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Information
| Unit | FACULTY OF SCIENCE AND LETTERS |
| MATHEMATICS PR. | |
| Code | MT493 |
| Name | Eliptic Curves |
| Term | 2019-2020 Academic Year |
| Semester | 7. Semester |
| Duration (T+A) | 3-0 (T-A) (17 Week) |
| ECTS | 5 ECTS |
| National Credit | 3 National Credit |
| Teaching Language | Türkçe |
| Level | Belirsiz |
| Type | Normal |
| Label | E Elective |
| Mode of study | Yüz Yüze Öğretim |
| Catalog Information Coordinator | Prof. Dr. YILMAZ DURĞUN |
| Course Instructor |
Prof. Dr. YILMAZ DURĞUN
(Güz)
(A Group)
(Ins. in Charge)
|
Course Goal / Objective
The aim of this course is to introduce the theory of elliptic curves and their use, and to help students understand the connection between important areas of mathematics such as number theory and cryptography. It is aimed to create the necessary infrastructure for the students to work on elliptic curves.
Course Content
Weierstrass Equations, The Group Law, Projective Space and the Point at Infinity, Proof of Associativity, The Theorems of Pappus and Pascal ,Other Equations for Elliptic Curves, Legendre Equation, Cubic Equations, Quartic Equations, Intersection of Two Quadratic Surfaces, Projective Coordinates, Jacobian Coordinates, Edwards Coordinates, The j-invariant, Elliptic Curves in Characteristic 2, Endomorphisms , Singular Curves, Elliptic Curves mod n, Torsion Points , Division Polynomials, The Weil Pairing ,The Tate-Lichtenbaum Pairing, Elliptic Curves over Finite Fields,The Frobenius Endomorphism, Determining the Group Order, Subfield Curves.
Course Precondition
Resources
Notes
Course Learning Outcomes
| Order | Course Learning Outcomes |
|---|---|
| LO01 | Use elliptic curves to solve some problems of mathematics. |
| LO02 | Learn the group structure of the points on the elliptic curves |
| LO03 | Learn the j-invariant of an elliptic curve and isomorphisms and endomorphisms of the curves; |
| LO04 | Learn the singular curves and determine group law of singular curves. |
| LO05 | Learn the torsion points of an elliptic curve and learn division polynomials of an elliptic curve. |
| LO06 | Learn elliptic curves over finite fields and counts the number of the points on these curves |
| LO07 | Learn the elliptic curves over Q and the torsion subgroup and the Lutz-Nagell theorem |
Relation with Program Learning Outcome
| Order | Type | Program Learning Outcomes | Level |
|---|---|---|---|
| PLO01 | - | Comprehend the ability to prove the mathematical knowledge gained in secondary education on the basis of theoretical basis. | 5 |
| PLO02 | - | Understands importance of basic consepts of Algebra, Analaysis and Topology. | 5 |
| PLO03 | - | Mathematical reasoning demonstrates the ability to develop and write mathematical proofs by gaining maturity. | 3 |
| PLO04 | - | Demonstrate the ability to express the basic theories of mathematics both correctly. | 1 |
| PLO05 | - | Understands the relationship between the different fields of mathematics and its relation to other disciplines. | 3 |
| PLO06 | - | Comprehends the ability to understand the relationships between the objects in the most understandable way while creating a model for any problem. | 1 |
| PLO07 | - | Comprehend and explain mathematical models such as formulas, graphs, tables and schema. | 1 |
| PLO08 | - | Demonstrate the ability to mathematically rearrange, analyze, and model the problems they encounter. | 2 |
| PLO09 | - | Comprehends at least one of the computer programming languages. | 2 |
| PLO10 | - | Demonstrate the ability to use scientific methods and appropriate technologies effectively in problem solving. | 1 |
| PLO11 | - | Understands sufficient knowledge of foreign language to be able to understand Mathematical concepts and communicate with other mathematicians | 1 |
| PLO12 | - | In addition to their professional development, they demonstrate their ability to continuously improve themselves by identifying their educational needs in scientific, cultural, artistic and social areas in line with their interests and abilities. | 3 |
| PLO13 | - | Understands the programming techniques and shows the ability to do programming. | 1 |
| PLO14 | - | Demonstrates the ability to study mathematics both independently and as a group. | 3 |
| PLO15 | - | Demonstrate an awareness of the universal and social impacts and legal consequences of mathematical applications in the field of study. | |
| PLO16 | - | Demonstrate the ability to select, use and develop effectively for contemporary mathematical applications. | |
| PLO17 | - | It has ability of lifelong learning awareness, access to information, monitoring developments in science and technology and self-renewal ability. | |
| PLO18 | - | Gains the ability to use information technologies effectively for contemporary mathematical applications. | |
| PLO19 | - | Gains the ability to design, conduct experiments, field work, data collection, analysis, archiving, text solving and / or interpretation according to mathematics fields. | |
| PLO20 | - | Gains the consciousness of prefesional ethics and responsibility. |
Week Plan
| Week | Topic | Preparation | Methods |
|---|---|---|---|
| 1 | Weierstrass Equation | Read the relevant sections of the course notes | |
| 2 | Use elliptic curves to solve some problems of mathematics. | Read the relevant sections of the course notes | |
| 3 | The group law on the elliptic curves and proof of associativity. | Read the relevant sections of the course notes | |
| 4 | Other equations for elliptic curves, Legendre equation, cubic equations and quartic equations | Read the relevant sections of the course notes | |
| 5 | The j-invariant of an elliptic curve and isomorphisms and endomorphisms of the curves. | Read the relevant sections of the course notes | |
| 6 | The singular curves and determining group law of singular curves | Read the relevant sections of the course notes | |
| 7 | Torsion points of elliptic curves and division polynomials of an elliptic curves | Read the relevant sections of the course notes | |
| 8 | Mid-Term Exam | Read the relevant sections of the course notes | |
| 9 | Elliptic curves over finite fields, counting the number of the points on these curves and the theorem of Hasse | Read the relevant sections of the course notes | |
| 10 | Determining the group structure of the points on the elliptic curves over finite fields and the group order. | Read the relevant sections of the course notes | |
| 11 | Some family of elliptic curves over finite fields. | Read the relevant sections of the course notes | |
| 12 | The elliptic curves over Q and the torsion subgroup and the Lutz-Nagell theorem. | Read the relevant sections of the course notes | |
| 13 | The method of descent of Fermat and the Mordell-Weil theorem. | Read the relevant sections of the course notes | |
| 14 | The elliptic curves over C. | Read the relevant sections of the course notes | |
| 15 | Overview on Fermat s last theorem | Read the relevant sections of the course notes | |
| 16 | Term Exams | Read the relevant sections of the course notes | |
| 17 | Term Exams | Read the relevant sections of the course notes |
Assessment (Exam) Methods and Criteria
| Assessment Type | Midterm / Year Impact | End of Term / End of Year Impact |
|---|---|---|
| 1. Midterm Exam | 70 | 28 |
| 1. Homework | 30 | 12 |
| General Assessment | ||
| Midterm / Year Total | 100 | 40 |
| 1. Final Exam | - | 60 |
| Grand Total | - | 100 |