MATH 568

Math 568 - Spring 2022, Topics in Number Theory: Elliptic curves, modular forms, arithmetic statistics.

Instructor:

Dinesh Thakur,

Hylan 1013, 275-7767, dinesh.thakur@rochester.edu

Class: MW 9.00-10.15 a.m., Hylan 206

Office hours: M 10.30 a.m. -11.30 am, W 1-2 pm or by appt.

Personal room zoom id : 566 385 6457

Textbook:

The textbook is 2nd edition of Silverman’s GTM: The arithmetic of elliptic curves, which covers the basics.

The best starting book (for undergraduates especially) is Silverman-Tate Springer-Verlag UTM: Rational points on elliptic curves.

There is also a book by J. S. Milne on elliptic curves and modular forms, which is freely available from his website.

Syllabus:

Elliptic curves, which are genus one curves as well as algebraic groups, are very important in number theory and are actively pursued theoretically, as well as in their practical uses in cryptology, factoring etc. Their theory over complex number is linked to rich theory of doubly periodic functions as well as modular functions. Their theory over number fields, with connections to Galois representations and modular forms was fundamental in Ribet-Wiles solution of Fermat problem.

We will study their main aspects over finite fields, local fields and global fields in detail.

We will also discuss some hard theorems, open problems, conjectures, recent arithmetic statistical results, and applications at the level of examples, heuristics, analogy, intuition.

I will try to adapt to student’s background and not enforce any strict {\bf prerequisites}. If you have had courses in complex analysis, algebraic number theory, or geometry, some things will seem familiar and feel easier!

We will also study some recent papers depending on the interest of students. With active participation from at least some of the students, I can cover some recent developments in addition to the foundations of the subject.

Exam:

There is no exam. Attendance is expected. Various projects, problems and homeworks will be suggested, but not required.