In the 1st hour, I will follow
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).
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.
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.