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

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

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 
• CRM Thematic semester: Number Theory – Cohomology in Arithmetic, Fall 2020
• 15th Conference of the Canadian Number Theory Association 
• Québec-Maine Number Theory Conference 
• Algebra and Number Theory Seminars at Laval University
• Study groups at Laval University 
• CMS winter meeting 2015 in Montreal, Algebraic Number Theory session 

Invited Talks

• 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
• 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
• Semi-ordinary Iwasawa theory for Rankin-Selberg products, L-values and Iwasawa theory, NCTS online conference, November 2020
• Bounded Euler systems for Rankin-Selberg products of modular forms, UBC Number Theory Seminar, March 2020
• Bounded Euler systems for Rankin-Selberg products of modular forms, Caltech Number Theory Seminar, February 2020
• Congruences of anticyclotomic p-adic L-functions, Maine-Québec Number Theory Conference, October 2019 
• 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 
• Lecture series on Iwasawa theory for elliptic curves with supersingular reduction, PIMS, March 2018 
• Mordell-Weil ranks of elliptic curves over a tower of extensions, University of Pennsylvania, January 2018 
• Second Chern Classes for Supersingular Elliptic Curves, Maine-Québec Number Theory Conference, October 2017 
• Mini course on cyclotomic units, Introductory Workshop on Euler Systems and Special Values of L-functions, Bernoulli Center, Lausanne, August 2017 
• Beilinson-Flach elements and finiteness of sha, Recent Developments on Elliptic Curves, Clay Mathematics Institute workshop, Oxford, September 2016 
• 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 
• 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 
• Iwasawa theory of elliptic curves over Zp^2-extensions, Joint Mathematics Meetings, San Antonio, January 2015 
• Universal norm of crystalline classes, CMS winter meeting, Hamilton, December 2014 
• Integral Iwasawa theory of p-adic representations at non-ordinary primes, AMS Fall Sectional Meeting, Halifax, October 2014 
• Iwasawa theory of elliptic curves over Zp^2-extensions, Québec-Maine Number Theory Conference, September 2014 
• Integral Iwasawa theory of p-adic representations at non-ordinary primes, University at Buffalo, September 2014 
• Siegel units and Euler systems, Princeton University/IAS Number Theory Seminar, February 2014 
• 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 
• Euler systems for modular forms over imaginary quadratic fields, McMaster University, December 2013 
• 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 

PhD students

• 2023-present: Flash Trevor Thompson (Ottawa)
• 2022-present: Tam Nguyen (UBC, co-supervision with R. Sujatha)
• 2021-present: Mihir Deo (Ottawa)
  
• 2020-present: Cédric Dion (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

• 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

• 2023-present: Erman Isik (Ottawa)
• 2022-2023: Anwesh Ray (CRM-Simons)
• 2021-2023: Katharina Muller (Laval)
• 2021-2023: Jiacheng Xia (Laval)
• 2020-2021: Jishnu Ray (CRM)
  
• 2020: Debanjana Kundu (CRM)
  
• 2018-2020: Antonio Cauchi (Laval)
• 2017-2019: Guhan Venkat (Laval)

Undergraduate research projects

• 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.

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 20 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.

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

Preprints

• 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 ”
• Lei A, Mueller K, “On ordinary isogeny graphs with level structure”
• Doyon A, Lei A “Lambda-invariants of Mazur–Tate elements attached to Ramanujan’s tau function and congruences with Eisenstein series”

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. 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
2. Lei A, Zhao L, “On the BDP Iwasawa main conjecture for modular forms”, Manuscripta Mathematica, Volume 173, pages 867–888, (2024)
3. Lei A, Mueller K, “On the zeta functions of supersingular isogeny graphs and modular curves”, Archiv der Mathematik, Volume 122, pages 285–294, (2024)  
4. 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
5. 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.   
6. 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
7. 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)
8. Kundu D, Lei A, “Growth of p-parts of ideal class groups and fine Selmer groups in Z_q-extensions with p≠q”, Acta Arithmetica, 207 (2023), 297-313
9. 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)
10. Hatley J, Lei A, “The vanishing of anticyclotomic mu-invariants for non-ordinary modular forms “, Comptes Rendus Mathématique, 361 (2023), 65-72
11. 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)
12. 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)
13. Lei A, Ray J, “Iwasawa theory of automorphic representations of GL_2n at non-ordinary primes”, Research in Mathematical Sciences, 10, 1, 2023
14. Buyukboduk K, Lei A, “Iwasawa theory for GL₂×Res_K/QGL₁“, Annales mathématiques du Québec (special volume in honour of Bernadette Perrin-Riou), 46 (2), 2022, 347-418
15. 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
16. Cauchi A, Lei A “On the analogue of Mazur-Tate type conjectures in the Rankin-Selberg setting”, 66 (2), 2022, 571-630, Publicacions Matemàtiques
17. 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
18. 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
19. Kundu D, Lei A, Ray A, “Arithmetic Statistics and noncommutative Iwasawa Theory”, Documenta Mathematica, 27, 2022, 89-149
20. 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
21. 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
22. 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
23. Buyukboduk K, Lei A, “Interpolation of Generalized Heegner Cycles in Coleman Families”, Journal of London Mathematical Society, 104 (4), 2021, 1682-1716
24. 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  
25. 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
    
26. 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
27. Buyukboduk K, Lei A, Venkat G, “Iwasawa theory for symmetric square of non-p-ordinary eigenforms”, Documenta Mathematica, 26, 2021, 1–63
28. Lei A, Sujatha R, “On Selmer groups in the supersingular reduction case”, Tokyo Journal of Mathematics, 43 (2), 2020, 455–479
29. 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
30. 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
31. 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
32. Lei A, Sprung F, “Ranks of elliptic curves over Z_p^2-extensions”, Israel Journal of Mathematics, 236 (1), 2020, 183–206  
33. Buyukboduk K, Lei A, “Functional Equation for p-adic Rankin-Selberg L-functions”, Annales mathématiques du Québec, 44 (1), 2020, 9–25  
34. 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
35. Buyukboduk K, Lei A, “Rank–two Euler systems for symmetric squares”, Transactions of the American Mathematical Society,  372 (12), 2019, 8605-8619  
36. Hatley J, Lei A, “Comparing anticyclotomic Selmer groups of positive coranks for congruent modular forms”, Mathematical Research Letters, 26 (4), 2019, 1115-1144  
37. Lei A, Palvannan B, “Codimension two cycles in Iwasawa theory and elliptic curves with supersingular reduction”, Forum of Mathematics, Sigma, 7, 2019, e25  
38. Hatley J, Lei A, “Arithmetic properties of signed Selmer groups at non-ordinary primes”, Annales de l’Institut Fourier, 69 (3) 2019, 1259-1294  
39. 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  
40. Lei A, Loeffler D, Zerbes S, “Euler systems for Hilbert modular surfaces”, Forum of Mathematics, Sigma, 6, 2018 , e23  
41. 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   
42. Buyukboduk K, Lei A, “Anticyclotomic p-ordinary Iwasawa theory of elliptic modular forms”, Forum Mathematicum, 30 (4), 2018, 887–913  
43. Lei A, “Bounds on the Tamagawa numbers of a crystalline representation over towers of cyclotomic extensions”, Tohoku Mathematical Journal, 69 (4), 2017, 497-524  
44. Dion, C, Lei, A “Plus and minus logarithms and Amice transform”, Comptes Rendus Mathématique, 355 (9), 2017, 942-948  
45. Lei A, “Estimating class numbers over metabelian extensions”, Acta Arithmetica, 180 (4), 2017, 347-364  
46. 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  
47. Buyukboduk K, Lei A, “Integral Iwasawa Theory of Galois Representations for non-ordinary primes”, Mathematische Zeitschrift, 286 (1), 2017, 361-398  
48. Lei A, Ponsinet G, “Functional equations for multi-signed Selmer groups”, Annales mathématiques du Québec, 41 (1), 2017, 155-167  
49. 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  
50. Ayotte D, Lei A, Rondy-Turcotte J-C, “On the parity of supersingular Weil polynomials”, Archiv der Mathematik, 106 (4), 2016, 345-353  
51. 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  
52. 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  
53. Delbourgo D, Lei A, “Transition formulae for ranks of abelian varieties”, Rocky Mountain Journal of Mathematics, 45 (6), 2015, 1807-1838  
54. Lei A, Loeffler D, Zerbes S, “Euler systems for modular forms over imaginary quadratic fields”, Compositio Mathematica, 151 (9), 2015, 1585-1625  
55. 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  
56. Lei A, “Factorisation of two-variable p-adic L-functions”, Canadian Mathematical Bulletin, 57(4), 2014, 845-852  
57. Lei A, Loeffler D, Zerbes S, “Euler systems for Rankin-Selberg convolutions of modular forms”, Annals of Mathematics, 180(2), 2014, 653-771  
58. 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  
59. Lei A, “Non-commutative p-adic L-functions for Supersingular Primes”, International Journal of Number Theory, 8(8), 2012, 1813-1830  
60. Lei A, Zerbes S, “Signed Selmer Groups over p-adic Lie Extensions”, Journal de théorie des nombres de Bordeaux, 24(2), 2012, 377-403  
61. Lei A, “Iwasawa Theory for the Symmetric Square of a CM Modular Forms at Inert Primes”, Glasgow Mathematical Journal, 54(2), 2012, 241-259  
62. Lei A, Loeffler D, Zerbes S, “Coleman Maps and p-adic Regulator”, Algebra and Number Theory, 5(8), 2011, 1095-1131  
63. Lei A, “Iwasawa Theory for Modular Forms at Supersingular Primes”, Compositio Mathematica, 147(3), 2011, 803-838  
64. Lei A, Loeffler D, Zerbes S, “Wach Modules and Iwasawa Theory for Modular Forms”, Asian Journal of Mathematics, 14(4), 2010, 475-528  
65. 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 

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.