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Regular version of the site
2020/2021

Research Seminar "The Weil Conjectures"

Type: Optional course (faculty)
When: 3, 4 module
Open to: students of one campus
Instructors: Vadim Vologodsky
Language: English
ECTS credits: 6

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

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

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

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

Assessment Elements

  • non-blocking Homeworks and Final Exam
  • non-blocking Homeworks and Final Exam
Interim Assessment

Interim Assessment

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

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