Research Seminar "Modular Forms and Number Theory 2"

2025/2026

Academic Year

ENG

Instruction in English

3

ECTS credits

Delivered at:

Faculty of Mathematics

Course type:

Elective course

When:

1 year, 3, 4 module

Instructors

Bolbachan, Vasily

Kalmynin, Alexander B.

Course Syllabus

Abstract

The goal of this seminar is to introduce interested students to various aspects of number theory, both algebraic and analytic. It is expected that most of the talks will be given by students themselves. The connecting theme this year is modular forms. Modular forms are a classical mathematical object that first arose in the context of the theory of elliptic functions and Riemann surfaces. As this field has developed, it has turned out that modular forms manifest themselves in a wide variety of areas of mathematics. Many very striking applications of the theory of modular forms are related to number theory. For example, the connection between theta functions and Eisenstein series can be used to prove formulas for the number of representations of a natural number by sums of squares. Parabolic forms associated with elliptic curves help in solving a wide class of Diophantine equations. The modularity of the Dedekind eta function allows one to prove the Hardy-Ramanujan formula for the number of partitions. Finally, studying the values ​​of the modular j-invariant allows one to solve the Gauss class number problem for imaginary quadratic fields. We welcome talks on any topics related to number theory and modular forms.