Arithmetic of Plane CurvesHHU Düsseldorf, Wintersemester 2026/27 Teacher: Jun.-Prof. Dr. Marta Pieropan
Lecture room: 25.22.00.72
ContentPlane curves are subsets of the projective plane defined by homogeneous polynomials in three variables. In this course you will study the set of solutions of those homogeneous polynomials over fields that are not algebraically closed and learn about their properties. Learning objectives:
In degree 2, you will learn about plane conics, their classification un to isomorphism and their connection to quaternion algebras. In degree 3, you will learn about elliptic curves, their group of points, the Mordell-Weil Theorem over the rational numbers, and the structure of the group of torsion points. In degree at least 4, you will learn about Fermat's descent and its application to prove some cases of Fermat's Last Theorem. Time permitting, we will get a glimpse about the theory behind Wile's proof of Fermat's Last Theorem.
Exercise classesThe exercise sessions take place on Fridays at 14:30 in Room 25.22.00.72. The exercise sheets are available on ILIAS. ExamAdmittance to the exam is based on oral presentations of homework during the exercise sessions as follows: at least one successful presentation between October 27 and November 20, at least one successful presentation between November 24 and December 18, at least one successful presentation between January 5 and February 5. The presentations are greaded pass/fail. The exam is an oral exam. LiteratureGille and Szamueli, Central simple algebras and Galois cohomology, Cambridge University Press 2017 ULB link Knapp, Elliptic curves, Princeton University Press 2018. ULB link | ||