## Moduli of elliptic curves

Lecturer |
Dr. Andreas Mihatsch |

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

Office |
Room 4.024 |

Lecture |
Mon 12:15 -- 13:45, Thu 14:15 -- 15:45 Kleiner Hörsaal, Wegelerstr. 10 |

Registration |
Please register on eCampus |

#### Lecture Notes and References

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

Elliptic Curves: Silverman's Arithmetic of Elliptic Curves or my previous course

Abelian Varieties: Mumford's book

Descent theory: Neron models Chapter 6 or Vistoli's Notes on Descent

Moduli of Elliptic Curves: Article by Deligne--Rapoport

1 - Introduction

2 - ECs over finite fields

3 - Abelian extensions from ECs

4 - Relative ECs

5 - Mordell--Weil

6 - Weierstrass moduli

7 - Fine moduli (away from 2 and 3)

8 - Weil Extension Thm

8.5 - Plan for next lectures

9 - Weil pairing

10 - Cartier Duality

11 - Representability of M_3

12 - M_n at primes 2, 3

13 - Examples of Descent

14 - Noetherian Approximation

15 - Shimura variety perspective

16 - Level K_0(p) (Notes from during lecture)

17 - Structure of M_0(p)

18 - Structure of M_0(p) II

19 - Structure of M_0(p) III

20 - Eichler--Shimura

Reference mentioned in lecture: Article of Diamond--Im

Example of the isogeny class of ECs of conductor 11 in LMFDB Project

Link to texed version of the lecture notes, written by Matthew Stevens.

Link to Kleine AG on Weil conjectures

Further Arithmetic Geometry resources:

Neron models by Bosch, Lütkebohmert, Raynaud, a most enjoyable read

Arithmetic geometry, edited by Cornell-Silverman

Kleine AG website with guides to various topics

#### Schedule

My (preliminary) plan is to cover the following topics this term.

- Elliptic curves, in particular the Theorems of Mordell--Weil and Hasse
- The Modular Curve, in particular its properties like compactification or reduction mod p
- The Modularity Theorem and its consequences

I plan to follow up with a Selected Topics Course on

- Intersection theory on the modular curve
- The Gross--Zagier Theorem

#### Motivation

Elliptic curves are algebraic curves with group structure. In number theory, one is especially interested in the situation of an elliptic curve E over a number field K. A famous result, due to Mordell--Weil, states that E(K) is always a finitely generated abelian group. Based on Computer heuristics, Birch and Swinnerton-Dyer conjectured in the 1960s that the rank of E(K) can be expressed through the L-function of E. This conjecture is still open in its generality and spurs many developments in number theory.

Gross and Zagier were able to construct non-torsion points on E(Q) for certain elliptic curves E/Q and, in this way, showed that they have rank at least 1. Their work is quite beautiful and involves the most shimmering objects from arithmetic geometry, above all modular forms and the modular curve.

The aim of this course and its continuation next term is to thoroughly introduce the above objects and to explain some aspects of the work of Gross and Zagier. Above all, we will introduce the modular curve as the moduli space of elliptic curves and we will study its arithmetic properties.

#### Prerequisites

- Firm knowledge of algebraic geometry, e.g. from the courses AG 1 and 2
- Basic algebraic number theory

#### Exam Dates

Oral exam weeks suggestion: February 14 -- 18, March 21 -- 25.