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Algebraic Geometry
| Credits |
8 credit points |
| Instructors |
Edixhoven, S.J. (Universiteit Leiden), Taelman, L. (Universiteit Leiden) |
| E-mail |
edix@math.leidenuniv.nl, Lenny@math.leidenuniv.nl |
| Aim |
To become acquainted with basic techniques of algebraic geometry, through the study of the proof of the Riemann Hypothesis for curves over finite fields. |
| Description |
Riemann'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 (Wikipedia) 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. |
| Organization |
Two 45 minute lectures and one 45 minute problem session, weekly. |
| Examination |
Each week, students hand in solutions to exercises that are given on the website of this course. The solutions are graded by one or two teaching assistants. The final grade is exclusively based on the results obtained for the weekly homework assignments. The solutions of some problems will be discussed, before the end of June, between the students and the instructors before the final grade is given (the list of these problems will be given in due time). |
| Literature |
R. Hartshorne. Algebraic Geometry. Springer GTM 52. |
| Prerequisites |
The standard undergraduate algebra courses on groups, rings and fields (for more details see the three algebra syllabi (in Dutch) available here)
, and some basic topology. No prior knowledge of algebraic geometry is necessary.
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| Remarks |
More information |
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