NMAK14034U Heights and Diophantine problems

Volume 2015/2016
Education

MSc Programme in Mathematics

Content

The aim of this course is to discover the notion of height in
arithmetic (i.e. the height of an algebraic number) and in arithmetic
geometry (i.e. the height of a point on a curve for example) and the
spectacular results that it enables to prove. We will focus on the
arithmetic of projective curves, elliptic curves and abelian varieties,
the Mordell conjecture and the Mordell-Weil theorem to be more specific.

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, abstract 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. In 
particular calculate the height of a point on a curve, decide if a 
rational point on an abelian variety is a torsion point, etc.

Algebra 3 or similar is an advantage
6 hours of lectures and 2 hours of tutorials each week for 7 weeks.
  • Category
  • Hours
  • Exam
  • 2
  • Exercises
  • 14
  • Lectures
  • 42
  • Preparation
  • 148
  • Total
  • 206
Credit
7,5 ECTS
Type of assessment
Written examination, 2 hours under invigilation
Continuous assessment
Two written assignments count each 10%. A final written exam counts the remaining 80% of the grade
Aid
All aids allowed

NB: If the exam is held at the ITX, the ITX will provide you a computer. Private computer, tablet or mobile phone CANNOT be brought along to the exam. Books and notes should be brought on paper or saved on a USB key.

Marking scale
7-point grading scale
Censorship form
No external censorship
One internal examiner
Re-exam

Resubmission of failed assignments and a 30 minutes oral exam with several internal examiners. The assignments must be resubmitted no later than two weeks before the beginning of the re-exam week.

Criteria for exam assesment

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