Physical Sciences and Mathematics Commons^™

Statistics For Iwasawa Invariants Of Elliptic Curves, Ii, Debanjana Kundu, Anwesh Ray

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

We study the average behavior of the Iwasawa invariants for Selmer groups of elliptic curves. These results lie at the intersection of arithmetic statistics and Iwasawa theory. We obtain lower bounds for the density of rational elliptic curves with prescribed Iwasawa invariants.

An Exploration Of Absolute Minimal Degree Lifts Of Hyperelliptic Curves, Justin A. Groves

Doctoral Dissertations

For any ordinary elliptic curve E over a field with non-zero characteristic p, there exists an elliptic curve E over the ring of Witt vectors W(E) for which we can lift the Frobenius morphism, called the canonical lift. Voloch and Walker used this theory of canonical liftings of elliptic curves over Witt vectors of length 2 to construct non-linear error-correcting codes for characteristic two. Finotti later proved that for longer lengths of Witt vectors there are better lifts than the canonical. He then proved that, more generally, for hyperelliptic curves one can construct a lifting over …

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 F_q-rational solutions, Tate modules, supersingular curves, and applications to elliptic curves over Q. In particular, we approach the topic largely through the use …

On The Iwasawa Invariants Of Nonordinary Modular Forms, Rylan J. Gajek-Leonard

Doctoral Dissertations

We extend known results on the behavior of Iwasawa invariants attached to Mazur-Tate elements for p-nonordinary modular forms of weight k=2 to higher weight modular forms with a_p=0. This is done by using a decomposition of the p-adic L-function due to R. Pollack in order to construct explicit lifts of Mazur-Tate elements to the full Iwasawa algebra. We then study the behavior of Iwasawa invariants upon projection to finite layers, allowing us to express the invariants of Mazur-Tate elements in terms of those coming from plus/minus p-adic L-functions. Our results combine with work of Pollack and Weston to relate the …

Elliptic Curves And Their Practical Applications, Henry H. Hayden Iv

MSU Graduate Theses

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

Journal of Humanistic Mathematics

I share some of my adventures in mathematical research and homeschooling in the time of COVID-19.

Torsion Subgroups Of Rational Elliptic Curves Over Odd Degree Galois Fields, Caleb Mcwhorter

Dissertations - ALL

The Mordell-Weil Theorem states that if K is a number field and E/K is an elliptic curve that the group of K-rational points E(K) is a finitely generated abelian group, i.e. E(K) = Z^{r_K} ⊕ E(K)_tors, where r_K is the rank of E and E(K)_tors is the subgroup of torsion points on E. Unfortunately, very little is known about the rank r_K. Even in the case of K = Q, it is not known which ranks are possible or if the ranks are bounded. However, there have been great strides in determining the sets E(K)_tors. Progress began in 1977 with …

On Elliptic Curves, Montana S. Miller

MSU Graduate Theses

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.

Classification Of Torsion Subgroups For Mordell Curves, Zachary Porat

Honors Theses

Elliptic curves are an interesting area of study in mathematics, laying at the intersection of algebra, geometry, and number theory. They are a powerful tool, having applications in everything from Andrew Wiles’ proof of Fermat’s Last Theorem to cybersecurity. In this paper, we first provide an introduction to elliptic curves by discussing their geometry and associated group structure. We then narrow our focus, further investigating the torsion subgroups of elliptic curves. In particular, we will examine two methods used to classify these subgroups. We finish by employing these methods to categorize the torsion subgroups for a specific family of elliptic …

Elliptic Curves And Power Residues, Vy Thi Khanh Nguyen

Doctoral Dissertations

Let E₁ x E₂ over Q be a fixed product of two elliptic curves over Q with complex multiplication. I compute the probability that the pth Fourier coefficient of E₁ x E₂, denoted as a_p(E₁) + a_p(E₂), is a square modulo p. The results are 1/4, 7/16, and 1/2 for different imaginary quadratic fields, given a technical independence of the twists. The similar prime densities for cubes and 4th power are 19/54, and 1/4, respectively. I also compute the probabilities without the technical …

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 Curve Cryptography: Extensions Of Subfield Curves In Characteristic 2, Joel Dearmond

Pence-Boyce STEM Student Scholarship

This paper examines subfield curve extensions on a number of elliptic curves over finite fields in characteristic 2. The data generated is aimed to assist further understanding into the nature of elliptic curves, and any possible characteristics or patterns that they share. The total rational points on base fields were found using C++, and points on their field extensions were calculated using Scientific Workplace. Different extensions were then categorized based on the factorization of their respective points. We found that the total number of points on a base field will divide the total number of points of any extension of …

Properties Of The Iterates Of The Weierstrass-℘ Function, Walter H. Chen, Michael S. Willis

Furman University Electronic Journal of Undergraduate Mathematics

This paper discusses several properties of the Weierstrass-℘ function, as defined on the fundamental parallelogram C/Γ, where C is the complex plane and Γ is the lattice generated by ω1 and ω2. Using the addition formula for ℘(z₁ + z₂), we develop a reccurence relation for ℘(nz) in terms of ℘(z). We then examine the degree of this expression, some coefficients, and patterns concerning the poles of this function. We also consider the geometric interpretation of taking an arbitrary z₀ and adding it to itself, both in the fundamental parallelogram C/Γ and …

Elliptic Curves, Trinity Mecklenburg

Electronic Theses, Projects, and Dissertations

The main focus of this paper is the study of elliptic curves, non-singular projective curves of genus 1. Under a geometric operation, the rational points E(Q) of an elliptic curve E form a group, which is a finitely-generated abelian group by Mordell’s theorem. Thus, this group can be expressed as the finite direct sum of copies of Z and finite cyclic groups. The number of finite copies of Z is called the rank of E(Q).

From John Tate and Joseph Silverman we have a formula to compute the rank of curves of the form …

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 E_n is 0. We then introduce L-functions and explain the Birch and …

Lattices From Elliptic Curves Over Finite Fields, Lenny Fukshansky, Hiren Maharaj

CMC Faculty Publications and Research

In their well known book Tsfasman and Vladut introduced a construction of a family of function field lattices from algebraic curves over finite fields, which have asymptotically good packing density in high dimensions. In this paper we study geometric properties of lattices from this construction applied to elliptic curves. In particular, we determine the generating sets, conditions for well-roundedness and a formula for the number of minimal vectors. We also prove a bound on the covering radii of these lattices, which improves on the standard inequalities.

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

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 …

Modular Forms And Elliptic Curves Over The Cubic Field Of Discriminant - 23, Paul E. Gunnells, Dan Yasaki

Paul Gunnells

Let F be the cubic field of discriminant –23 and let O Ϲ F be its ring of integers. By explicitly computing cohomology of congruence subgroups of 〖GL〗_2(O) , we computationally investigate modularity of elliptic curves over F.

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, y^₂ = 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 …

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

Sunil Chetty

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.

Computing Local Constants Of Cm Elliptic Curves, Sunil Chetty, Lung Li

Sunil Chetty

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.

Elliptic Curves: Minimally Spanning Prime Fields And Supersingularity, Travis Mcgrath

Senior Projects Spring 2011

Elliptic curves are cubic curves that have been studied throughout history. From Diophantus of Alexandria to modern-day cryptography, Elliptic Curves have been a central focus of mathematics. This project explores certain geometric properties of elliptic curves defined over finite fields.

Fix a finite field. This project starts by demonstrating that given enough elliptic curves, their union will contain every point in the affine plane. We then find the fewest curves possible such that their union still contains all these points. Using some of the tools discussed in solving this problem, we then explore what can be said about the number …

Class Numbers Of Ray Class Fields Of Imaginary Quadratic Fields, Omer Kucuksakalli

Open Access Dissertations

Let K be an imaginary quadratic field with class number one and let [Special characters omitted.] be a degree one prime ideal of norm p not dividing 6 d K . In this thesis we generalize an algorithm of Schoof to compute the class number of ray class fields [Special characters omitted.] heuristically. We achieve this by using elliptic units analytically constructed by Stark and the Galois action on them given by Shimura's reciprocity law. We have discovered a very interesting phenomena where p divides the class number of [Special characters omitted.] . This is a counterexample to the elliptic …

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 …