sequence as follows: If a number is even, divide it by two. Else, multiply by three and add one. Repetition of this process will eventually reach the value 1. Proof or disproof of this seemingly simple conjecture have remained elusive. However, it is known that if the generated Collatz Sequence reaches a cycle other than 4, 2, 1, the conjecture is disproven. This fact has motivated our search for occurrences of 4, 2, 1, and ...

Computing Boundary Extensions Of Conformal Maps, 2014 Iowa State University

#### Computing Boundary Extensions Of Conformal Maps, Timothy H. Mcnicholl

*Mathematics Publications*

We show that a computable and conformal map of the unit disk onto a bounded domain D has a computable boundary extension if D has a computable boundary connectivity function.

Constructing Carmichael Numbers Through Improved Subset-Product Algorithms, 2013 Illinois Wesleyan University

#### Constructing Carmichael Numbers Through Improved Subset-Product Algorithms, W.R. Alford, Jon Grantham, Steven Hayman, Andrew Shallue

*Andrew Shallue*

The Kronecker-Weber Theorem: An Exposition, 2013 Lawrence University

#### The Kronecker-Weber Theorem: An Exposition, Amber Verser

*Lawrence University Honors Projects*

This paper is an investigation of the mathematics necessary to understand the Kronecker-Weber Theorem. Following an article by Greenberg, published in *The American Mathematical Monthly* in 1974, the presented proof does not use class field theory, as the most traditional treatments of the theorem do, but rather returns to more basic mathematics, like the original proofs of the theorem. This paper seeks to present the necessary mathematical background to understand the proof for a reader with a solid undergraduate background in abstract algebra. Its goal is to make what is usually an advanced topic in the study of algebraic number ...

Quotients Of Gaussian Primes, 2013 Pomona College

#### Quotients Of Gaussian Primes, Stephan Ramon Garcia

*Pomona Faculty Publications and Research*

It has been observed many times, both in the Monthly and elsewhere, that the set of all quotients of prime numbers is dense in the positive real numbers. In this short note we answer the related question: "Is the set of all quotients of Gaussian primes dense in the complex plane?"

Wilson Theorem, 2013 Selected Works

Rock Art Tallies: Mathematics On Stone In Western North America, 2013 Millikin University

#### Rock Art Tallies: Mathematics On Stone In Western North America, James V. Rauff

*Journal of Humanistic Mathematics*

Western North America abounds with rock art sites. From Alberta to New Mexico and from Minnesota to California one can find the enigmatic rock paintings and rock carvings left by the pre-Columbian inhabitants. The images left behind on the rocks of the American plains and deserts are those of humanoids and animals, arrows and spears, and a variety of geometric shapes and abstract designs. Also included, in great numbers, are sequences of repeated shapes and marks that scholars have termed "tallies." The tallies are presumed to be an ancient accounting of something or some things. This article examines rock art ...

Factoring The Duplication Map On Elliptic Curves For Use In Rank Computations, 2013 Scripps College

#### Factoring The Duplication Map On Elliptic Curves For Use In Rank Computations, Tracy Layden

*Scripps Senior Theses*

This thesis examines the rank of elliptic curves. We first examine the correspondences between projective space and affine space, and use the projective point at infinity to establish the group law on elliptic curves. We prove a section of Mordell’s Theorem to establish that the abelian group of rational points on an elliptic curve is finitely generated. We then use homomorphisms established in our proof to find a formula for the rank, and then provide examples of computations.

Slicing A Puzzle And Finding The Hidden Pieces, 2013 Olivet Nazarene University

#### Slicing A Puzzle And Finding The Hidden Pieces, Martha Arntson

*Honors Program Projects*

The research conducted was to investigate the potential connections between group theory and a puzzle set up by color cubes. The goal of the research was to investigate different sized puzzles and discover any relationships between solutions of the same sized puzzles. In this research, first, there was an extensive look into the background of Abstract Algebra and group theory, which is briefly covered in the introduction. Then, each puzzle of various sizes was explored to find all possible color combinations of the solutions. Specifically, the 2x2x2, 3x3x3, and 4x4x4 puzzles were examined to find that the 2x2x2 has 24 ...

Aliquot Cycles For Elliptic Curves With Complex Multiplication, 2013 Washington University in St Louis

#### Aliquot Cycles For Elliptic Curves With Complex Multiplication, Thomas Morrell

*Undergraduate Theses—Unrestricted*

We review the history of elliptic curves and show that it is possible to form a group law using the points on an elliptic curve over some field *L*. We review various methods for computing the order of this group when *L* is finite, including the complex multiplication method. We then define and examine the properties of elliptic pairs, lists, and cycles, which are related to the notions of amicable pairs and aliquot cycles for elliptic curves, defined by Silverman and Stange. We then use the properties of elliptic pairs to prove that aliquot cycles of length greater than two ...

Some Contributions To The Sociology Of Numbers, 2013 Saint Mary's University - Canada

#### Some Contributions To The Sociology Of Numbers, Robert Dawson

*Journal of Humanistic Mathematics*

Those who work with numbers eventually realize that they all have different personalities (the word "numbers" can of course be replaced by any number of other nouns here.) Here is one view of the issue.

Supercharacters, Exponential Sums, And The Uncertainty Principle, 2013 Pomona College

#### Supercharacters, Exponential Sums, And The Uncertainty Principle, J.L. Brumbaugh '13, Madeleine Bulkow '14, Patrick S. Fleming, Luis Alberto Garcia '14, Stephan Ramon Garcia, Gizem Karaali, Matt Michal '15, Andrew P. Turner '14

*Pomona Faculty Publications and Research*

The theory of supercharacters, which generalizes classical character theory, was recently introduced by P. Diaconis and I.M. Isaacs, building upon earlier work of C. Andre. We study supercharacter theories on $(Z/nZ)^d$ induced by the actions of certain matrix groups, demonstrating that a variety of exponential sums of interest in number theory (e.g., Gauss, Ramanujan, and Kloosterman sums) arise in this manner. We develop a generalization of the discrete Fourier transform, in which supercharacters play the role of the Fourier exponential basis. We provide a corresponding uncertainty principle and compute the associated constants in several cases.

Pointless Hyperelliptic Curves, 2013 Gettysburg College

#### Pointless Hyperelliptic Curves, Ryan P. Becker, Darren B. Glass

*Math Faculty Publications*

In this paper we consider the question of whether there exists a hyperelliptic curve of genus g which is defined over but has no rational points over for various pairs . As an example of such a result, we show that if p is a prime such that is also prime then there will be pointless hyperelliptic curves over of every genus.

Galois Representations From Non-Torsion Points On Elliptic Curves, 2013 Bard College

#### Galois Representations From Non-Torsion Points On Elliptic Curves, Matthew Phillip Hughes

*Senior Projects Spring 2013*

Working from well-known results regarding *l*-adic Galois representations attached to elliptic curves arising from successive preimages of the identity, we consider a natural deformation. Given a non-zero point P on a curve, we investigate the Galois action on the splitting fields of preimages of P under multiplication-by-*l* maps. We give a group-theoretic structure theorem for the corresponding Galois group, and state a conjecture regarding composita of two such splitting fields.