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Articles 1 - 10 of 10
Full-Text Articles in Number Theory
Elliptic Curves Over Finite Fields, Christopher S. Calger
Elliptic Curves Over Finite Fields, Christopher S. Calger
Honors Theses
The goal of this thesis is to give an expository report on elliptic curves over finite fields. We begin by giving an overview of the necessary background in algebraic geometry to understand the definition of an elliptic curve. We then explore the general theory of elliptic curves over arbitrary fields, such as the group structure, isogenies, and the endomorphism ring. We then study elliptic curves over finite fields. We focus on the number of Fq-rational solutions, Tate modules, supersingular curves, and applications to elliptic curves over Q. In particular, we approach the topic largely through the use …
Elliptic Curves And Their Practical Applications, Henry H. Hayden Iv
Elliptic Curves And Their Practical Applications, Henry H. Hayden Iv
Graduate Theses/Dissertations
Finding rational points that satisfy functions known as elliptic curves induces a finitely-generated abelian group. Such functions are powerful tools that were used to solve Fermat's Last Theorem and are used in cryptography to send private keys over public systems. Elliptic curves are also useful in factoring and determining primality.
Arithmetics, Interrupted, Matilde Lalín
Arithmetics, Interrupted, Matilde Lalín
Journal of Humanistic Mathematics
I share some of my adventures in mathematical research and homeschooling in the time of COVID-19.
On Elliptic Curves, Montana S. Miller
On Elliptic Curves, Montana S. Miller
Graduate Theses/Dissertations
An elliptic curve over the rational numbers is given by the equation y2 = x3+Ax+B. In our thesis, we study elliptic curves. It is known that the set of rational points on the elliptic curve form a finitely generated abelian group induced by the secant-tangent addition law. We present an elementary proof of associativity using Maple. We also present a relatively concise proof of the Mordell-Weil Theorem.
A Variation On The Theme Of Nicomachus, Florian Luca, Geremías Polanco, Wadim Zudilin
A Variation On The Theme Of Nicomachus, Florian Luca, Geremías Polanco, Wadim Zudilin
Mathematics Sciences: Faculty Publications
In this paper, we prove some conjectures of K. Stolarsky concerning the first and third moments of the Beatty sequences with the golden section and its square.
Elliptic Curves And The Congruent Number Problem, Jonathan Star
Elliptic Curves And The Congruent Number Problem, Jonathan Star
CMC Senior Theses
In this paper we explain the congruent number problem and its connection to elliptic curves. We begin with a brief history of the problem and some early attempts to understand congruent numbers. We then introduce elliptic curves and many of their basic properties, as well as explain a few key theorems in the study of elliptic curves. Following this, we prove that determining whether or not a number n is congruent is equivalent to determining whether or not the algebraic rank of a corresponding elliptic curve En is 0. We then introduce L-functions and explain the Birch and …
Computing Local Constants For Cm Elliptic Curves, Sunil Chetty, Lung Li
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.
Aliquot Cycles For Elliptic Curves With Complex Multiplication, Thomas Morrell
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 …
Elliptic Curves Of High Rank, Cecylia Bocovich
Elliptic Curves Of High Rank, Cecylia Bocovich
Mathematics, Statistics, and Computer Science Honors Projects
The study of elliptic curves grows out of the study of elliptic functions which dates back to work done by mathematicians such as Weierstrass, Abel, and Jacobi. Elliptic curves continue to play a prominent role in mathematics today. An elliptic curve E is defined by the equation, y2 = x3 + ax + b, where a and b are coefficients that satisfy the property 4a3 + 27b2 = 0. The rational solutions of this curve form a group. This group, denoted E(Q), is known as the Mordell-Weil group and was proved by Mordell to be isomorphic …
Galois Structure And De Rhan Invariants Of Elliptic Curves, Darren B. Glass, Sonin Kwon
Galois Structure And De Rhan Invariants Of Elliptic Curves, Darren B. Glass, Sonin Kwon
Math Faculty Publications
Let K be a number field with ring of integers OK. Suppose a finite group G acts numerically tamely on a regular scheme X over OK. One can then define a de Rham invariant class in the class group Cl(OK[G]), which is a refined Euler characteristic of the de Rham complex of X. Our results concern the classification of numerically tame actions and the de Rham invariant classes. We first describe how all Galois etale G-covers of a K-variety may be built up from finite Galois extensions of K and from geometric covers. When X is a curve of positive …