2020/2021

## Research Seminar "The Weil Conjectures"

Type:
Optional course (faculty)

Delivered by:
Faculty of Mathematics

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

- 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 groupInfinitesimal criterion, Henselian rings.
- Grothendieck's topologyFaithfully flat descent
- The etale cohomology of curvesTsen's theorem, Poincare duality for curves.
- Proper base change.Proof of the proper base change theorem
- Poincare dualityGrothendieck-Lefschetz fixed point formula
- Cohomological dimension. Purity.The Gysin sequence
- Smooth base changeProof of the smooth base change theorem
- Vanishing cyclesPicard-Lefschetz formula
- Lefschetz PencilsKazhdan-Margulis Theorem
- Proof of the Weil conjecturesRankin's trick

#### 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