2020/2021

Research Seminar "The Weil Conjectures"

Category 'Best Course for Career Development'

Category 'Best Course for Broadening Horizons and Diversity of Knowledge and Skills'

Category 'Best Course for New Knowledge and Skills'

Type: Optional course (faculty)

Delivered by: Faculty of Mathematics

Where: Faculty of Mathematics

When: 3, 4 module

Instructors: Vadim Vologodsky

Language: English

ECTS credits: 6

Contact hours: 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