1  Course Title:  THEORY OF ELLIPTIC CURVES AND ITS APPLICATIONS II 
2  Course Code:  MAT6112 
3  Type of Course:  Optional 
4  Level of Course:  Third Cycle 
5  Year of Study:  1 
6  Semester:  2 
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. OSMAN BİZİM 
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  Algebraic varieties and maps between varieties, algebraic curves and maps between them.  
2  The RiemannRoch theorem, the geometry of elliptic curves, Weiestrass’s equations, isogenies, dual isogenies.  
3  Endomorphism rings and the automorphism groups, the formal group of an elliptic curve, formal logarithm.  
4  Formal groups in characteristic p, elliptic curves over finite fields, the Weil conjecture, calculating the Hasse invariant.  
5  Elliptic curves over C, construction of elliptic functions, analytic and algebraic maps.  
6  Elliptic curves over local fields, minimal Weierstrass equations, reductions and points of finite order.  
7  Elliptic curves over global fields, heights on elliptic curves, the rank of an elliptic curve.  
8  Siegel’s theorem, Shafarevich’s theorem and Roth’s theorem.  
9  Computing the MordellWeil group an examples.  
10  The Selmer and ShafarevichTate groups.  
11  The twists of elliptic curves and applications over some family of elliptic curves.  
12  Algorithmic aspects of elliptic curves and Lenstra’s elliptic curve algorithm.  
13  Elliptic curves in characteristics 2 and 3.  
14  Cohomology of finite groups and Galois cohomology, non abelian cohomology. 
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 