Then we shall examine an addition law on the group of points on an elliptic curve, before we proceed by looking at maps between elliptic curves. We focus on elliptic curves, which are the simplest case, and also the only curves used in practice. Determining fundamental periods from an elliptic curve over c. Applications of the theory of elliptic functions in number. While not as general as miyake and others, it explains the fundamental concepts and driving ideas behind modular forms and elliptic curves in a down to earth fashion not seen in modern text. To attach an elliptic curve e b via kle93 to the quintic used in buh78.
Explicit descent on elliptic curves by thomas womack, mmath. A modular elliptic curve is an elliptic curve e that admits a parametrisation x 0 n e by a modular curve. I spend my days doing math, more precisely number theory, although recently ive been looking for excuses to think about geometry instead. All existing implementations of pairingbased cryptosystems are built with elliptic curves. One of these applications is the explicit determination of all imaginary quadratic fields with classnumber one, famous as the 10th discriminant problem. Counting elliptic curves with an nrational isogeny. Elliptic curves, modular forms, and their lfunctions is a marvelous addition to the literature. The essential fact about elliptic curves with a 5torsion point sketched in the above paragraphs is that for any scheme s, there is a bijection between, on the one hand, isomorphism classes of pairs e,p of an elliptic curve esand a section p of exact order 5 in all geometric. Elliptic curves and modular forms 2 elliptic curves and modular curves are one of the most important objects studied in number theory. This thesis is concerned with the modular icosahedral representation constructed by buhler in buh78. Ecdsa optimizations on an arm processor for a nist curve over gfp abstract approved. Mathematical foundations of elliptic curve cryptography. Springer new york berlin heidelberg hong kong london milan paris tokyo. I have made them public in the hope that they might be useful to others, but these are not o cial notes in any way.
Side channel attack resistant elliptic curves cryptosystem on multicores for power efficiency abstract approved. Eis isogenous to an elliptic curve with jinvariant 0, 1123, or 529325 and so from tables of modular elliptic curves of low conductor is modular. One characterization is that if c is smooth then given p,q 2 ck there exists a unique r 2 ck such that there exists p2k with \c fp,q,rg. In this thesis we utilize integer moduli of special form, which is reminiscent of lowhamming weight polynomials. Explicit descent on elliptic curves by thomas womack. The fundamental critical points of modular elliptic curves. Zeros of the modular parametrization of rational elliptic curves a thesis submitted to the department of mathematics of the university of hawaii in partial fulfillment of plan b for the master s degree in mathematics. The aim of this thesis is to show the tools and some steps to prove the case of. Quadratic points on modular curves universiteit leiden. In this thesis, we investigate modularity of elliptic curves over a general totally real number field. An abstract of the thesis of haluk kent tanik for the degree of master of science in computer science presented on june 19, 2001. Their calculation of this quantity for all elliptic curves of conductor less than 424 confirmed that, with only 16 exceptions, the bound is sharp.
The smallest integer m satisfying h gm is called the logarithm or index of h with respect to g, and is denoted. We begin with a reasonably comprehensive introduction to the theory of elliptic curves, including proofs of most of the relevant results. In her master s thesis zeros of the modular parametrization of rational elliptic curves, univ. Algant master thesis july 2018 elliptic curves and modular forms candidate francesco bruzzesi advisor prof. Torsion subgroups of elliptic curves over number elds. We show the feasibility of the implementation and use of this cryptography in the iot by a thorough evaluation of the solution by analyzing the performance using. An elliptic curve over a eld f is a complete algebraic group over f of dimension 1.
For more details about elliptic curves and missing proofs, the reader may refer to 4,5. Some of the projects are rather ambitious and may very well be the topic of a master thesis. Elliptic curves of odd modular degree by frank calegari. Sutherland december 24, 2012 abstract this is an extended version of an expository talk given at a seminar1 on mazurs torsion theorem, summarizing work on generalizations to number elds and related results. Iwasawa theory of elliptic curves with complex multiplication. The overall goal of this thesis is to show how such a representation is attached to an elliptic curve. It is absolutely voluntary, but if you decide to give us your.
The langlandskottwitz method for the modular curve uvafnwi. And in this objective lozanorobledo succeeds admirably. In this master thesis we present a lightweight bsdbased implementation of the elliptic curve cryptography ecc for the contiki os and its evaluation. Both conjectures were proved by gerd mersmann in a brilliant masters thesis. The star of this thesis is the modular curve x0n and we will examine its life from several. The goal of this diploma thesis is to provide such a background. Modularity of some elliptic curves over totally real. In the end of this chapter, we will consider alternative forms of. However, the primary objects of study, the statements of the main theorems, and their corollaries are within the grasp of advanced undergraduates. This is an introduction to the arithmetic theory of modular functions and modular forms, with a greater emphasis on the geometry than most accounts. The aim of this thesis is to present some striking applications of number theory, essentially based on the powerful machinery of elliptic modular functions and class field theory. Modular functions and modular forms elliptic modular curves j. Gunnells joint work with farshid hajir, dinakar ramakrishnan, dan yasaki let.
Louis on march 29, 1972, at the ams symposium on analytic number theory and related paris of analysis 48. Speci cally we will be interested in the set of all elliptic curves with integral modulus je and also the set of all elliptic curves with complex multiplication. In my discussion of this problem, i have followed the work. The modular curve x 0n is very important as it is one of the objects which links the world of elliptic curves with the world of modular forms. For additional links to online elliptic curve resources, and for other material, the reader is invited to visit the arithmetic of. An introduction to the theory of elliptic curves the discrete logarithm problem fix a group g and an element g 2 g. Etd an algorithm for numerically computing preimages of. Projects on elliptic curves and modular forms math 480, spring 2010 in the following are 11 projects for this course. If an elliptic curve over qwith a given jinvariant is modular then it is easy to see that all elliptic curves with the same jinvariant are modular in which case we say that the j. General requirements for graduate degrees at rutgers are governed by the rules of the graduate school new brunswick and are listed in the current catalog.
The star of this thesis is the modular curve x 0n and we will examine its life from several di erent angles. As the title suggests, this thesis is about elliptic curve cryptography. An ellipticcurve, e, is a nonempty, smooth variety vfwhere deg f 3. Equivalently, an elliptic curve is a smooth projective curve of genus one over f equipped with a distinguished frational point, the identity element for the algebraic group law. Zeros of the modular parametrization of rational elliptic. The theories of elliptic curves, modular forms, and \l\functions are too vast to be covered in a single volume, and their proofs are outside the scope of the undergraduate curriculum.
Torsion subgroups of elliptic curves over number elds andrew v. Theses, see here so that math department senior theses can more easily benefit other undergraduate, we would like to exhibit more senior theses online while all theses are available through harvard university archives, it would be more convenient to have them online. My thesis work was in the drinfeld setting, which offers for function fields analogs of elliptic curves, modular forms, and modular curves. As everybody knows, the theory is a base of the proof by wiles through ribets work of fermats last theorem, it supplies a fast prime factorization algorithm cf. It turns out that it would help a lot if one relates elliptic curves to another class of objects. Iwasawa theory for elliptic curves and bsdp algant. This thesis revolves around a second 2descent, which we call a fourdescent. Eichler integrals play an integral part in the modular parametrizations of elliptic curves. An abstract of the thesis of oregon state university. We know additional information about the elliptic curve eby considering its corresponding modular curve, x 049.
Online senior thesis harvard mathematics department. Of particular note are two free packages, sage 275 and pari 202, each of which implements an extensive collection of elliptic curve algorithms. Tate curves 33 references 36 in this notes, we hope to go through basics of elliptic curves and modular curves in three steps. Elliptic curves, modular forms, and their lfunctions. This is a very well thought out and accessible introduction to modular forms that paves the way to further study and research. We give some generalizations to other modular curves, and to noncusp. Fishers part iii course on elliptic curves, given at cambridge university in lent term, 20. However, these questions about elliptic curves are still too hard to solve. In my thesis, i studied weierstrass points on drinfeld modular curves, and along the way, i proved.
An elliptic curve is a nonsingular projective curve, given by a cubic equation over an arbitrary eld. Sl2z also give moduli spaces of elliptic curves but include some extra. In this paper, we prove a conjecture of coleman, kaskel, and ribet about the qvalued points of the modular curve x 0n which map to torsion points on j 0n via the cuspidal embedding. Any such elliptic curve has the property that its hasseweil zeta function has an analytic continuation and satis. The connection between elliptic curves and modular forms is a very special case of a vast enterprise known as the langlands program. Your reportpresentation do not need to carry the projects all the way through.
The goal of this chapter will be to introduce elliptic curves, and present how we consider points on elliptic curves. The other equivalence class of cusps on x 049 is sent via this map to the. Introduction the thesis has the aim to study the eichlershimura construction associating elliptic curves to weight2 modular forms for. The book is full of examples and exercises of such appeal that a properly disposed rookie should go after nighon all of them. Accordingly, we provide a brief overview of elliptic curves, and functions. Implementation and evaluation of bsd elliptic curve.
Equations for modular curves university of auckland. Since the torsion subgroup of an elliptic curve over a number eld is a nite abelian group with. We will denote by z the usual ring of integers, by q its eld of. Introduction to elliptic curves to be able to consider the set of points of a curve cknot only over kbut over all extensionsofk. P rational points on certain elliptic modular curves. In their paper, arithmetic of weil curves, mazur and swinnertondyer prove that the number of fundamental critical points of the normalized weight two modular form associated with an elliptic curve is an upper bound on the analytic rank of the curve. Elliptic functions and equations of modular curves. This is not the same as a modular curve that happens to be an elliptic curve, something that could be called an elliptic modular curve. The rst part, consisting of chapters 14 is a purely mathematical introduction to elliptic curves. In this talk we describe recent computational work that investigates the modularity of elliptic curves over f. This thesis is aimed at master level students who have taken an introductory course in algebraic geometry and elliptic curves. In order to achieve this goal we first present some background on elliptic curves in the xy plane. Rational points on modular elliptic curves henri darmon. Then, relationships between hypergeometric functions over f p and modular forms are studied in chapter iv.