Udi De Shalit "Modularity of elliptic curves" (80844 in shnaton). 


Recordings: 

https://huji.cloud.panopto.eu/Panopto/Pages/Sessions/List.aspx?folderID=a29a25b2-ba64-437d-90e7-af090084a19e 

Notes: 

http://www.ma.huji.ac.il/~deshalit/new_site/files/Modularity_Notes_2023.pdf


Abstract: In this seminar we shall go over Wiles' famous theorem from
1995 that semi-stable elliptic curves over Q are modular.
It is well known that this theorem implies Fermat's Last Theorem.

We shall start by a brief explanation of the Galois representations
attached to elliptic curves and modular forms, and give an overview of
the proof.

We shall then spend most of the time unfolding it. Necessary
background on Galois cohomology of number fields, deformations of
Galois representations, p-adic Hodge theory, modular forms for GL_2(Q), the Langlands
correspondence and commutative algebra will be surveyed along the way.

Warning for the experts: there have been a lot of new ideas introduced
into the area by many authors, many more cases of modularity are
known, and even the original case treated by Wiles and Taylor-Wiles can be
approached today in several different ways. 

Partly for pedagogical reasons, and partly for my own lack of knowledge, our approach will be close to the original proof from 1995 (with improvements mainly on the
commutative algebra front).