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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
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 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

#### 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