The Weil conjectures are famous conjectures formulated by André Weil in 1949. They have driven a large part of the development of algebraic geometry in the following decades, until they were proved by Pierre Deligne in the 1970s. The proof uses, in an essential way, the language of schemes and the theory of étale cohomology developed by Alexander Grothendieck and his school.

The Weil conjectures (now properly speaking the Theorem of Deligne, but the old name is still in use) describe in an elegant way an astonishing regularity of the numbers of solutions of systems of polynomial equations over finite fields $\mathbb F_q$, where $q$ runs through all powers of a fixed prime $p$. Part of the fascination is caused by the appearance of so-called zeta functions which are defined in closed analogy to the famous Riemann zeta function. The most difficult part of the Weil conjectures asserts that for these functions the analogue of the Riemann hypothesis holds true.

For algebraic curves, the situation is much simpler than in the general case. In this situation the conjecture habe been formulated already in 1924 by Emil Artin, and had been proved by Weil. Understanding this theorem and its proof by Bombieri is the topic of the seminar.

Prerequisites: Good knowledge of Lineare Algebra 1, 2 and Algebra. Knowledge of commutative algebra is helpful, similarly as the course on Algebraic Geometry

Credits: The seminar is a joint Bachelor/Master seminar. For a successful Bachelor seminar talk, you earn 6 ECTS points. For a successful Master seminar talk you earn 6 or 9 ECTS points (depending on the version of the regulations (Prüfungsordnung) which appplies to you). Several of the talks can serve as the basis of a Bachelor’s thesis; if you are interested, then it would be useful to also learn some commutative algebra and algebraic geometry. Most of the talks in the last third of the seminar have the level of Master seminar talks.

Seminar program: (in German —- I will be happy to assist if that is a problem; all the references are in English) pdf

Date: Tuesday, 2-4pm, S-U-3.01, First meeting: Oct., 16

Contact/further information: Ulrich Görtz, ulrich.goertz@uni-due.de

References

E. Bombieri, Counting points on curves over finite fields, Séminaire Bourbaki, Exposé no. 430 (1972/73).

U. Görtz, T. Wedhorn, Algebraic Geometry I. Schemes, Vieweg+Teubner, 2010.

S. H. Hansen, Rational Points on Curves over Finite Fields, Lect. Notes Ser., Aarhus Univ. Mat. Institute, 1995.

R. Hartshorne, Algebraic Geometry, Springer Graduate Texts in Mathematics 52, 1977.

D. Lorenzini, An invitation to Arithmetic Geometry, Grad. Studies in Math. 9, Amer. Math. Soc., 1996.

H. Stichenoth, Algebraic Function Fields and Codes, Springer Graduate Texts in Math. 254, 2nd ed., 2009.

Talks

 0 16.10. Introduction Ulrich Görtz 1 23.10. Discrete valuation rings 1 Ulrich Görtz 2 30.10. Discrete valuation rings 2 Felix Gora 3 6.11. Affine varieties Supplementary notes Ulrich Görtz 4 13.11. Projective varieties Tristan Kurz 5 20.11. Algebraic curves Felix Gora 6 27.11. Plane curves Ulrich Görtz 7 4.12. Divisors Mike Farniok 8 11.12. The zeta function of a curve Alexander Graf 9 18.12. Linear equivalence Gregor Kremers 10 8.1. The theorem of Riemann-Roch Michael Ingelski 11 15.1. Rationality of the zeta function and functional equation Felix Gora 12 22.1. The theorem of Bombieri Ahmed Elashry 13 29.1. Proof of the Riemann hypothesis for zeta functions curves Ulrich Görtz