**Bas Edixhoven:**

In the 1st hour, I will follow

http://images.math.cnrs.fr/Representations-galoisiennes-et.html?lang=fr

So, I will talk about complex numbers, field automorphisms (not

necessarily continuous), how these permute the roots of a polynomial

with rational coefficients, what 1 and 2-dimensional Galois

representations with coefficients in Z/nZ are. And I will discuss how

elliptic curves over Q give us 2 dimensional Galois representations.

In the 2nd hour I will talk about how an elliptic curve over Q gives

its Hasse-Weil L-function, and that modularity means that this is also

the L-function of an eigenform. This has been proved via Galois

representations. So I will also say a bit about how eigenforms give

Galois representations, and how Fermat follows from the theorem of

Khare and Wintenberger (formerly Serre’s modularity conjecture).

**İlhan İkeda**

**In Lecture 1 (1 hour)**, we plan to recall the definitions and the very basic properties of global and local fields;

the restricted direct products: the ring of adeles and the group of ideles; statements of (global and local)

class field theory; and Artin L-functions.

**In Lecture 2 (1 hour)**, we plan to state the reciprocity and the functoriality principles of Langlands; the hypothetical

automorphic Langlands group of a global field; a new approach to reciprocity and functoriality principles.

**Winfried Kohnen:**

I plan to give one lecture to introduce and motivate modular forms through quadratic forms and representation numbers. Then the 2nd lecture will concretely introduce modular forms and give basic facts, just as cusp forms, fundamental domain, Eisenstein series, valence formula, L-functions. Each talk 60 minutes.

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