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.

**Exam**

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.

**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.

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

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

**Schedule **

- Tu. April 19. Algebraically closed fields.
- Th. April 21. Algebraic sets and the Zariski topology on affine space
- Tu. April 26. Hilbert's Nullstellensatz, dictionary between geometry and algebra. [Homework 1]
- Th. April 28. Cayley-Hamilton, Noetherian rings, Hilbert's Basis Theorem, Irreducible components
- Tu. May 3. Proof of Hilbert's theorems [Homework 2] (Reference: p. 11-14 of Looijenga)
- Th. May 5.
**NO LECTURE** - Tu. May 10. Quasi-affine varieties, regular maps, morphisms of quasi-affine varieties [Homework 3]
- Th. May 12. Intermezzo: Categories, functors, and equivalences of categories (by M. Preisinger)
- Tu. May 17. Proof of equivalence of categories, products of quasi-affine varieties [Homework 4]
- Th. May 19. Products continued (Moonen: Prop. 2.27), P^n (Edixhoven-Taelman, Lecture 4)
- Tu. May 24. Quasi-projective varieties and morphisms [Homework 5]
- Th. May 26.
**NO LECTURE** - Tu. May 31. Projective varieties are complete [Homework 6] (Gathmann 2003, paragraph 3)
- Th. June 2. Dimensions via projection from a point (Gathmann 2003, paragraph 4)
- Tu. June 7. Dimensions of hypersurfaces and smoothness (Looijenga) [Homework 7]
- Th. June 9. Singular locus, tangent spaces, rational maps. (Moonen 6.18, Ed-Ta Ch. 9.1, Silv. II.2.1)

- Tu. June 14. Exercise Session (by R. Wilms) [Homework 8]
- Th. June 16. Exercise Session (by R. Wilms)
- Tu. June 21. Riemann-Roch [Homework 9]
- Th. June 23. Riemann-Roch and Serre duality
- Tu. June 28. Riemann-Roch and Serre duality summarized [Homework 10] (by R. Wilms)
- Th. June 30. Exercise session (by R. Wilms)
- Tu. July 5. The group law (by R. Wilms)
- Th. July 7. Exercise session (by R. Wilms)

- Tu. July 12. Elliptic curves and Weierstrass equations [Homework 11]
- Th. July 14. The group law and the Picard group
- Tu. July 19. Isogenies
- Th. July 21. Exercise session
- Tu. July 26th. NO LECTURE. [Homework 12]

For Part II of this course click here.