Algebraische Geometrie II, Winter 2015-2016

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.  



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.



  1. To be handed in on 2015 November 2nd    : HW1 AG II 2015-2016
  2. To be handed in on 2015 November 16th   : HW2 AG II 2015-2016
  3. To be handed in on 2015 November 30th   : HW3 AG II 2015-2016
  4. To be handed in on 2015 December 14th   : HW4 AG II 2015-2016
  5. To be handed in on 2016 January 11th       : HW5 AG II 2015-2016
  6. To be handed in on  2016 January 25th      : HW6 AG II 2015-2016
  7. To be handed in on 2016 February 15th       : HW7 AG II 2015-2016
  8. 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)

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

Mo. November 9.  NO LECTURE

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

We. November 25. NO LECTURE.

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

Mo. December 7. NO LECTURE.

We. December 9. UBUNGSGRUPPE (Axel Stäbler)

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