M8195 Number theory seminar

Faculty of Science
Autumn 2022
Extent and Intensity
0/2/0. 2 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Type of Completion: z (credit).
Teacher(s)
Mgr. Pavel Francírek, Ph.D. (lecturer)
prof. RNDr. Radan Kučera, DSc. (lecturer)
Guaranteed by
prof. RNDr. Radan Kučera, DSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Thu 10:00–11:50 M6,01011
Prerequisites
The knowledge of Galois theory and basic informations concerning elliptic curves in the range of M8190 Algorithms of Number Theory will be useful. However, even the students who did not attend the lecture M8190 can acquire the mentioned knowledge of elliptic curves on their own easily.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
Understanding the basic theory of elliptic curves.
Learning outcomes
At the end of this course, students should be able to:
define basic notions of the studied theory;
explain learned theoretical results;
apply learned methods to concrete exercises.
Syllabus
  • We shall follow Washington's book at a suitable speed, so it is not clear if we cover all the following plan during this semester:
  • Weierstrass Equation;
  • The Group Law;
  • Other Equations for Elliptic Curves;
  • Other Coordinate;
  • 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;
  • Determining the Group Order;
  • Supersingular Curves.
Literature
  • WASHINGTON, Lawrence C. Elliptic curves : number theory and cryptography. Second edition. Boca Raton: Chapmann & Hall/CRC, 2008, xviii, 513. ISBN 9781420071467. info
Bookmarks
https://is.muni.cz/ln/tag/PříF:M8195!
Teaching methods
In this semester, we will meet in person if the current epidemiological situation allows it. Home preparation will play an important role.
Assessment methods
Credit will be given in the case of active work in seminars: the study of the mentioned book during the term, regular delivering of solved homeworks. To get the credit, students must deliver at least 50% of homeworks.
Language of instruction
Czech
Further Comments
Study Materials
The course is taught each semester.
The course is also listed under the following terms Autumn 2007 - for the purpose of the accreditation, Spring 2008 - for the purpose of the accreditation, Spring 2011 - only for the accreditation, Autumn 2010 - only for the accreditation, Spring 2005, Autumn 2005, Spring 2006, Autumn 2006, Spring 2007, Autumn 2007, Spring 2008, Autumn 2008, Spring 2009, Autumn 2009, Spring 2010, Autumn 2010, Spring 2011, Autumn 2011, Spring 2012, Autumn 2011 - acreditation, spring 2012 - acreditation, Autumn 2012, Spring 2013, Autumn 2013, Spring 2014, Autumn 2014, Spring 2015, Autumn 2015, Spring 2016, Autumn 2016, Spring 2017, autumn 2017, spring 2018, Autumn 2018, Spring 2019, Autumn 2019, Spring 2020, Autumn 2020, Spring 2021, autumn 2021, Spring 2022, Spring 2023, Autumn 2023, Spring 2024, Autumn 2024, Spring 2025.
  • Enrolment Statistics (Autumn 2022, recent)
  • Permalink: https://is.muni.cz/course/sci/autumn2022/M8195