Kursinformation

In this seminar, we will study modular curves from an algebraic point of view, Eichler-Shimura relations, Galois representations and formulate the Modularity Theorem for elliptic curves over the rational numbers Q. We will follow Chapters 6-9 of Diamond-Shurman.

Prerequistes: the course Modular Forms-1 (chapters 1-5 of Diamond-Shurman, A first course in modular forms; in particular the definitions of modular forms and Hecke operators).

Time and Room: Tuesday 12h15-13h45, WSC-S-U-3.01

Tentative program:

1. [DS]6.1-6.2, Jacobians of Riemann surfaces
2. [DS]6.3-6.5, Hecke algebras
3. [DS]6.6, Modular abelian varieties and modularity theorem over complex numbers C
4. [DS]7.1-7.2, Elliptic curves as algebraic curves
5. [DS]7.3-7.4, Weil pairing on elliptic curves
6. [DS]7.5-7.6, Function fields of modular curves over C and over Q
7. [DS]7.7-7.9, Modular curves and algebraic curves and modularity theorem over Q
8. [DS]8.1-8.2, Elliptic curves and algebraic curves in arbitrary characteristic
9. [DS]8.3-8.4, Reductions mod p of elliptic curves
10. [DS]8.5-8.6, Reductions mod p of modular curves
11. [DS]8.7-8.8, Eichler-Shimura relation
12. *[DS]9.1-9.2, Galois number fields
13. [DS]9.3-9.4, Galois representations and elliptic curves
14. [DS]9.5-9.6, Galois representations and modular forms

*: maybe skipped depending on the backgrounds of the participants

References: 

1. [DS] Diamond and Shurman, A first course in modular forms (online access through university library here)