This is the Homepage for the Vertiefungsvorlesung Algebraische Geometrie II, given in Mainz during the Wintersemester of 2015-2016.

**Practical information**

**Teacher:** Ariyan Javanpeykar

**Course: **Monday, 14-16h, Wednesday 14h-16h

**Room: **04-230

**Course material and description**

The spoken language during the course is **German**, but all the references we will use are in English.

Preliminaries for this course are a basic knowledge of Algebraic Geometry (varieties over fields, schemes, sheaves, etc.). For instance, both the course Algebraische Geometrie I given by Stefan Müller-Stach during the Sommersemester 2014-2015 and the course of Ronan Terpereau on Algebraic Groups given during the Sommersemester 2014-2015 are more than sufficient.

During the semester we will assign homework precisely **seven** times. The "exam" is given in the form of a last (**eighth**) homework sheet. You can find all the homework sheets below.

We will follow these __notes on Algebraic Geometry__ by Bas Edixhoven and Lenny Taelman very closely.

In the last part of the course we will use Silverman's book *Arithmetic of Elliptic Curves. *

**Homework**

Below you will find all the Homework sheets. We recommend you write your solutions in LateX. This will increase the quality of your homework, and it's a great way to practice writing mathematics with LateX. To use LateX, download MikTeX and an editor (like TeXmaker or TeXworks). A good tutorial for using LateX can be found under __this link__. I prefer you e-mail me your homework solutions. You can of course also just give them to me.

- To be handed in on 2015 November 2nd : HW1 AG II 2015-2016
- To be handed in on 2015 November 16th : HW2 AG II 2015-2016
- To be handed in on 2015 November 30th : HW3 AG II 2015-2016
- To be handed in on 2015 December 14th : HW4 AG II 2015-2016
- To be handed in on 2016 January 11th : HW5 AG II 2015-2016
- To be handed in on 2016 January 25th : HW6 AG II 2015-2016
- To be handed in on 2016 February 15th : HW7 AG II 2015-2016
- To be handed in on 2016 March 28th :
**Exam (Homework 8) AG II 2015-2016**

**Your homework will be part of your final grade. To get the Klausurzulassung you will need to pass at least 4 out of the first 7 homework sheets**.

Your final grade is determined by your homework grades, and your grade for the exam (to be handed in on March 20th 2016). Grades are given from 1 to 10. You pass a homework sheet if you have at least 5,5 out of 10 points.

If c is your grade for the exam (i.e., the eighth homework sheet), and h_i is your grade for the i-th homework (i=1,...,7), then, depending on how the *Nachbesprechung* goes,

**"Final grade" = 1/10 (3c + h_1+...+h_7)**

**A brief outline of the course**

We will prove the Riemann hypothesis for zeta functions of curves over finite fields. Here are some of the Topics we will discuss.

- Zeta functions of finitely generated
**Z**-algebras. - Hasse-Weil's theorem + rationality of the zeta function + functional equation for the zeta function imply the Riemann hypothesis for curves over finite fields.
- Algebraic sets. Projective space. Bezout's theorem. Segre embedding.
- Algebraic varieties. Regular functions. Affine varieties. Products.
- Presentations. Smooth varieties. Rational functions. Tangent spaces and 1-forms.
- Riemann-Roch and Serre duality.
- Varieties over
**F_q**and their zeta function. - Intersection theory on smooth projective surfaces
- Elliptic curves: Silverman's Chapter III
- Elliptic curves over finite fields: Schoof's algorithm

**The schedule (details will be added during Semester)**

- Mo. October 19. Hasse-Weil I: Zeta functions of finitely generated
**Z**-algebras. - We. October 21. Hasse-Weil II: Zeta function
- Mo. October 26. Hilbert's Nullstellensatz. Cayley-Hamilton.
- We. October 28. Irreducible components of affine varieties. Dimension of an irreducible affine variety.
- Mo. November 2. Projective varieties. [Hand in HW1]
- We. November 4. The category of varieties, and smooth varieties. I

Mo. November 9. NO LECTURE

- We. November 11. Crash course: Categories, functors and natural transformations. (Lecture by M. Preisinger)
- Mo. November 16. The category of varieties, and smooth varieties. II [Hand in HW2]
- We. November 18. Products. Separatedness. Rational functions.
- Mo. November 23. Divisors on curves. (Note: Lecture ends earlier at 15:15h)

We. November 25. NO LECTURE.

- Mo. November 30. H0, H1 and Riemann-Roch for smooth projective irreducible curves. [Hand in HW3]
- We. December 2. Glueing data. Tangent space of a variety at a point.

Mo. December 7. NO LECTURE.

We. December 9. UBUNGSGRUPPE (Axel Stäbler)

- Mo. December 14. Derivations and Differentialforms [Hand in HW4]
- We. December 16. Serre duality for smooth projective irreducible curves.
- Mo. January 4. Rationality of the zeta function
- We. January 6. The functional equation for the zeta function + the Picard group of a variety
- Mo. January 11. Proof of Hasse-Weil via intersection theory on C x C [Hand in HW5]
- We. January 13. Finish proof of Hasse-Weil.
- Mo. January 18. Weierstrass Equations. Silverman III.1. III.3
- We. January 21. Homework session
- Mo. January 25. The group law I. Silverman III.1,2,3. [Hand in HW6]
- We. January 27. The group law II. Silverman III.1,2,3.
- Mo. February 1. The group law III. Silverman III.2. Isogenies. Silverman III.4,6,8
- We. February 3.
- Mo. February 8.
- We. February 10.
- Mo. February 15. Extra lecture. [Hand in HW7]