Elliptische Kurven I, Sommer 2015-2016

This is the Homepage for the Vertiefungsvorlesung Elliptische Kurven I, given in Mainz during the Sommersemester of 2015-2016. If you're taking this course, please enroll on Jogustine.


Practical information

Teacher: Ariyan Javanpeykar

Course: Tuesday 10-12h, Thursday 10h-12h

Room: 04-422

Course material and description

We will follow Chapter I of Hartshorne's Algebraic Geometry. We will also use Silverman's The Arithmetic  of Elliptic Curves. We will also sometimes use Silverman's sequel to the previous book Advanced Topics in the Arithmetic of Elliptic Curves

Other useful references are Edixhoven-Taelman's notes on Algebraic Geometry and Ben Moonen's notes on Algebraic Geometry. Also, we will occasionally use Gathmann's notes.

One of the goals of this course is to prove the Mordell-Weil theorem for elliptic curves over number fields. We will get to this only in the next part of this course.


Everybody should hand in the 12 homework sheets. These won't be graded, but rather awarded with a minus or plus. If you have enough +'s, you can do the Take Home exam at the end of the course.

There will also be a Mundliche Prüfung (Oral Exam) discussing the homework sheets, the Take Home exam, and some of the course material that didn't make it into the homework.



You will find here 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 2016 April 26th    : Homework 1
  2. To be handed in 2015 May 3rd      : Homework 2
  3. To be handed in 2016 May 10th     : Homework 3
  4. To be handed in 2016 May 17th    :  Homework 4
  5. To be handed in 2016 May 24th     : Homework 5
  6. To be handed in 2016 May 31st     : Homework 6
  7. To be handed in 2016 June 7th      : Homework 7
  8. To be handed in 2016 June 14th    : Homework 8
  9. To be handed in 2016 June 21st    : Homework 9
  10. To be handed in 2016 June 28th    : Homework 10
  11. To be handed in 2016 July 12th     : Homework 11
  12. To be handed in 2016 July 26th     : Homework 12

Take Home Exam, to be handed in on August 2nd (or before). Click here


  1. Tu. April 19. Algebraically closed fields.
  2. Th. April 21. Algebraic sets and the Zariski topology on affine space
  3. Tu. April 26. Hilbert's Nullstellensatz, dictionary between geometry and algebra. [Homework 1]
  4. Th. April 28.  Cayley-Hamilton, Noetherian rings, Hilbert's Basis Theorem, Irreducible components
  5. Tu. May 3.  Proof of Hilbert's theorems [Homework 2] (Reference: p. 11-14 of Looijenga)
  6. Th. May 5. NO LECTURE
  7. Tu. May 10.  Quasi-affine varieties,  regular maps, morphisms of quasi-affine varieties [Homework 3]
  8. Th. May 12.  Intermezzo: Categories, functors, and equivalences of categories (by M. Preisinger)
  9. Tu. May 17. Proof of equivalence of categories, products of quasi-affine varieties  [Homework 4]
  10. Th. May 19. Products continued (Moonen: Prop. 2.27), P^n (Edixhoven-Taelman, Lecture 4)
  11. Tu. May 24. Quasi-projective varieties and morphisms [Homework 5]
  12. Th. May 26. NO LECTURE
  13. Tu. May 31. Projective varieties are complete [Homework 6] (Gathmann 2003, paragraph 3)
  14. Th. June 2.  Dimensions via projection from a point (Gathmann 2003, paragraph 4)
  15. Tu. June 7.  Dimensions of hypersurfaces and smoothness (Looijenga) [Homework 7]
  16. Th. June 9.  Singular locus, tangent spaces, rational maps. (Moonen 6.18, Ed-Ta Ch. 9.1, Silv. II.2.1)
  17. Tu. June 14.  Exercise Session (by R. Wilms) [Homework 8]
  18. Th. June 16.  Exercise Session (by R. Wilms)
  19. Tu. June 21.  Riemann-Roch [Homework 9]
  20. Th. June 23. Riemann-Roch and Serre duality
  21. Tu. June 28.  Riemann-Roch and Serre duality summarized [Homework 10] (by R. Wilms)
  22. Th. June 30.  Exercise session (by R. Wilms)
  23. Tu. July 5.  The group law  (by R. Wilms)
  24. Th. July 7.  Exercise session  (by R. Wilms)
  25. Tu. July 12. Elliptic curves and Weierstrass equations [Homework 11]
  26. Th. July 14. The group law and the Picard group
  27. Tu. July 19. Isogenies
  28. Th. July 21. Exercise session
  29. Tu. July 26th. NO LECTURE. [Homework 12]

For Part II of this course click here.