2020/2021

Научно-исследовательский семинар "Гипотезы Вейля"

Лучший по критерию «Полезность курса для Вашей будущей карьеры»

Лучший по критерию «Полезность курса для расширения кругозора и разностороннего развития»

Лучший по критерию «Новизна полученных знаний»

Статус: Дисциплина общефакультетского пула

Кто читает: Факультет математики

Где читается: Факультет математики

Когда читается: 3, 4 модуль

Преподаватели: Вологодский Вадим Александрович

Язык: английский

Кредиты: 6

Контактные часы: 72

Course Syllabus

Abstract

Given a system of polynomial equations with integer coefficients one can compute the number N_n of solutions to the system in a finite field of p^n elements, for some fixed prime number p. One of the Weil conjectures asserts that the exponential of the generating series for the sequence N_1/1, N_2/2, ... is rational. That is the numbers N_n satisfy a certain recurrence relation. The course will discuss basic ideas in the proof of the Weil conjectures due Grothendieck and Deligne. Prerequisites: basic algebraic geometry (first 3 chapters of Hartshorne's book.)

Learning Objectives

• Students will learn basics of the etale cohomology theory, including a proof of the Weil conjectures and its applications to Number Theory and Topology.

Expected Learning Outcomes

• Students will learn how to compute zeta function of projective spaces and elliptic curves.
• Students will learn how to compute the etale fundamental group of projective spaces and elliptic curves.
• Students will learn Hilbert's 90 theorem
• Students will learn how to compute 0th, 2nd, and the rank of the first cohomology groups for curves.
• Students will learn how to compute the cohomology of punctured curves
• Students will learn about the cycle class map
• Students will learn that the cohomology of a smooth proper family form a local system over the base
• Students will learn about the Weil pairing for curves
• Students will learn how to compute the monodromy
• Students will learn how to show that the monodromy is big
• Students will learn the basic ideas in Deligne's proof of the last Weil conjecture.

Course Contents

• Zeta function of varieties over finite fields.
  Statement of the Weil conjectures. Proof for curves. Kahler's analogue.
• Etale morphisms. The etale fundamental group
  Infinitesimal criterion, Henselian rings.
• Grothendieck's topology
  Faithfully flat descent
• The etale cohomology of curves
  Tsen's theorem, Poincare duality for curves.
• Proper base change.
  Proof of the proper base change theorem
• Poincare duality
  Grothendieck-Lefschetz fixed point formula
• Cohomological dimension. Purity.
  The Gysin sequence
• Smooth base change
  Proof of the smooth base change theorem
• Vanishing cycles
  Picard-Lefschetz formula
• Lefschetz Pencils
  Kazhdan-Margulis Theorem
• Proof of the Weil conjectures
  Rankin's trick

Assessment Elements

• Homeworks and Final Exam
• Homeworks and Final Exam

Interim Assessment

• Interim assessment (4 module)
  0.3 * Homeworks and Final Exam + 0.7 * Homeworks and Final Exam

Bibliography

Recommended Core Bibliography

• Dolgachev, I. (2012). Classical Algebraic Geometry : A Modern View. Cambridge: Cambridge University Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=473170

Recommended Additional Bibliography

• Hartshorne, R., & American Mathematical Society. (1975). Algebraic Geometry, Arcata 1974 : [proceedings]. Providence: AMS. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=772699