2025/2026
Научно-исследовательский семинар "Модулярные формы и теория чисел 1"
Статус:
Дисциплина общефакультетского пула
Кто читает:
Факультет математики
Где читается:
Факультет математики
Когда читается:
1, 2 модуль
Охват аудитории:
для всех кампусов НИУ ВШЭ
Язык:
английский
Кредиты:
3
Контактные часы:
30
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.
Learning Objectives
- Овладеть базовыми понятиями теории чисел. Получить навык подготовки доклада по теме теории чисел.
Expected Learning Outcomes
- Уметь вычислять с операторами Гекке
- Уметь вычислять род модулярной кривой для классических модулярных кривых
- Student able to make his own talk connected with modular forms and number theory
Course Contents
- Hyperbolic plane, the group SL(2,Z) and its fundamental domain
- The Eisenstein Series
- Hecke operators
- Modular curves
- Проведение доклада
Assessment Elements
- Проведение доклада на семинареТребуется выступить на семинаре с докладом по теме модулярных форм
- ЭкзаменЭкзамен для тех студентов, кто не сделал доклад
Bibliography
Recommended Core Bibliography
- Geometric modular forms and elliptic curves, Hida, H., 2012
- Introduction to elliptic curves and modular forms, Koblitz, N., 1993
- J. S. Milne. (2009). Algebraic Number Theory. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.CB7FD32F
- Zagier, D., & Skoruppa, N.-P. (1988). Jacobi forms and a certain space of modular forms. Zagier, Don; Skoruppa, Nils-Peter: Inventiones Mathematicae. 94 1988. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsdzs&AN=edsdzs.GDZPPN002105705
- Курс арифметики, Серр, Ж.-П., 1972
Recommended Additional Bibliography
- Elementary dirichlet series and modular forms, Shimura, G., 2007