2022/2023
Research Seminar "Gauss Class Number Problem"
Type:
Optional course (faculty)
Delivered by:
Faculty of Mathematics
When:
1, 2 module
Open to:
students of all HSE University campuses
Language:
English
ECTS credits:
3
Contact hours:
32
Course Syllabus
Abstract
Set $p(x)=x^2+x+41$. It is well-known that all the values $p(0),p(1),\ldots,p(39)$ are prime. It is also known that the number $\exp(\sqrt{163}\pi)$ is very close to an integer. These facts are connected to the uniqueness of prime factorization in the ring of integers of the field $\mathbb Q(\sqrt{-163})$. The problem of describing all the imaginary quadratic fields with this property was first stated by Gauss, who also listed 9 discriminants of such fields. The question of existence of the 10th discriminant remained open until the middle of 20th century. In this course, we are going to discuss several different solutions of this problem that rely on various techniques of number theory, from modular forms and class field theory to methods of transcendent number theory. In context of this problem, we will also discuss a number of fundamental conjectures, including the Riemann hypothesis and Birch and Swinnerton-Dyer conjecture.