###### 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 | We reviewed the definition of isogenies and some basic properties. We studied the number of points on an elliptic curve over a finite field and defined ordinary and supersingular primes. We have started a discussion on Selmer structures.
Homework: Read these notes. |

Sep 27 | We have discussed dual Selmer structures, defined Bloch-Kato’s Selmer conditions and Selmer groups. We have stated a few conjectures on L-functions of geometric representations as well Bloch-Kato’s conjecture.
Homework: Review the examples in Section 2 of Skinner’s notes. |

Oct 4 | We reviewed the module structure of dual groups. We have started Section 3 of Skinner’s notes, reviewing the definition of Iwasawa algebras and several Selmer groups of elliptic curves over Z_p^d-extensions that will be useful later.
Homework: Study Chapter V, Section 1 of the book of Neukirch-Schmidt-Wingberg. |

Oct 11 | Thanksgiving (no class) |

Oct 18 | Guest lecture given by Katharina: We studied the Lambda-structure of Selmer groups over a Z_p^d-extension. We have also studied a few different versions of control theorems of Selmer groups, including a control theorem for Kobayashi’s plus and minus Selmer groups in the supersingular case.
Homework: Study Greenberg’s proof of the control theorem, P22-25 of these notes. |

Oct 25 | Reading week (no class) |

Nov 1 | Guest lecture given by Luochen Zhao on the construction of p-adic L-functions attached to modular forms via modular symbols.
Homework: Exercises in Luochen’ snotes |

Nov 8 | We defined modular forms over p-adic rings, introduced Serre’s p-adic modular forms and Lambda-adic modular forms. We have studied Eisenstein series as examples.
Homework: Study Section 3 of Serre’s paper on p-adic modular forms. |

Nov 15 | We discussed results of Hida on ordinary Lambda-adic modular forms, defined theta series and CM forms. We have also begun studying p-adic L-functions of Rankin products.
Homework: Read Chapter 7.6 of Hida’s book. |

Nov 22 | We have defined three-variable p-adic L-functions of Hida and also several two-variable p-adic L-functions attached to an elliptic curve. We has also stated both 1-variable and 2-variable Iwasawa main conjectures, as well as variants that do not involve p-adic L-functions.
Homework: Read this paper. |

Nov 29 | We went through some of the ingredients in Skinner-Urban’s proof of the Iwasawa main conjecture, following these notes by Xin Wan.
Homework: Read this paper. |

Dec 13 |
We discussed several applications of Iwasawa main conjectures on Birch and Swinnerton-Dyer conjectures. We have also discussed the p-converse theorem and a Lambda-adic version of the the Gross-Zagier formula due to Castella and Howard. |

###### References

- Arizona Winter School 2018
- Christian Wuthrich: Overview of some Iwasawa Theory
- Haruzo Hida: Elementary Theory of L-functions and Eisenstein seires
- James Milne: Class Field Theory
- James Milne: Arithmetic Duality Theorems
- Jürgen Neukirch, Alexander Schmidt, Kay Wingberg:

Cohomology of Number Fields - Joseph H. Silverman: The arithmetic of elliptic curves
- Notes by Pete L. Clark and Allan Lacy
- Notes by Joel Beillaiche on Bloch-Kato conjectures