1  Course Title:  THEORY OF ELLIPTIC CURVES AND ITS APPLICALTIONS I 
2  Course Code:  MAT6111 
3  Type of Course:  Optional 
4  Level of Course:  Third Cycle 
5  Year of Study:  1 
6  Semester:  1 
7  ECTS Credits Allocated:  5 
8  Theoretical (hour/week):  3 
9  Practice (hour/week) :  0 
10  Laboratory (hour/week) :  0 
11  Prerequisites:  none 
12  Recommended optional programme components:  None 
13  Language:  Turkish 
14  Mode of Delivery:  Face to face 
15  Course Coordinator:  Prof. Dr. İSMAİL NACİ CANGÜL 
16  Course Lecturers:  Prof. Dr. Osman Bizim 
17  Contactinformation of the Course Coordinator: 
Uludağ Üniversitesi, FenEdebiyat Fakültesi Matematik Bölümü, Görükle BursaTÜRKİYE 0 224 294 17 57 / obizim@uludag.edu.tr 
18  Website:  
19  Objective of the Course:  The theory of elliptic curves brings important areas of mathematics such as abstract algebra, number theory and related fields. The aim of this course is to make the students get all connections among all these areas. The goal is to teach the elementary theory of elliptic curves. So students can bring new ideas the theory of elliptic curves and have the ability conduct original research and independent publication. 
20  Contribution of the Course to Professional Development 
21  Learning Outcomes: 

22  Course Content: 
Week  Theoretical  Practical 
1  Basic concepts of elliptic curves, the group law on the elliptic curves and proof of associativity.  
2  Other equations for elliptic curves, Legendre equation, cubic equations and quartic equations. The jinvariant of an elliptic curve and isomorphisms and endomorphisms of the curves.  
3  Torsion points of elliptic curves and division polynomials of an elliptic curve. Weil paring, TateLicthenbaum pairing.  
4  Elliptic curves over finite fields, counting the number of the points on these curves and the theorem of Hasse, The frobenius endomorphism, Schoof’s Algorithm.  
5  Determining the group structure of the points on the elliptic curves over finite fields and the group order. Some family of elliptic curves over finite fields, singular and supersingular curves.  
6  The discrete logarithm problem, general attacks on discerete logs, baby step, giant step, Pollard’s method, the PohlingHellman method.  
7  MOV attack, FreyRück attack and other attacks.  
8  The basic concepts in the elliptic curve cryptography, DiffieHellman key Exchange, MasseyOmura encryption.  
9  ElGamal public key encryption and the digital signatures, the digital signature algorithm.  
10  The elliptic curves over Q and the torsion subgroup and the LutzNagell theorem, the method of descent, the Mordell Weil theorem.  
11  2Selmer groups, ShafarevichTate grous, a nontrivial ShafarevichTate groups, Galois cohomology.  
12  The elliptic curves over C, doubly periodic functions, tori are elliptic curves, the arithmetic and geometric mean.  
13  Division poliynomials and the torsion subgroups, Doud’s method, complex multiplication and numerical examples, the integrality of jinvariants.  
14  Hyperelliptic curves, Cantor’s algorithm, zeta functions, Fermat’s last theorem, sketch of Wiles’s proof. 
23  Textbooks, References and/or Other Materials: 
[1] Rational Points on Elliptic Curves, J. H. Silverman ve J. Tate, [2]The Arithmetic of Elliptic Curves, J. H. Silverman, [3]Elliptic Curves, L. C. Washington. [4] Introduction to Elliptic Curves and Modular Forms, N. Koblitz. 
24  Assesment 
TERM LEARNING ACTIVITIES  NUMBER  PERCENT 
Midterm Exam  0  0 
Quiz  0  0 
Homeworks, Performances  0  0 
Final Exam  1  100 
Total  1  100 
Contribution of Term (Year) Learning Activities to Success Grade  0  
Contribution of Final Exam to Success Grade  100  
Total  100  
Measurement and Evaluation Techniques Used in the Course  
Information 
25  ECTS / WORK LOAD TABLE 
Activites  NUMBER  TIME [Hour]  Total WorkLoad [Hour] 
Theoretical  14  3  42 
Practicals/Labs  0  0  0 
Self Study and Preparation  14  5  70 
Homeworks, Performances  0  0  0 
Projects  0  0  0 
Field Studies  0  0  0 
Midtermexams  0  0  0 
Others  14  5  70 
Final Exams  1  13  13 
Total WorkLoad  195  
Total workload/ 30 hr  6,5  
ECTS Credit of the Course  6,5 
26  CONTRIBUTION OF LEARNING OUTCOMES TO PROGRAMME QUALIFICATIONS  

LO: Learning Objectives  PQ: Program Qualifications 
Contribution Level:  1 Very Low  2 Low  3 Medium  4 High  5 Very High 