Faculty of Science

Elliptic Curves
Provider: Faculty of Science

Activity no.: 5565-23-07-31
	Enrollment deadline: 06/02/2023

Place

Department of Mathematical Sciences

Date and time

06.02.2023, at: 08:00 - 14.04.2023, at: 16:00

Regular seats

ECTS credits

7.50

Contact person

Nina Weisse E-mail address: weisse@math.ku.dk

Enrolment Handling/Course Organiser

Fabien Pazuki E-mail address: fpazuki@math.ku.dk

Written language

English

Teaching language

English

Semester/Block

Block 3

Scheme group

Exam form

Continuous assessment

Exam details

Written examination, 3 hours under invigilation Two written assignments count each 20%. A final written exam counts the remaining 60% of the grade.

Exam aids

Writen aids allowed

Grading scale

7 point grading scale. For PhD students: Passed / Not Passed

Criteria for exam assessment

The student must in a satisfactory way demonstrate that he/she has mastered the learning outcome of the course.

Exam re-examination

Re-exam: 30 minutes oral exam without preparation time, several internal examiners, all written aids allowed, counting for 100% of the grade.

Course workload

Course workload category	Hours
Lectures	42.00
Preparation	147.00
Exercises	14.00
Exam	3.00

Sum	206.00

Content

The aim of this course is to discover the beautiful theory of elliptic curves. Elliptic curves are objects at the crossroads between geometry, analysis, algebra and number theory. They constitute one of the key ingredient in the proof of Fermat’s Last Theorem for instance, and famous open conjectures -for example the Birch and Swinnerton-Dyer conjecture- focus on these special curves. Studying compact Riemann surfaces, lattice theory and periodic functions, rational points and diophantine problems, projective and affine geometry of curves, schemes, higher Galois theory, modular forms and L functions, abelian varieties, local fields, global fields, finite fields, modern cryptography, each time these curves show up at a central place.

As these objects really appear as a corner stone in the modern mathematical landscape, we offer a course presenting in details their various definitions and basic properties and focus on some modern applications.

Learning outcome

Knowledge: The student should be familiar with the main results of the topics of the course.

Skills: At the end of the course the student is expected to be able to follow and reproduce arguments at a high level corresponding to the contents of the course.

Competences: The student should be able to apply the theory to solve problems of moderate difficulty within the topics of the course.

Literature

The Arithmetic of Elliptic Curves, GTM 106, Springer, by Joseph Silverman.
Rational points on elliptic curves, UTM, Springer, by Joseph Silverman and John Tate.

Teaching and learning methods

6 hours of lectures and 2 hours of tutorials each week for 7 weeks.

Remarks

Elliptic Curves definitely fits in the circle of ideas presented in these other courses: Algebra 3, Algebraic Number Theory, Analytic Number Theory and Algebraic Geometry. Nevertheless, these courses are not requirements, the course will be self-contained.

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