About
This is an undergraduate student seminar on geometry and topology.
This semester, our main theme will be modular curves, with Diamond and Shurman's A first course in modular forms (from Chapter 6 on) as a main reference.
Other useful resources include Milne's notes as well as a recent introductory, motivating
article on modular forms by Cepelewicz of Quanta Magazine.
There will also be two colloquiums in a row on the topic in October.
Still, Chao Li's recent survey is recommended.
We meet on Thursdays 9–11 am in M5024.
Here is a course description for those who wish to register for credit.
Talks
Sep 29, '23,
Yifan Wu,
An overview of modular forms and modular curves
Oct 12, '23,
Rongming Yin,
An introduction to Jacobian varieties of algebraic curves
Oct 12, '23 (Colloquium, 4:30–5:30 pm, M1001),
Don Zagier (Max Planck Institute for Mathematics),
Modular forms and quantum modular forms
Modular forms are one of the most beautiful and fundamental notions in modern number theory and
play a role in almost every domain of pure mathematics and physics. The famous specialist
Martin Eichler once said, "There are five fundamental operations of mathematics:
addition, subtraction, multiplication, division, and modular forms."
In the first talk I will explain this notion and present some highlights of the theory and its applications.
The last part, about the relatively new notion of "quantum modular forms" and its variants, is based on
joint work with Stavros Garoufalidis (SUSTech) during the last several years in which
many beautiful interconnections between number theory and 3-dimensional topology, and in particular the
so-called quantum invariants of 3-dimensional topology, were found.
Oct 19, '23,
Zian Zhao,
Double coset maps between space of modular forms and modular Jacobians
Nov 2, '23,
Jinghua Xi,
Modular Jacobians and Hecke operators
Nov 2, '23 (Colloquium, 4:30–5:30 pm, M1001),
Ruochuan Liu (Peking University),
Congruences between modular forms and geometry of the eigencurve
The study of congruences between modular forms is a classic and vital subject in number theory.
Hida was the first to establish the theory of $p$-adic modular forms with slope zero; Hida theory
is a core tool in Wiles' proof of the Iwasawa main conjecture, and also one of the main tools in
proving Fermat's Last Theorem. Coleman extended Hida's theory to $p$-adic modular forms of
general finite slopes and, together with Mazur, constructed the eigencurve. The geometry of the
eigencurve is essential for understanding the congruence properties of modular forms, yet our
knowledge about geometry of the eigencurve is very limited. In a recent joint work with Nha Xuan
Truong, Xiao Liang and Zhao Bin, we prove the finiteness of irreducible components of the
eigencurve under a generic condition.
Nov 9, '23,
Jinghua Xi,
Algebraic eigenvalues
Nov 16, '23,
Yifan Wu,
More geometry and examples
Nov 23, '23,
Rongming Yin,
Eigenforms and Abelian varieties
Nov 30, '23,
Rongming Yin,
Algebraic curves and their function fields
Dec 7, '23,
Rongming Yin,
Isogenies between elliptic curves
Dec 14, '23,
Yifan Wu,
Function fields of modular curves
Dec 21, '23 (Science Lecture, 4:30–5:30 pm, CS1142),
Ruochuan Liu (Peking University),
An introduction to elliptic curves and modular forms
In this lecture, I will introduce modular forms and elliptic curves, two important notions from number theory, as well as their close relationship.
In particular, I will explain how they played a key role in the proof of Fermat's Last Theorem.
Moreover, I will discuss applications of modular forms and elliptic curves to other disciplines, such as cryptography and quantum computing.
Dec 28, '23,
Yifan Wu,
Function fields of modular curves, cont'd