# Elliptic Curves

## Course at FU Berlin, Winter Term 15/16

### Content

An elliptic curve over a field $$K$$ is a smooth projective plane cubic curve with a rational point. If the characteristic of $$K$$ is not 2 or 3 such a curve is always isomorphic to the closure in $$\mathbb{P}^2_K$$ of a curve in the 2 dimensional affine space given by the vanishing of $$y^2-x^3-ax-b$$, $$a,b \in K$$ with $$4a^3+27b^2\neq 0$$. What makes elliptic curves so fascinating is that within all smooth projective curves they are precisely those which are commutative $$K$$-group schemes, i.e. if $$E\subset \mathbb{P}^2_K$$ is an elliptic curve given by a homogenous cubic polynomial $$F$$ and $$A$$ is any $$K$$-algebra, then the vanishing set of $$F$$ in $$\mathbb{P}^2(A)$$ is an abelian group, denoted by $$E(A)$$, and a homomorphism of $$K$$-algebras $$A\to B$$ induces a homomorphism of groups $$E(A)\to E(B)$$. For example if $$K$$ is the field of complex numbers $$\mathbb{C}$$, then there is an isomorphism of groups $$E(\mathbb{C})\cong \mathbb{C}^2/\Lambda$$, where $$\Lambda\subset\mathbb{C}$$ is a free $$\mathbb{Z}$$-module of rank 2.

In the course we will establish some first basic properties of elliptic curves, like a cohomological characterization, for which we will also quickly review the theory of cohomology of cohrenet sheaves on a variety and in particular on curves. Our main goal will then be the famous Theorem of Mordell-Weil which states the following: Let $$K$$ be a number field, i.e. a finite extension of $$\mathbb{Q}$$, then the group $$E(K)$$ is a finitely generated $$\mathbb{Z}$$-module. In particular it can be written as a direct sum of a finitely generated torsion group and a free $$\mathbb{Z}$$-module of finite rank.

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

### Data

Lecturer Time Room
Lecture Kay Rülling Mi 10-12 055/T9 Seminarraum (Takustr. 9)
Exercise Lei Zhang Mi 14-16 SR 130/A3 Seminarraum (Hinterhaus) (Arnimallee 3-5)

The first lecture is on October 14, 2015. The first exercise session is on October 21.
If you have any questions concerning the lectures or the exercises feel free to contact one of us.

### Exercises

Every Wednesday there will be a new exercise sheet, the solutions are discussed in the exercise session on the Wednesday of the following week.

1. Exercise sheet October 14, 2015. (2nd ver 19.10.)
2. Exercise sheet October 21, 2015.
3. Exercise sheet October 28, 2015. (2nd ver 4.11.)
4. Exercise sheet November 4, 2015.
5. Exercise sheet November 11, 2015. (2nd ver 17.11)
6. Exercise sheet November 18, 2015.
7. On December 2 there will be a lecture in the exercise session (additionally to the lecture in the morning).
The week after will be as usual: one lecture/one exercise and on December 16 there will be two exercise sessions.
8. Exercise sheet December 2, 2015. (2nd ver 18.01.)
This exercise sheet is to be discussed on Dec. 9 and in the two exercise sessions on Dec. 16.
I suggest to prepare 7.1 - 7.4 for Dec. 9.
9. On January 20 there will be a lecture in the exercise session (additionally to the lecture in the morning).
10. On January 27 there will be a lecture in the exercise session (additionally to the lecture in the morning).
11. Exercise sheet January 29, 2016.

### References

Here is a list of references for the course:

1. J. H. Silverman, Arithmetic of Elliptic Curves , Graduate Texts in Mathematics, vol 106 , Springer.
2. J.S. Milne, Elliptic Curves , BookSurge Publishers. Available at http://www.jmilne.org/math/Books/index.html.
3. R. Hartshorne Algebraic Geometry , Graduate Texts in Mathematics, vol 57, Springer.