## Elliptic curves and their moduli spaces

Lecturer |
Dr. Andreas Mihatsch |

Email |
mihatsch (add ''at''math.uni-bonn.de) |

Office |
Room 4.024 |

Lecture |
Wed 14:15 - 16:00 (Kleiner Hörsaal) Fri 10:15 - 12:00 (Zeichensaal) First Exam: Monday, August 5 at 9:00, Großer Hörsaal Post exam review (Einsicht): Tuesday, August 6 during 10:00 - 11:00, Room 0.006 Second Exam: September 16 click to get to eCampus |

Exercises |
Mon 14:15 - 15:45 (0.011) Tue 12:15 - 13:45 (1.008) |

#### References

Lecture notes will be written and updated regularly. Last change: July 17.

Algebraic Geometry: Course notes for AG 1 and AG 2 as well as the Stacks Project

Elliptic Curves: Arithmetic of Elliptic Curves by Silverman

Group schemes, smoothness: Neron models by Bosch, Lütkebohmert and Raynaud

#### Exercise Sheets

Sheet 1 (due April 17)

Sheet 2 (due April 24)

Sheet 3 (due May 3)

Sheet 4 (due May 8)

Sheet 5 (due May 17)

Sheet 6 (due May 29)

Sheet 7 (due June 5)

Sheet 8 (due June 12)

Sheet 9 (due June 19)

Sheet 10 (due June 28)

Sheet 11 (due July 5)

Sheet 12 (due July 12)

#### Elliptic curves and their moduli spaces

An elliptic curve is a proper smooth connected algebraic curve of genus 1 with a fixed rational point. We will prove during the course that every such curve can be realized as a plane cubic curve. We will also show that every elliptic curve has a unique group structure for which the fixed point becomes the identity element. In that, elliptic curves are the same as 1-dimensional abelian varieties.

It is the group structure that distinguishes the theory of elliptic curves from that of curves of higher genus. For example, if E is an elliptic curve over a field k, then the set of rational points E(k) forms an abelian group. If k is the field of complex numbers, this group is isomorphic to a complex torus C/Λ, where Λ denotes a lattice in the complex plane, which relates elliptic curves with the theory of modular forms. If k is instead a finite extension of Q, then E(k) is a finitely generated abelian group that is a central object of study in modern number theory (see e.g. the Birch and Swinnerton-Dyer Conjecture).

In general, moduli spaces are spaces that parametrize isomorphism classes of a certain type of objects. For example, the points of the moduli space of elliptic curves are the same as the isomorphism classes of elliptic curves. Studying the moduli space of some type of object is hence equivalent to studying all objects of that type in a single family.

The moduli space of elliptic curves is especially interesting because it provides a construction of the Shimura variety for GL_2, also called the Modular curve. Its cohmology is related to modular forms, Fermat's theorem, and the Langlands program.

#### This Course

Our aim will be the construction of the moduli space of elliptic curves with level structure as scheme over the integers. We will use the language of schemes throughout, so a solid background in algebraic geometry (e.g. from the course Algebraic Geometry 1) is necessary. At several points, we will also take some time to discuss more advanced topics in scheme theory such as smoothness, group schemes, cohomology and base change, and the formalism of representable functors. This makes the course an ideal companion to Algebraic Geometry 2.