Home > Algebraic Geometry
DescriptionRiemann's zeta function has a natural generalisation to zeta functions associated to finitely generated (commutative) rings, and more generally, to schemes of finite type. For nonsingular projective curves over finite fields the Riemann hypothesis has been proven by Hasse (elliptic curves) and Weil (arbitrary genus, 1940's). The case of higher dimensional varieties over finite fields was proved by Deligne (1974), building on the work of Grothendieck. In this course we will treat the case of curves over finite fields, using intersection theory on surfaces. The course will start with some explicit examples of zeta functions, including Riemann's and those of curves over finite fields. Then slowly we will develop those techniques necessary to treat Weil's proof, from Hartshorne's book Algebraic Geometry. Finally, we will present Weil's proof. Our goal is to provide a good overview of Weil's proof. Obviously, it is not desirable nor possible to treat all of Hartshorne's book.
This course is part of the Dutch Master Program in Mathematics. They maintain a page for this course here.
|The lectures will be given by|
|Bas Edixhoven||edix (at) math (dot) leidenuniv (dot) nl|
|Lenny Taelman||taelman (at) math (dot) leidenuniv (dot) nl|
|The problem sessions are taught by:|
|Sylvain Brochard||brochard (at) math (dot) leidenuniv (dot) nl|
|Arjen Stolk||astolk (at) math (dot) leidenuniv (dot) nl|
HomeworkEvery week during the lecture some homework will be assigned. This homework has to be handed in the next week before the start of the lecture. While cooperation is encouraged when thinking about the problems, we require students to write up and hand in the homework individually. Copying from others is not cooperation and will not be condoned. Homework exercises will be corrected and marked pass or fail. Late homework is not accepted.
ExamAt the end of the course, each student is required to take an oral exam. These oral exams will take place at Leiden University. A list of possible dates will be given in due time. The deadline for the oral exams is June 30th. On the exam, students will be questioned about the homework exercises. In order to be admitted into the exam, a student has to pass at least 7 of the homework sets. Questions may be asked about all the homework exercises, not just those for which the student has received a pass.
Oral examsThe oral exams will take place at the start of June. Contact the instructors if you want to take part in the exam. The exams will be held in room 236 of the Snellius building (in Leiden). This is prof. Edixhoven's room.
Tuesday, June 2
|11:00||Matthijs van Duin|
|14:00||Milan de Nijs|
|16:00||Johannes van der Horst|
Wednesday, June 3
|13:30||Monique van Beek|
Thursday, June 4
NotesThe course is based on material from Hartshorne's book Algebraic Geometry. We will pick and choose from the book and at times will cover material which is not presented in it. For the convenience of the students, we therefore want to compile notes of the course. We are looking for student volunteers to make notes during the class and typeset them.
The notes are now available.
ScheduleThere is an extra problem session on Tuesday May 26. It will be held in Leiden, in the Snellius building (where the math department is) in room 412, from 13:00 to 16:00.
|Feb. 3||Introduction: zeta functions, Riemann hypothesis||Set 1|
|Feb. 10||Introduction: examples of plane curves, genus, Hasse-Weil bound||Set 2|
|Feb. 17||Affine algebraic varieties ([AG. I.1])||Set 3|
|Feb. 24||Projective space||Set 4|
|Mar. 3||Geometry in projective space: examples||Set 5|
|Mar. 10||Regular functions, category of abstract varieties||Set 6|
|Mar. 17||Affine varieties, product varieties, presentation of varieties.||Set 7|
|Mar. 24||No course|
|Mar. 31||Gluing, presentation, smoothness and rational functions.||Set 8|
|Apr. 7||The Riemann-Roch theorem||Set 9|
|Apr. 14||Tangent spaces, 1-forms, residues||Set 10|
|Apr. 21||Serre duality, Varieties over finite fields, rationality of zeta.||Set 11|
|Apr. 28||Rationality and functional equation of zeta.||Set 12|
|May 5||No course|
|May 12||Curves on surfaces.||Set 13|
|May 19||The Hasse-Weil inequality.||None|