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Technische Universität München

Algebraic curves and the Weil Conjectures

Time and venue

Lecture: Di 10:15 - 11:45, in 02.06.020, Rechnerraum (5606.02.020)

Exercises: Mi 8:30 - 10:00 (every second week)

Content

Let X be a projective variety over a finite field k, in particular it is given by finitely many homogenous polynomials F1,…, Fr in n variables and with coefficients in k. Denote by ka the algebraic closure of k. Then the ka-rational points of X are the common solutions of the equations Fi=0 in Pn-1(ka). If L/k is an extension of finite fields, we say that a ka-rational point x of X is L-rational if it has a representative in Ln\{0}. Denote by N(X(L)) the number of L-rational points in X. It is an interesting and in general very hard question to compute N(X(L)) or to decide whether it is non-zero or not, i.e., whether the equations Fi=0 have a common non-trivial solution in L. The data of all the numbers N(X(L)), for L running through all finite extensions of k is encoded in the zeta function of X. It was conjecture by Weil in 1949 and proved by him in the curve case and later in general by Deligne in 1974, that the zeta function of a smooth projective variety defined over a finite field has certain nice properties which provide a very deep and systematic machinery to analyze the number of rational points of X. For example, although the zeta function is a priori defined as a power series it turns out that it is in fact determined by certain polynomials whose roots are algebraic integers and there is a surprisingly simple and general formula for the absolute values of this roots. (The analog in number theory of this last formula is the famous and until today unproven Riemann Hypothesis.) As an application of Weil's theorem on curves one obtains for example that if C is a smooth projective and geometrically connected curve of genus g over the finite field with q elements Fq, then one gets the following estimation of the number of Fq-rational points of C: 1 + q - 2g√q ≤ N(C(Fq))) ≤ 1 + q + 2g√q.

In the course we will go through the proof of the Weil conjectures for smooth projective curves over a finite field. To this end we will discuss

  • divisors and line bundles on varieties
  • cohomology of quasi-coherent sheaves
  • the Riemann-Roch theorem 
  • Serre duality for curves
  • a bit of intersection theory on surfaces
  • the Hodge index theorem

There won't be enough time to discuss all this in detail and we will use some results without proof.

Prerequisites

Basic knowledge of commutative algebra (as in Atiyah-Macdonald) and algebraic geometry, in particular the language of sheaves and schemes (as, e.g., in Hartshorne's book, II, 1-5). Some familiarity with the cohomology of sheaves on schemes might be useful, is however not required.

Exam

There will be an oral exam on 23.08.2019 (Friday) during 10 - 13. Each oral exam will last about 30 minutes. The students will receive the time and place of their examination by email.

Exercises

  1. Exercise Sheet  (will be discussed on Wed, May 8)
  2. Exercise Sheet(v2) (will be discussed on Wed, May 15)
  3. Exercise Sheet (will be discussed on Wed, May 29)
  4. Exercise Sheet (will be discussed on Wed, June 12)
  5. Exercise Sheet (will be discussed on Wed, July 3)
  6. Exercise Sheet     (v2, will be discussed on Wed, July 24)

Lecture Notes

Here you find the handwritten notes for the course (they are numbered as the sections in the course).

  1. Zeta functions of arithmetic schemes
  2. The Weil Conjectures
  3. Properties of varieties
  4. Invertible Sheaves and Cartier Divisors
  5. Weil Divisors and the first Chern class of invertible Sheaves
  6. Chow groups and proper pushforward
  7. Riemann-Roch (curves)
  8. Rationality of the Zeta Function and the Functional Equation
  9. Intersecting Divisors
  10. Riemann-Roch for Surfaces and Hodge Index Theorem
  11. Proof of the Riemann Hypothesis for curves over finite fields

Literature

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