2025/2026
Мотивы и L-функции
Статус:
Дисциплина общефакультетского пула
Кто читает:
Факультет математики
Где читается:
Факультет математики
Когда читается:
1, 2 модуль
Охват аудитории:
для всех кампусов НИУ ВШЭ
Преподаватели:
Гайсин Ильдар Маратович
Язык:
английский
Кредиты:
3
Контактные часы:
30
Course Syllabus
Abstract
Grotendieck originally introduced the idea of a ""motif"" in a letter to Serre in the early 1960s. Motives are interesting in their own right and much is still unknown about them, as indicated for instance by Grothendieck's ""standard conjectures"".
On the other hand an L-function is a holomorphic function which encodes certain properties of an arithmetic-geometric object (e.g. a finite type scheme over a number field or a motive). The classical Riemann-zeta function is perhaps the earliest example of an L-function.
In this course we will discuss Deligne's conjecture (and it's variants e.g. Bloch-Kato) which is a statement relating the values of L-functions at integer points (so called special values) to the ""regulator"" of the motive. This is a vast generalization of the classical class number formula.
Learning Objectives
- The Goal of the course is to understand the definition given by Voevodsky’s seminal work of the derived category of mixed motives. In addition we will understand how motivic cohomology relates to the various (co)homology theories given by algebraic cycles.
Expected Learning Outcomes
- The development of Grothendieck's theory of pure motives will lead to a deeper understanding of the intrinsic structure of algebraic varieties across different cohomological frameworks. It is expected to provide powerful new methods for addressing long-standing problems in number theory and algebraic geometry by unifying diverse approaches into a coherent categorical setting.
- Proving Beilinson's conjectures would establish precise relationships between special values of L-functions and arithmetic data, significantly advancing our understanding of the interplay between analysis, algebra, and geometry in number theory.
- Advancing Algebraic K-theory will yield deeper insights into the structure of algebraic varieties, improve computational techniques for classifying vector bundles, and strengthen connections between algebraic geometry, topology, and number theory
Course Contents
- Grothendieck's theory of pure motives
- Voevodsky's approach via transfers for motivic complexes
- Beilinson's conjectures around special values of L-functions
- Algebraic K-theory
Bibliography
Recommended Core Bibliography
- Algebraic K-theory. Vol.2: "Classical" algebraic K-theory, and connections with arithmetic, , 1973
- Algebraic K-theory. Vol.3: Hermitian K-theory and geometric applications, , 1973
- Atiyah, M. F., & Anderson, D. W. (2018). K-theory. Boca Raton: CRC Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1728843
Recommended Additional Bibliography
- Algebraic K-theory. Vol.1: Higher K-theories, , 1973