# Winter 2014-2015: Elliptische Kurven und Komplexe Multiplikation

This is the website for the course "Elliptische Kurven und Komplexe Multiplikation" given at the Institut für Mathematik in Mainz during the Wintersemester of 2014-2015. This is an Erganzungsvorlesung.

Practical information

Teachers: Ariyan Javanpeykar and Stefan Muller-Stach.

Course: Wednesday, 10-12h. (First class on 29th of October)

Room: 04-432

Program of course

1. We introduced lattices L in C. We explained that the set of endomorphisms of the complex torus C/L is the set of x in C such that xL is contained L. We then proved that this subring of C is in fact equal to Z, or equal to an order in a quadratic imaginary number field. As an application, we computed the automorphism group of a complex torus (which is almost always isomorphic to Z/2Z). We defined complex tori with Complex Multiplication (CM). We finished with a discussion of orders in quadratic imaginary number fields.
2. We introduced the class group of a number field, and proved that, for K an imaginary quadratic number field, the class group of K acts freely and transitively on the set of isom classes of complex tori with CM by K. In particular, the number of isom classes of complex tori with CM by K is the class number of K. (Note: we did not treat the analogous theorem for orders in K.)
3. We explained that isomorphic (resp. isogenous) complex tori are in the same SL_2(Z) (resp. GL_2^+(Q)) orbit, and that the set of isomorphism (resp. isogeny) classes of lattices in C identifies naturally with the Riemann surface SL_2(Z)\H (resp. the set GL_2^+(Q)\H). We gave the standard fundamental domain for SL_2(Z), and discussed the modular curves Y_0(n), Y_1(n) and Y(n) very briefly (more on these later). Then we introduced the j-invariant of a lattice, and proved it induces an isomorphism from SL_2(Z)\H to the complex points of the affine line C. We also showed that the j-function is a modular form for the modular group Gamma(1) = PSL_2(Z) of weight zero.
4. The j-function parametrizes the coarse moduli space of all lattices. What are the properties of the j-invariants of a CM complex torus? We wish to prove that it is an algebraic integer, i.e., for all CM lattices L in C, there exists a monic polynomial F(x) with integer coefficients such that F(j(L)) =0. Before we discuss the ideas of the proof we explain some of its consequences. For instance, in this class we showed that every CM elliptic curve can be defined over some number field.
5. We proved that, if E is an elliptic curve over C with CM by O_K, then the degree of j=j(E) is the class number of K, the number field K(j) is the Hilbert class field of K and K(j) is of degree h_K over K. We used some class field theory and basic properties of the Hilbert class field. We then continued to sketch the proof of the integrality of j via Neron-Ogg-Shafarevich. We explained briefly how the Galois representation attached to the ell-adic Tate module of E over a p-adic field is potentially unramified.
6. Continuation of 5.
7. We explain what the possible Q-algebras arising as End(E)_Q are, where E is an elliptic curve (over an arbitrary field).
8. We show that an elliptic curve is supersingular if and only if multiplication by p is purely inseparable.
9. We prove the injectivity of the map End(E)_{Z_\ell} \to End(T_\ell E) for all prime numbers \ell \in k*. We end with a brief discussion of Tate's conjecture, and applications of the isogeny conjecture (e.g., Serre's theorem on the irreducibility of GL_2(E[ell]) as a Galois representation associated to an elliptic curve over a number field).
10. We explain the statement of the Andre-Oort conjecture for a product of two modular curves. To do so, we introduced the modular curves Y(n), Y_1(n) and Y_0(n) and formulated the AO conjecture as follows: An irreducible curve in C^2 is special if and only if its set of special points is infinite (ie. dense).
11. We explain Baker's theory of linear forms and give some applications to elliptic curves.
12. We give an idea of the proof of the AO conjecture.

Course description

Here is a rough outline of the course indicating the topics treated in this course. We will mostly follow Silverman's book Advanced topics in the arithmetic of elliptic curves.

1. We introduce the notion of "complex multiplication" for complex tori.
2. We prove that the class group of a quadratic imaginary number field K acts freely and transitively on the set of  complex tori with complex multiplication by K.
3. We show that   complex tori over are classified by their j-invariant.
4. We classify all CM complex tori  with j-invariant a rational number.
5. We show that a complex torus C/L can be given by a homogeneous Weierstrass equation.
6. We show that a compact Riemann surface of genus one can be given by a homogeneous Weierstrass equation.
7. We introduce elliptic curves over arbitrary fields and study their endomorphisms. We introduce the notion of supersingularity. We discuss Tate's conjectures on endomorphisms of elliptic curves and endomorphisms of the Tate module.
8. We introduce the moduli spaces Y(n) for elliptic curves with level n structure.
9. We classify all curves in Y(1) x Y(1) with a dense set of special points.
10. We prove the Andre-Oort conjecture for Y(1) x Y(1) using Baker's theory of linear forms in logarithms, equidistribution results and some CM theory.