# Algebraic curves and the Weil conjectures

## Course at HU Berlin, Winter Term 16/17

### Content

Let X be a projective variety over a finite field $$k$$, in particular it is given by finitely many homogenous polynomials $$F_1,\ldots, F_r$$ in $$n$$ variables and with coefficients in $$k$$. Denote by $$\bar{k}$$ the algebraic closure of $$k$$. Then the $$\bar{k}$$-rational points of $$X$$ are the common solutions of the equations $$F_i=0$$ in $$\mathbb{P}^{n-1}(\bar{k})=(\bar{k}^n\setminus\{0\})/ \bar{k}^\times$$. If $$L/k$$ is an extension of finite fields, we say that a $$\bar{k}$$-rational point $$x$$ of $$X$$ is $$L$$-rational if $$x$$ has a representative in $$L^n\setminus\{0\}$$. 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 $$F_i=0$$ 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 $$\mathbb{F}_q$$, then one gets the following estimation of the number of $$\mathbb{F}_q$$-rational points of $$C$$: $$1+q-2g\sqrt{q}\le N(C(\mathbb{F}_q)))\le 1+q+2g\sqrt{q}$$.

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 and Serre duality for curves, a bit of intersection theory on surfaces and the Hodge index theorem.

Prerequisites: Basic knowledge in algebra, commutative algebra and algebraic geometry.

### Data

Lecturer Time Room
Lecture Kay Rülling Fr 11-13 Seminarraum 1304 (RUD26)
Exercise Kay Rülling Fr 13-15 (every second week) Seminarraum 1304 (RUD26)

The first lecture is on October 21, 2016. The first exercise session is on October 28.
If you have any questions concerning the lectures or the exercises feel free to contact me.

### Exercises

Every second Friday there will be a new exercise sheet on this web page, the solutions are discussed in the exercise session on the Friday of the following week.

1. Exercise sheet October 21, 2016
2. Exercise sheet November 4, 2016
3. Exercise sheet November 18, 2016 (ver 3, 29.11., Danke an Claudius)
4. Exercise sheet December 2, 2016 (This exercise will be discussed on December 16. For Exercise 4.3 and 4.4 you will need the material discussed in the lecture on December 7.)
5. Exercise sheet Christmas 2016 (This exercise will be discussed on January 6.)
6. Exercise sheet January 13, 2017
7. Exercise sheet January 27, 2017

### Exam

The exam was on Friday, February 17, 11:00-14:00, in room 1.304, RUD26.
The second exam will be on Friday, March 17, 11:00-14:00, in room 1.303, RUD26.

You can take 4 one-sided (or 2 two-sided) hand written DIN A4 paper sheets to the exam. Besides this and a pen you are not allowed to bring any other material to the exam.

### References

Here is a list of references for the course:

1. B. Kahn, Fonctions zêta et $$L$$ de variétés et de motifs . Available at https://arxiv.org/abs/1512.09250.
2. S. Raskin, Weil Conjectures for curves (follwing lectures by Beilinson) . Available at http://www.math.uchicago.edu/~mitya/beilinson/SamREU07.pdf.
3. R. Hartshorne, Algebraic Geometry , Graduate Texts in Mathematics, vol 57, Springer.
4. J.-P. Serre, Local Fields , Graduate Texts in Mathematics, vol 67, Springer.
5. W. Fulton, Intersection Theory , Springer.