A21-MAT7395

About

This is the webpage for my graduate topic course MAT7395 in Fall 2021. The goal of this course is to study Hida theory and Iwasawa theory of modular forms. We will first review basic arithmetic properties of modular forms and elliptic curves. We will study L-functions, Hecke operators and Galois representations attached to these objects. We will study the theory of deformation theory of modular forms, Hecke algebras and Hida theory, as well as the Iwasawa main conjecture of modular forms. We will explore different elements and techniques used in the proofs of different versions of the Iwasawa main conjectures developed in recent years.

The course takes place on Mondays from 13:30 to 16:20, from August 30th to December 6th. It is in hybrid mode. The in-person class room is VCH-3840 and the Zoom link is available on request.

Lecture notes used it class are available here. Some class recordings can be found here.

Weekly summary
Aug 30 Introduction – We have given an overview of classical Iwasawa Theory of class groups over Zp-extensions of number fields, following the first chapter of Wuthrich’s notes. We have also quickly reviewed group cohomology and group homology.

Homework: Read Chapter II, Section 1 of Milne’s notes on Class Field Theory.

Sep 6 Labour day (no class)
Sep 13  We have discussed Section 2.1 of Skinner’s notes. In particular, we studied several properties of Galois cohomology of elliptic curves and defined Selmer groups.

Homework: Read Chapter VII of Silverman’s book.

Sep 20
Sep 27
Oct 4
Oct 11  Thanksgiving (no class)
Oct 18
Oct 25  Reading week (no class)
Nov 1
Nov 8
Nov 15
Nov 22
Nov 29
Dec 6
References
  • Arizona Winter School 2018
  • Christian Wuthrich: Overview of some Iwasawa Theory
  • James Milne: Class Field Theory
  • Joseph H. Silverman: The arithmetic of elliptic curves
  • Notes by Pete L. Clark and Allan Lacy