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.
| 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) |
Every Wednesday there will be a new exercise sheet, the solutions are discussed in the exercise session on the Wednesday of the following week.
Here is a list of references for the course: