Uludağ Üniversitesi, Fen-Edebiyat Fakültesi Matematik Bölümü, Görükle Bursa-TÜRKİYE 0 224 294 17 50 / obizim@uludag.edu.tr
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Website:
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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 know the basic theory of elliptic curves.
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Contribution of the Course to Professional Development
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Learning Outcomes:
1
Use elliptic curves to solve some problems of mathematics. ;
2
Learn the group structure of the points on the elliptic curves.;
3
Learn the j-invariant of an elliptic curve and isomorphisms and endomorphisms of the curves;
4
Learn the singular curves and determine group law of singular curves. ;
5
Learn the torsion points of an elliptic curve and learn division polynomials of an elliptic curve. ;
6
Learn elliptic curves over finite fields and counts the number of the points on these curves.;
7
Give some results about the numbers of the points of the elliptic curves over finite fields;
8
Learn the elliptic curves over Q and the torsion subgroup and the Lutz-Nagell theorem.;
9
Learn the method of descent of Fermat and the Mordell-Weil theorem.;
10
Learn the elliptic curves over C.;
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Course Content:
Week
Theoretical
Practical
1
Basic concepts on groups, rings and fields.
2
Use elliptic curves to solve some problems of mathematics.
3
The group law on the elliptic curves and proof of associativity.
4
Other equations for elliptic curves, Legendre equation, cubic equations and quartic equations.
5
The j-invariant of an elliptic curve and isomorphisms and endomorphisms of the curves.
6
The singular curves and determining group law of singular curves.
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Torsion points of elliptic curves and division polynomials of an elliptic curve.
8
Elliptic curves over finite fields, counting the number of the points on these curves and the theorem of Hasse.
9
Determining the group structure of the points on the elliptic curves over finite fields and the group order.
10
Some family of elliptic curves over finite fields.
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The elliptic curves over Q and the torsion subgroup and the Lutz-Nagell theorem.
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The method of descent of Fermat and the Mordell-Weil theorem.
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The elliptic curves over C.
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Overview on Fermat’s last theorem.
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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.
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Assesment
TERM LEARNING ACTIVITIES
NUMBER
PERCENT
Midterm Exam
1
40
Quiz
0
0
Homeworks, Performances
0
0
Final Exam
1
60
Total
2
100
Contribution of Term (Year) Learning Activities to Success Grade
40
Contribution of Final Exam to Success Grade
60
Total
100
Measurement and Evaluation Techniques Used in the Course
Information
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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
1
15
15
Others
14
1
14
Final Exams
1
9
9
Total WorkLoad
150
Total workload/ 30 hr
5
ECTS Credit of the Course
5
26
CONTRIBUTION OF LEARNING OUTCOMES TO PROGRAMME QUALIFICATIONS