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.
||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.
||Labour day (no class)
||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.
||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.
||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.
||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.
|| Thanksgiving (no class)
||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.
|| Reading week (no class)
||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
||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.
||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.
||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.
||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.
|6 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.
- 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
- 2021-present: Mihir Deo
- 2020-present: Cédric Dion
- 2017-2021: A Arthur Bonkli Razafindrasoanaivolala (Sur les diviseurs milieux d’un entier , co-supervision with Jean-Marie De Koninck)
- 2016-2020: Jesse Larone (Résolution de certaines équations diophantiennes et propriétés de certains polynômes , co-supervision with Omar Kihel and Claude Levesque )
- 2014-2018: Gautier Ponsinet (On the algebraic side of the Iwasawa theory of some non-ordinary Galois representations )
- 2022-present: Tam Nguyen (PhD student from UBC, co-supervision with R. Sujatha)
- 2021-present: Luochen Zhao (PhD student from John Hopkins)
- 2016: José Ibrahim Villanueva Gutierrez (exchange PhD student from Bordeaux 1)
- 2021-present: Anthony Doyon
- 2019-2021: Fenomila Naivoarilala (Calcul de lambda invariant de la fonction L p-adique d’un caractère de Dirichlet sur un corps quartique)
- 2018-2019: Cédric Dion (Fonction L p-adique d’une forme modulaire )
- 2017-2019: David Ayotte (Relations entre le nombre de classes et les formes modulaires )
- 2021-present: Katharina Muller
- 2021-present: Jiacheng Xia
- 2020-2021: Jishnu Ray
- Fall 2020: Debanjana Kundu
- 2018-2020: Antonio Cauchi
- 2017-2019: Guhan Venkat
Undergraduate research projects
- 2022: Sai Sanjeev Balakrishnan, Jeanne Laflamme
- 2021: Anthony Doyon
- 2020: Maxime Cinq-Mars, Anthony Doyon, Antoine Poulin, Michaël Rioux
- 2019: Hugues Bellemare, Anthony Doyon, Sarthak Gupta
- 2018: Hugues Bellemare, Arihant Jain, Antoine Poulin, Mathieu Trudelle
- 2017: Cédric Dion
- 2016: David Ayotte
- 2015: David Ayotte, Jean-Christophe Rondy-Turcotte
Research environment in Ottawa
University of Ottawa is a bilingual university. You can carry out your studies and research in English or in French here. If you study with me, you will be part of the Ottawa-Carleton Number Theory research group. Several scholarships are available to Canadian and international students and postdoctoral fellows. More information can be found here.
If you are interested in working with me as a graduate students, you can expect to work on one of the following projets.
1. Factorization of p-adic L-functions
An elliptic curves is a cubic curve that is equipped with a group structure. They are used heavily in modern-day cryptography. In this project you will study the p-adic L-function of an elliptic curve, which is a power series defined over p-adic numbers. It tells us about arithmetic properties of the curve over extensions of the rational numbers that are given by p-power roots of unity. You will investigate how the coefficients of these functions behave. By analysing values of these functions evaluated at a set of special points, you will study whether they factor into functions that are easier to understand. This will result in new arithmetic information on the elliptic curve.
2. Iwasawa theory of modular forms
A modular form is an analytic function that satisfies invariant properties under certain Mobius transforms. In order to prove Fermat’s Last Theorem, Andrew Wiles showed that an elliptic curve is in fact a special case of modular forms. The analytic structure of modular forms allow us to obtain new information about elliptic curves. In this project, you will study certain representations of the Galois group of the rational numbers that are described by the Fourier coefficients of a modular form. More precisely, you will look at the behaviour of these representations as you restrict them to a tower of extensions and understand their cohomology groups.
3. Construction of Euler systems
An Euler system is a family of cohomology classes of a representation that satisfy certain compatibility condition. It is a very powerful tool in Iwasawa Theory because they give us a link between algebraic objects and to analytic objects. In this project, you will study the construction of Euler systems for a certain family of representations and apply them to study asymptotic behaviour of these representations. The construction will rely on understanding the geometry of modular curves and modular surfaces.
Hello! My name is Antonio Lei. I am a mathematician with research interests in algebraic number theory. To find out more about my research, click here.
Since 2022, I am an associate professor at the Department of Mathematics and Statistics of University of Ottawa.
Currently, Javad Mashreghi and I are Editors-in-Chief of the Canadian Mathematical Bulletin. I am also a member of the editorial board of Annales mathématiques du Québec. I encourage you to submit your articles to these journals (note that CMB only accepts articles of at most 18 pages long)!
Previously, I have been a faculty member of Laval University (2014-2022), a postdoctoral researcher at McGill University (2011-2014, supervisor: Prof Henri Darmon) and Monash University (2010-2011, supervisor: Dr Daniel Delbourgo). I received my doctorate under the supervision of Prof Tony Scholl at University of Cambridge in 2010. I am originally from Macau, a former overseas province/colony of Portugal.
If you are interested in doing a summer internship, a master, a PhD or a postdoc with me, here is some information. If you have any questions, feel free to contact me.
antonio (DOT) lei (AT) uottawa (DOT) ca
Department of Mathematics and Statistics
150 Louis-Pasteur Pvt
Canada K1N 6N5
Broadly speaking, I am interested in Algebraic Number Theory. More particularly, the problems I study come from Iwasawa Theory, which is a study of behaviours of mathematical objects over a series of algebraic domains. For example, if E is an elliptic curve with coefficients in Q (the field of rational numbers), we could ask how many points on E have coordinates in Q. But we could equally ask, how many points there are if we allow the coordinates to be expressions involving square roots of some number? What if we allow fourth roots? Eight roots? What if we continue forever?
In Iwasawa Theory, we are able to answer this type of questions explicitly.
You can learn more about the subject by reading one of the introductory articles written by Ralph Greenberg. One powerful tool to study this type of problems is p-adic Hodge Theory, which allows us to study the behaviour of representations over cyclotomic extensions (algebraic structures that are defined using roots of unity) at a fixed prime number using Linear Algebra. This often allows us to solve a complex problem by very explicit calculations. The pioneer in this area is Bernadette Perrin-Riou who has developed many powerful machineries which I make use of in my works. Results of Laurent Berger has also motivated many of my works. His notes on the theory of p-adic representations introduce many of the tools used in p-adic Hodge Theory.
To find out more about my research, have a look at my list of publications and academic activities.