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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 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 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.
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

Teaching

Session/Semester Rôle/Role Cours/Course Université/University
2024 Winter Instructor Topics in number theory/algebra – Diophantine equations and elliptic curves (MAT4167/5327) Ottawa
2023 Fall Coordinator and instructor Introduction to linear algebra (MAT1341) Ottawa
2023 Winter Coordinator and instructor Introduction to linear algebra (MAT1341) Ottawa
2023 Winter Instructor Elementary real analysis (MAT2125) Ottawa
2022 Hiver Enseignant Algèbre II (MAT-3300) Laval
2022 Hiver Enseignant Algèbre linéaire avancée (MAT-2200) Laval
2021 Automne Enseignant Modular forms and elliptic curves (MAT-7395) Laval
2021 Hiver Enseignant Algèbre linéaire avancée (MAT-2200) Laval
2020 Hiver Enseignant Algèbre linéaire avancée (MAT-2200) Laval
2019 Automne Enseignant Algèbre I (MAT-2300) Laval
2019 Automne Enseignant et coordinateur Introduction à l’algèbre linéaire (MAT-1200) Laval
2019 Hiver Enseignant Algèbre linéaire avancée (MAT-2200) Laval
2019 Hiver Enseignant Analyse p-adique et corps locaux (MAT-7395/MAT-2700) Laval
2018 Automne Enseignant et coordinateur Introduction à l’algèbre linéaire (MAT-1200) Laval
2018 Hiver Enseignant Théorie algébrique des nombres (MAT-7345) Laval
2017 Automne Enseignant Algèbre I (MAT-2300) Laval
2017 Automne Enseignant Introduction à l’algèbre linéaire (MAT-1200) Laval
2017 Hiver Enseignant Mathématiques discrètes (MAT-1310) Laval
2016 Automne Enseignant Algèbre I (MAT-2300) Laval
2016 Automne Enseignant Introduction à l’algèbre linéaire (MAT-1200) Laval
2016 Hiver Enseignant Analyse p-adique et les groupes de Lie p-adiques (MAT-7390) Laval
2015 Automne Enseignant Algèbre I (MAT-2300) Laval
2015 Hiver Enseignant Algèbre linéaire avancée (MAT-2200) Laval
2014 Automne Enseignant Algèbre commutative et théorie de Galois (MAT-7200) Laval
2014 Winter Course coordinator and instructor Mathematics for Education Students (MATH111) McGill
2013 Fall Course coordinator and instructor Enriched Linear Algebra and Geometry (MATH134) McGill
2013 Winter Instructor Fundamental Mathematics II (MATH209) Concordia
2012 Fall Course coordinator and instructor Calculus 2 (MATH141) McGill
2012 Winter Instructor Calculus 2 (MATH141) McGill
2011 Semester 2 Lecturer and tutor Diophantine Equations (M4361) Monash
2009 Michaelmas Tutor Number Theory Cambridge
2009 Lent Tutor Groups, Rings and Modules Cambridge
2008 Michaelmas Tutor Number Theory Cambridge
2008 Lent Tutor Complex Analysis Cambridge
2007 Michaelmas Tutor Galois Theory Cambridge

Academic activities

Conferences/Seminars Organization

  • Carleton-Ottawa Number Theory Seminar
  • CMS winter meeting 2023, Montreal, Scientific Organizing Committee
  • CMS summer meeting 2023, Ottawa, Arithmetic Aspects of Automorphic Forms session
  • CMS winter meeting 2021 in Vancouver, Galois representations and L-functions  session
  • CMS 75th+1 anniversary summer meeting, Ottawa, Algebraic Number Theory session https://summer20.cms.math.ca
  • CRM Thematic semester: Number Theory – Cohomology in Arithmetic, Fall 2020 http://www.crm.umontreal.ca/2020/Nombres2020/index_e.php
  • 15th Conference of the Canadian Number Theory Association http://www.math.umaine.edu/numbertheory/mainequebec.html
  • Québec-Maine Number Theory Conference http://www.math.umaine.edu/numbertheory/mainequebec.html
  • Algebra and Number Theory Seminars at Laval University https://researchseminars.org/seminar/ANTULaval
  • Study groups at Laval University https://sites.google.com/site/formesmodulaires/
  • CMS winter meeting 2015 in Montreal, Algebraic Number Theory session https://cms.math.ca/Events/winter15/schedule_session#ant

Invited Talks

  • Iwasawa theory of isogeny graphs and zeta functions, CRM Number Theory Seminars, August 2024
  • Asymptotic growth of elliptic curves over anticyclotomic extensions – applications of plus and minus Iwasawa theory, CNTA XVI, Fields Institute, Toronto, June 2024 
  • Iwasawa theory of isogeny graphs with level structures, Vistas in Number Theory, Brin Mathematics Research Center, University of Maryland, June 2024
  • Anticyclotomic Iwasawa theory of elliptic curves, UTRGV Algebra and Number Theory Seminar, May 2024
  • Iwasawa Theory and the Birch and Swinnerton-Dyer Conjecture, UTRGV Colloquium, April 2024
  • Iwasawa theory of isogeny graphs with level structures, UCD Algebra and Number Theory Seminar, February 2024
  • Iwasawa theory of isogeny graphs with level structures, Fields Number Theory Seminar, February 2024
  • Anticyclotomic Iwasawa theory of elliptic curves, Queen’s Number Theory Seminar, November 2023
  • Asymptotic formula for Tate–Shafarevich groups of p-supersingular elliptic curves over anticyclotomic extensions, UT Austin Number Theory Seminars, November 2023
  • Mini-course on Iwasawa theory, Université Laval, October 2023
  • On ordinary isogeny graphs with level structure, Maine-Quebec Number Theory Conference, September 2023
  • Anticyclotomic Iwasawa theory for elliptic curves at supersingular primes, Johns Hopkins Number Theory Seminar, February 2023
  • Constructing local points on supersingular elliptic curves, Number Theory Seminars, Concordia University, February 2023
  • 2 talks on The Perrin-Riou Map and its Use in Iwasawa Theory, Introductory Workshop: Algebraic Cycles, L-Values, and Euler Systems, MSRI-SLMath, January 2023
  • La théorie d’Iwasawa anticyclotomique pour les courbes elliptiques : le cas supersingulier, séminaire arithmétique, l’Université de Lille, November 2022
  • 2 talks on Introduction to the Iwasawa theory of elliptic curves, Algebra and Number Theory seminars, University College Dublin, October 2022
  • Colloquium on Birch and Swinnerton-Dyer Conjecture via Iwasawa Theory, University College Dublin, October 2022
  • Mini-course on Euler systems and Beilinson-Flach elements, Elliptic curves and the special values of L-functions, ICTS, August 2022
  • Asymptotic formula for Tate–Shafarevich groups of p-supersingular elliptic curves over anticyclotomic extensions, International conference on p-adic L-functions and eigenvarieties, University of Notre Dame, July 2022
  • Asymptotic formula for Tate–Shafarevich groups of p-supersingular elliptic curves over anticyclotomic extensions, Iwasawa theory Virtual Seminar, March 2022
  • Iwasawa theory over imaginary quadratic fields for inert primes, Automorphic Forms and Arithmetic Seminar, Columbia University, December 2021
  • Anticyclotomic Selmer groups for CM elliptic curves at inert primes, Maine-Québec Number Theory Conference, October 2021
  • Some recent development on the Iwasawa theory of fine Selmer groups, Number Theory in the Americas, special session at the Mathematical Congress of the Americas, July 2021 https://www.mca2021.org/en/tools/view-abstract?code=3021
  • Sur la sturcture algébrique du groupe de Mordell-Weil fin, Amicale de théorie des nombres en hommage à Robert Langlands, CMS 75th+1 anniversary summer meeting, June 2021 https://www2.cms.math.ca/Events/summer21/abs/arl#al
  • Semi-ordinary Iwasawa theory for Rankin-Selberg products, L-values and Iwasawa theory, NCTS online conference, November 2020 https://researchseminars.org/seminar/nctsnumbertheory
  • Bounded Euler systems for Rankin-Selberg products of modular forms, UBC Number Theory Seminar, March 2020 http://www.math.ubc.ca/Dept/Events/index.shtml?period=2020-03&series=69
  • Bounded Euler systems for Rankin-Selberg products of modular forms, Caltech Number Theory Seminar, February 2020 https://www.google.com/calendar/event?eid=MjZyMmJua2txYWY5YXBrc21pYWpsYmd2ZjdfMjAyMDAyMjhUMDAwMDAwWiBudGNhbHRlY2hAbQ&ctz=America/Los_Angeles
  • Congruences of anticyclotomic p-adic L-functions, Maine-Québec Number Theory Conference, October 2019 https://archimede.mat.ulaval.ca/QUEBEC-MAINE/19/mq19.html#Lei
  • Codimension two cycles and tensor products of Hida families, Recent advances in the arithmetic of Galois representations, Genova, July 2019 
  • Codimension two cycles and tensor products of Hida families, p–adic modular forms, satellite conference to Journées Arithmétiques XXXI, Istanbul, July 2019 
  • Growth of Mordell-Weil ranks of elliptic curves, Journées Arithmétiques XXXI, Istanbul, July 2019 
  • Pseudo-null modules and codimension two cycles for supersingular elliptic curves, Iwasawa 2019, Bordeaux, June 2019 
  • An algebraic toolbox in Iwasawa theory, University College Dublin, November 2018 
  • Towards a rank-two Euler system via Wach modules, Iwasawa Theory and Related Topics, Heidelberg, May 2018 https://www.mathi.uni-heidelberg.de/~heidelberg2018/
  • Lecture series on Iwasawa theory for elliptic curves with supersingular reduction, PIMS, March 2018 https://www.pims.math.ca/scientific-event/180305-ralal
  • Mordell-Weil ranks of elliptic curves over a tower of extensions, University of Pennsylvania, January 2018 https://www.math.upenn.edu/events/mordell-weil-ranks-elliptic-curves-over-tower-extensions
  • Second Chern Classes for Supersingular Elliptic Curves, Maine-Québec Number Theory Conference, October 2017 http://www.math.umaine.edu/numbertheory/mq17.html#Lei
  • Mini course on cyclotomic units, Introductory Workshop on Euler Systems and Special Values of L-functions, Bernoulli Center, Lausanne, August 2017 http://claymath.org/events/recent-developments-elliptic-curves
  • Beilinson-Flach elements and finiteness of sha, Recent Developments on Elliptic Curves, Clay Mathematics Institute workshop, Oxford, September 2016 http://claymath.org/events/recent-developments-elliptic-curves
  • Iwasawa theory of modular forms over imaginary quadratic fields, 14th Meeting of the Canadian Number Theory Association, Calgary, June 2016
  • Asymptotic behaviour of the Shafarevich-Tate groups of modular forms, University of Texas at Austin, October 2015
  • Asymptotic behaviour of the Shafarevich-Tate groups of modular forms, Maine-Québec Number Theory Conference, October 2015 http://www.math.umaine.edu/numbertheory/mq15.html#Lei
  • Beilinson-Flach elements on products of modular curves and modular surfaces, Workshop “p-adic methods in the theory of classical automorphic forms”, CRM, Montreal, March 2015 http://www.crm.umontreal.ca/2015/Automorph15/pdf/lei.pdf
  • Iwasawa theory of elliptic curves over Zp^2-extensions, Joint Mathematics Meetings, San Antonio, January 2015 http://jointmathematicsmeetings.org/amsmtgs/2168_abstracts/1106-11-894.pdf
  • Universal norm of crystalline classes, CMS winter meeting, Hamilton, December 2014 http://www.ams.org/amsmtgs/2223_abstracts/1103-11-145.pdf
  • Integral Iwasawa theory of p-adic representations at non-ordinary primes, AMS Fall Sectional Meeting, Halifax, October 2014 http://www.ams.org/amsmtgs/2223_abstracts/1103-11-145.pdf
  • Iwasawa theory of elliptic curves over Zp^2-extensions, Québec-Maine Number Theory Conference, September 2014 http://www.math.umaine.edu/numbertheory/qm14.html
  • Integral Iwasawa theory of p-adic representations at non-ordinary primes, University at Buffalo, September 2014 https://www.google.com/calendar/render?eid=bjRuNmNjaXZtczk5dWYzY2hmMzEyYzAzMzggdWJtYXRoY2FsZW5kYXJAbQ&ctz=America/New_York&sf=true&output=xml
  • Siegel units and Euler systems, Princeton University/IAS Number Theory Seminar, February 2014 http://www.math.princeton.edu/events/seminars/princeton-universityias-number-theory-seminar/sigel-units-and-euler-systems
  • Lecture series on Siegel units and Euler systems for modular forms, National Taiwan University, January 2014
  • Iwasawa theory of abelian varieties at supersingular primes, CMS Winter meeting, Ottawa, December 2013 http://cms.math.ca/Events/winter13/abs/nt#al
  • Euler systems for modular forms over imaginary quadratic fields, McMaster University, December 2013 http://www.math.mcmaster.ca/images/ArithGeoSem_Antonio_Lei.pdf
  • On the finiteness of Selmer groups, Waikato University, May 2013 
  • Euler system for Rakin-Selberg convolution, Québec-Vermont Number Theory Seminar, January 2013  
  • Beilinson-Flach elements for the Rankin-Selberg convolution, CMS winter meeting, Montreal, December 2012 
  • p-adic L-functions for symmetric powers of modular forms of CM type, Québec-Vermont Number Theory Seminar,March 2012 
  • p-adic L-functions for symmetric powers of modular forms of CM type, Université Laval, February 2012 
  • Signed Selmer groups via p-adic Hodge theory, University of Nottingham, July 2011 
  • A reformulation of Kato’s main conjecture for modular forms, Algebraische Zahlentheorie, Oberwolfach, June 2011 
  • Elementary divisors and Iwasawa theory for modular forms, University of Warwick, May 2011 
  • Aspects of non-commutative Iwasawa theory at supersingular prime, Université Bordeaux 1, April 2011 
  • Elementary divisors and p-adic Hodge theory, Victorian Algebra Conference, November 2010 
  • Iwasawa theory for modular forms, University of Melbourne, October 2010 
  • Iwasawa theory for modular forms and Wach modules II, London Number Theory Seminar, May 2010 
  • Plus/Minus p-adic L-functions at a supersingular prime, l’Institut de Mathématiques de Jussieu, March 2010
  • Iwasawa theory for modular forms and Wach modules, Université Bordeaux 1, March 2010 
  • p-adic L-functions of modular forms at supersingular primes, University of Exeter, February 2010 
  • Iwasawa theory for modular forms at supersingular primes, University of Cambridge, October 2008 

 Visits

  • University of Texas Rio Grande Valley, April-May 2024
  • University College Dublin, February 2024
  • University of Texas at Austin, November 2023
  • MSRI-SLMath, April-May 2023
  • University College Dublin, September-December 2022
  • PIMS, mars 2020
  • Caltech, février 2020
  • University College Dublin, November 2018
  • PIMS, March 2018
  • University of Warwick, November 2016
  • University of Texas at Austin, October 2015
  • University of Regensburg, July 2015
  • National Taiwan University, January 2014
  • Waikato University, May 2013
  • Université Bordeaux 1, April 2011
  • University of Warwick, March-July 2011
  • Institut Henri Poincaré, January-March 2010

Supervision

PhD students

  • 2024-present: Juan Pablo Llerena (Ottawa)
  • 2024-present: Raiza Corpuz (Waikato, co-supervision with Daniel Delbourgo)
  • 2023-present: Flash Trevor Thompson (Ottawa)
  • 2022-present: Tam Nguyen (UBC, co-supervision with R. Sujatha)
  • 2021-present: Mihir Deo (Ottawa)
  • 2020-2024: Cédric Dion (Topics in Iwasawa theory, Laval)
  • 2021-2023: Luochen Zhao (Topics in p-adic analysis , Johns Hopkins, co-supervision with Yiannis Sakellaridis)
  • 2017-2021: A Arthur Bonkli Razafindrasoanaivolala (Sur les diviseurs milieux d’un entier , Laval, co-supervision with Jean-Marie De Koninck)
  • 2016-2020: Jesse Larone (Résolution de certaines équations diophantiennes et propriétés de certains polynômes , Laval, co-supervision with  Omar Kihel and Claude Levesque )
  • 2016: José Ibrahim Villanueva Gutierrez (exchange student from Bordeaux 1)
  • 2014-2018: Gautier Ponsinet (On the algebraic side of the Iwasawa theory of some non-ordinary Galois representations , Laval)

Master students

  • 2024-present: Naman Partap (Pune IISER)
  • 2023: Antoine Velut (exchange student from ENS Lyon)
  • 2021-2022: Anthony Doyon (Quelques approches p-adiques pour étudier les formes modulaires et leur fonction L , Laval)
  • 2019-2021: Fenomila Naivoarilala (Calcul de lambda invariant de la fonction L p-adique d’un caractère de Dirichlet sur un corps quartique , Laval)
  • 2018-2019: Cédric Dion (Fonction L p-adique d’une forme modulaire , Laval)
  • 2017-2019: David Ayotte (Relations entre le nombre de classes et les formes modulaires , Laval)

Postdocs

  • 2024-present: Muammad Manji (co-supervision with Giovanni Rosso)
  • 2023-present: Erman Isik (Fields-Ottawa)
  • 2022-2023: Anwesh Ray (CRM-Simons)
  • 2021-2023: Katharina Muller (Laval)
  • 2021-2023: Jiacheng Xia (CRM-Laval)
  • 2020-2021: Jishnu Ray (CRM)
  • 2020: Debanjana Kundu (CRM)
  • 2018-2020: Antonio Cauchi (CRM-Laval)
  • 2017-2019: Guhan Venkat (CRM-Laval)

Undergraduate research projects

  • 2024: Christopher Komban Subhas (Ottawa-MITACS), Ying Xue (Ottawa-WISE), Carolina Scassa (Ottawa)
  • 2023: Naman Pratap, Rik Sarkar (Ottawa-MITACS)
  • 2022: Sai Sanjeev Balakrishnan (Laval-MITACS), Jeanne Laflamme (Laval)
  • 2021: Anthony Doyon (Laval)
  • 2020: Maxime Cinq-Mars, Anthony Doyon, Antoine Poulin, Michaël Rioux (Laval)
  • 2019: Hugues Bellemare, Anthony Doyon (Laval), Sarthak Gupta (Laval-MITACS)
  • 2018: Hugues Bellemare, Antoine Poulin, Mathieu Trudelle (Laval), Arihant Jain (Laval-MITACS)
  • 2017: Cédric Dion (Laval)
  • 2016: David Ayotte (Laval)
  • 2015: David Ayotte, Jean-Christophe Rondy-Turcotte (Laval)

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, which organizes regular seminars and other academic activities. We are also members of the CRM and the Fields Institute, so you will have the opportunity to participate in many of their activities (seminars, workshops, thematic programmes, etc). Several scholarships are available to Canadian and international students and postdoctoral fellows. More information can be found here.

Research Projects

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.

Welcome

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 a faculty member 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 20 pages long)!

Previously, I have been a faculty member at Laval University (2014-2022), and 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.

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.

Email address
antonio (DOT) lei (AT) uottawa (DOT) ca

Office
STEM 658

Postal address
Department of Mathematics and Statistics
STEM Complex
150 Louis-Pasteur Pvt
Ottawa, ON
Canada K1N 6N5

Publications

Preprints

  • Lei A, Mastella L, Zhao L, “On the structure of the Bloch–Kato Selmer groups of modular forms over anticyclotomic Zp-towers”
  • Isik E, Lei A, “The growth of Tate-Shafarevich groups of p-supersingular elliptic curves over anticyclotomic Zp-extensions at inert primes”
  • Lei A, “Artin formalism for p-adic L-functions of modular forms at non-ordinary primes”
  • Gajek-Leonard R, Hatley J, Kundu D, Lei A, “On a conjecture of Mazur predicting the growth of Mordell–Weil ranks in Zp-extensions”
  • Burungale A, Buyukboduk K, Lei A, “Anticyclotomic Iwasawa theory of abelian varieties of GL2-type at non-ordinary primes II”
  • Lei A, Mueller K, “On towers of isogeny graphs with full level structure”

Articles accepted for publication

  • Kundu D, Lei A, Sprung F, “Studying Hilbert’s 10th problem via explicit elliptic curves”, Mathematische Annalen
  • Balakrishnan S S, Lei A, Palvannan B “On computing the Newton polygons of plus and minus p-adic L-functions”, Involve – a Journal of Mathematics
  • Buyukboduk K, Lei A, “Iwasawa theory of twists of elliptic modular forms over imaginary quadratic fields at inert primes”, Mémoires de la société mathématique de France
  • Kundu D, Lei A, “Non-vanishing modulo p of Hecke L-values over imaginary quadratic fields”, Israel Journal of Mathematics

Published Articles

  1. Lei A, Mueller K, “On ordinary isogeny graphs with level structure”, Expositiones Mathematicae, Volume 42, Issue 5, 2024, 125589  
  2. Doyon A, Lei A “Lambda-invariants of Mazur–Tate elements attached to Ramanujan’s tau function and congruences with Eisenstein series”, Research in Number Theory, Volume 10, article number 53, (2024)
  3. Dion C, Lei A, Ray A, Vallières D, “On the distribution of Iwasawa invariants associated to multigraphs”, Nagoya Mathematical Journal, Volume 253 , 2024 , 48 – 90
  4. Lei A, Zhao L, “On the BDP Iwasawa main conjecture for modular forms”, Manuscripta Mathematica, Volume 173, pages 867–888, (2024)
  5. Lei A, Mueller K, “On the zeta functions of supersingular isogeny graphs and modular curves”, Archiv der Mathematik, Volume 122, pages 285–294, (2024)  
  6. Burungale A, Buyukboduk K, Lei A, “Anticyclotomic Iwasawa theory of abelian varieties of GL2-type at non-ordinary primes”, Advances in Mathematics, Volume 439, March 2024, 109465
  7. Lei A, Mueller K, Xia J, “On the Iwasawa invariants of BDP Selmer groups and BDP p-adic L-functions”, Forum Mathematicum, 36 (1) 2024, 153-172.   
  8. Lei A, “A remark on the characteristic elements of anticyclotomic Selmer groups of elliptic curves with complex multiplication at supersingular primes”, The Quarterly Journal of Mathematics, 74 (4), 2023, 1435–1487
  9. Lei A, Lim M F, Mueller K, “Asymptotic formula for Tate-Shafarevich groups of p-supersingular elliptic curves over anticyclotomic extensions”, Advances in Mathematics, volume 434, Article number: 109320 (2023)
  10. Kundu D, Lei A, “Growth of p-parts of ideal class groups and fine Selmer groups in Zq-extensions with p≠q”, Acta Arithmetica, 207 (2023), 297-313
  11. Lei A, Vallières D, “The non-ℓ-part of the number of spanning trees in abelian ℓ-towers of multigraphs”, Research in Number Theory, 9, 18 (2023)
  12. Hatley J, Lei A, “The vanishing of anticyclotomic mu-invariants for non-ordinary modular forms “, Comptes Rendus Mathématique, 361 (2023), 65-72
  13. Hatley J, Kundu D, Lei A, Ray J, “Control theorems for fine Selmer groups, and duality of fine Selmer groups attached to modular forms”, the Ramanujan Journal, 60, 237–258 (2023)
  14. Lei A, “Algebraic structure and characteristic ideals of fine Mordell–Weil groups and plus/minus Mordell–Weil groups”, Mathematische Zeitschrift, volume 303, Article number: 14 (2023)
  15. Lei A, Ray J, “Iwasawa theory of automorphic representations of GL2n at non-ordinary primes”, Research in Mathematical Sciences, 10, 1, 2023
  16. Buyukboduk K, Lei A, “Iwasawa theory for GL2×ResK/QGL1“, Annales mathématiques du Québec (special volume in honour of Bernadette Perrin-Riou), 46 (2), 2022, 347-418
  17. Lei A, Palvannan B, “Codimension two cycles in Iwasawa theory and tensor product of Hida families”, Mathematische Annalen, 383, no.1-2, 2022, 39–75
  18. Cauchi A, Lei A “On the analogue of Mazur-Tate type conjectures in the Rankin-Selberg setting”, 66 (2), 2022, 571-630, Publicacions Matemàtiques
  19. Doyon A, Lei A “Congruences between Ramanujan’s tau function and elliptic curves, and Mazur–Tate elements at additive primes”, The Ramanujan Journal, 58, 2022, 505–522
  20. Lei A, Lim M F, ” On fine Selmer groups and signed Selmer groups of elliptic modular forms”, Bulletin of the Australian Mathematical Society, 105 (3), 2022, 419 – 430
  21. Kundu D, Lei A, Ray A, “Arithmetic Statistics and noncommutative Iwasawa Theory”, Documenta Mathematica, 27, 2022, 89-149
  22. Lei A, Lim M F, “Mordell-Weil ranks and Tate-Shafarevich groups of elliptic curves with mixed-reduction type over cyclotomic extensions”, International Journal of Number Theory, 18 (2), 2022, 303-330
  23. Hatley J, Lei A, Vigni S, “Λ-submodules of finite index of anticyclotomic plus and minus Selmer groups of elliptic curves”, Manuscripta Mathematica, 167, 2022, 589-612
  24. Lei A, Lim M F, “Akashi series and Euler characteristics of signed Selmer groups of elliptic curves with semistable reduction at primes above p”, Journal de théorie des nombres de Bordeaux (proceeding of Iwasawa 2019), 33 (3.2), 2021, 997-1019
  25. Buyukboduk K, Lei A, “Interpolation of Generalized Heegner Cycles in Coleman Families”, Journal of London Mathematical Society, 104 (4), 2021, 1682-1716
  26. Buyukboduk K, Lei A, “Iwasawa theory of elliptic modular forms over imaginary quadratic fields at non-ordinary primes”, International Mathematics Research Notices,  2021 (14), 2021, 10654–10730  
  27. Hatley J, Lei A, “Comparing anticyclotomic Selmer groups of positive coranks for congruent modular forms — Part II”, Journal of Number Theory, 229, 2021, 342–363
  28. Lei A, Sujatha R, “On fine Selmer groups and the greatest common divisor of signed and chromatic p-adic L-functions”, Proceedings of the American Mathematical Society, 149 (8), 2021, 3235–3243
  29. Buyukboduk K, Lei A, Venkat G, “Iwasawa theory for symmetric square of non-p-ordinary eigenforms”, Documenta Mathematica, 26, 2021, 1–63
  30. Lei A, Sujatha R, “On Selmer groups in the supersingular reduction case”, Tokyo Journal of Mathematics, 43 (2), 2020, 455–479
  31. Bellemare H, Lei A, “Explicit uniformizers for certain totally ramified extensions of the field of p-adic numbers”, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 90 (1), 2020, 73–83
  32. Lei A, Poulin A, “On certain probabilistic properties of polynomials over the ring of p-adic integers”, American Mathematical Monthly, 127 (6), 2020, 519–529
  33. Delbourgo D, Lei A, “Heegner cycles and congruences between anticyclotomic p-adic L-functions over CM-extensions”, The New York Journal of Mathematics, 26, 2020, 496–525
  34. Lei A, Sprung F, “Ranks of elliptic curves over Z_p^2-extensions”, Israel Journal of Mathematics, 236 (1), 2020, 183–206  
  35. Buyukboduk K, Lei A, “Functional Equation for p-adic Rankin-Selberg L-functions”, Annales mathématiques du Québec, 44 (1), 2020, 9–25  
  36. Lei A, Ponsinet G, “On the Mordell-Weil ranks of supersingular abelian varieties in cyclotomic extensions”, Proceedings of the American Mathematical Society Series B, 7, 2020, 1-16
  37. Buyukboduk K, Lei A, “Rank–two Euler systems for symmetric squares”, Transactions of the American Mathematical Society,  372 (12), 2019, 8605-8619  
  38. Hatley J, Lei A, “Comparing anticyclotomic Selmer groups of positive coranks for congruent modular forms”, Mathematical Research Letters, 26 (4), 2019, 1115-1144  
  39. Lei A, Palvannan B, “Codimension two cycles in Iwasawa theory and elliptic curves with supersingular reduction”, Forum of Mathematics, Sigma, 7, 2019, e25  
  40. Hatley J, Lei A, “Arithmetic properties of signed Selmer groups at non-ordinary primes”, Annales de l’Institut Fourier, 69 (3) 2019, 1259-1294  
  41. Buyukboduk K, Lei A, Loeffler D, Venkat G, “Iwasawa theory for Rankin–Selberg products of p-non-ordinary eigenforms”, Algebra & Number Theory, 13(4), 2019, 901-941  
  42. Lei A, Loeffler D, Zerbes S, “Euler systems for Hilbert modular surfaces”, Forum of Mathematics, Sigma, 6, 2018 , e23  
  43. Delbourgo D, Lei A, “Congruences modulo p between rho-twisted Hasse-Weil L-values”, Transactions of the American Mathematical Society, 370 (11), 2018, 8047-8080   
  44. Buyukboduk K, Lei A, “Anticyclotomic p-ordinary Iwasawa theory of elliptic modular forms”, Forum Mathematicum, 30 (4), 2018, 887–913  
  45. Lei A, “Bounds on the Tamagawa numbers of a crystalline representation over towers of cyclotomic extensions”, Tohoku Mathematical Journal, 69 (4), 2017, 497-524  
  46. Dion, C, Lei, A “Plus and minus logarithms and Amice transform”, Comptes Rendus Mathématique, 355 (9), 2017, 942-948  
  47. Lei A, “Estimating class numbers over metabelian extensions”, Acta Arithmetica, 180 (4), 2017, 347-364  
  48. Lei A, Loeffler D, Zerbes S, “On the asymptotic growth of Bloch-Kato-Shafarevich-Tate groups of modular forms over cyclotomic extensions”, Canadian Journal of Mathematics, 69 (4), 2017, 826-850  
  49. Buyukboduk K, Lei A, “Integral Iwasawa Theory of Galois Representations for non-ordinary primes”, Mathematische Zeitschrift, 286 (1), 2017, 361-398  
  50. Lei A, Ponsinet G, “Functional equations for multi-signed Selmer groups”, Annales mathématiques du Québec, 41 (1), 2017, 155-167  
  51. Delbourgo D, Lei A, “Estimating the growth in Mordell-Weil ranks and Shafarevich-Tate groups over Lie extensions”, The Ramanujan Journal, 43 (1), 2017, 29-68  
  52. Ayotte D, Lei A, Rondy-Turcotte J-C, “On the parity of supersingular Weil polynomials”, Archiv der Mathematik, 106 (4), 2016, 345-353  
  53. Delbourgo D, Lei A, “Non-commutative Iwasawa theory for elliptic curves with multiplicative reduction”, Mathematical Proceedings of the Cambridge Philosophical Society, 160 (1), 2016, 11-38  http://dx.doi.org/10.1017/S0305004115000535
  54. Buyukboduk K, Lei A, “Coleman-adapted Rubin-Stark Kolyvagin systems and supersingular Iwasawa theory of CM abelian varieties”, Proceedings of the London Mathematical Society, 111 (6), 2015, 1338-1378  http://dx.doi.org/10.1112/plms/pdv054
  55. Delbourgo D, Lei A, “Transition formulae for ranks of abelian varieties”, Rocky Mountain Journal of Mathematics, 45 (6), 2015, 1807-1838  http://projecteuclid.org/euclid.rmjm/1411945718
  56. Lei A, Loeffler D, Zerbes S, “Euler systems for modular forms over imaginary quadratic fields”, Compositio Mathematica, 151 (9), 2015, 1585-1625  http://dx.doi.org/10.1112/S0010437X14008033
  57. Harron R, Lei A, “Iwasawa theory for symmetric powers of CM modular forms at non-ordinary primes”, Journal de théorie des nombres de Bordeaux, 26(3), 2014, 673-707  
  58. Lei A, “Factorisation of two-variable p-adic L-functions”, Canadian Mathematical Bulletin, 57(4), 2014, 845-852  http://dx.doi.org/10.4153/CMB-2013-044-2
  59. Lei A, Loeffler D, Zerbes S, “Euler systems for Rankin-Selberg convolutions of modular forms”, Annals of Mathematics, 180(2), 2014, 653-771  https://doi.org/10.4007/annals.2014.180.2.6
  60. Lei A, Loeffler D, Zerbes S, “Critical Slope p-adic L-functions of CM Modular Forms”, Israel Journal of Mathematics, 198(1), 2013, 261-282  
  61. Lei A, “Non-commutative p-adic L-functions for Supersingular Primes”, International Journal of Number Theory, 8(8), 2012, 1813-1830  
  62. Lei A, Zerbes S, “Signed Selmer Groups over p-adic Lie Extensions”, Journal de théorie des nombres de Bordeaux, 24(2), 2012, 377-403  
  63. Lei A, “Iwasawa Theory for the Symmetric Square of a CM Modular Forms at Inert Primes”, Glasgow Mathematical Journal, 54(2), 2012, 241-259  
  64. Lei A, Loeffler D, Zerbes S, “Coleman Maps and p-adic Regulator”, Algebra and Number Theory, 5(8), 2011, 1095-1131  
  65. Lei A, “Iwasawa Theory for Modular Forms at Supersingular Primes”, Compositio Mathematica, 147(3), 2011, 803-838 acroread 
  66. Lei A, Loeffler D, Zerbes S, “Wach Modules and Iwasawa Theory for Modular Forms”, Asian Journal of Mathematics, 14(4), 2010, 475-528  
  67. Lei A, “Coleman Maps for Modular Forms at Supersingular Primes over Lubin-Tate Extensions”, Journal of Number Theory, 130(10), 2010, 2293-2307  

Other articles

  • Lei A, “Iwasawa Theory for Modular Forms at Supersingular Primes”, PhD thesis, University of Cambridge, 2010 
  • Lei A, “A reformulation of Kato’s main conjecture for modular forms”, Oberwolfach Reports, 8(2), 2011, 1754-1756
  • Lei A, “Local Class Field Theory”, essay for the Part III Mathematical Tripos at University of Cambridge, 2007 
  • Lei A, “Efficient Parametrisation of Rational Normal Curves”, report on a summer research project, 2006 
  • Lei A, “What so perfect about perfect numbers?”, Eureka, 58  
  • Lei A, “Brauer groups”, entry for the Yeats Essay Prize at Trinity College Cambridge, 2005 

Research

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.