Math 608R: Etale Cohomology and the Weil conjectures

Fall 2007

MWF 1pm -1:50pm, PHY4208

Professor Niranjan Ramachandran, 4115, x5-5080

Textbooks:

• (MEC) Milne’s notes on “Etale Cohomology”. supplemented by
• (ECW) E. Freitag and R. Kiehl “Etale Cohomology and the Weil conjectures”

(available online in djvu format) (BAMS Review by N. Katz)

Description: The conjectures of André Weil have influenced (or directed) much of 20th century algebraic geometry.

These conjectures generalize the Riemann hypothesis (RH) for function fields (alias curves over finite fields), conjectured

(and verified in some special cases) by Emil Artin. Helmut Hasse proved RH for elliptic function fields.

RH for general function fields was finally proved by Weil who then formulated his conjectures

for higher dimensional algebraic varieties over finite fields. The last of this set of conjectures

directly generalizes RH.

The Weil conjectures are now known to be true (by work of Alexandre Grothendieck, Michael Artin,

Pierre Deligne, et al). This course will provide an overview of the methods and ideas which have

led to the formulation and the proof of the Weil conjectures.

In particular, we hope to cover

• Weil conjectures: formulation, proof in special cases, consequences (Ramanujan conjecture)
• the basic concepts of étale cohomology
• why étale cohomology generalizes Galois cohomology
• the Tate module of an elliptic curve is an étale cohomology group
• the generalization of the notion of topology (due to Grothendieck)
• l-adic cohomology for each prime l≠p for varieties over a field of characteristic p.
• comparison with singular cohomology
• interpretation of H¹
• Cohomology of curves (divisors and G_m)
• the smooth and proper base change theorems
• Poincaré duality
• Lefschetz fixed-point formula
• Lefschetz pencils
• Rough sketch of proof of RH in higher dimensions (Deligne’s Weil I)
• Brief sketch of motives and Grothendieck’s standard conjectures

Deligne’s “Cohomologie étale: les points de départ” ([Arcata] pp. 4-75 in SGA 4 1/2 LNM 569)

is a beautiful introduction; it cannot be recommended highly enough!

The course should be of interest to aspiring number theorists and algebraic geometers.

Basic material from commutative algebra, homological algebra, and manifold theory will be assumed.

See MEC for specific information about background.

Grading: There will be many exercises assigned during the semester.

Students will give one in-class presentation. Some possible topics are:

· Deligne-Lusztig theory (notes of Yoshida, wiki)

· Estimates for exponential sums (notes of Kowalski)

· Independence of the prime l in l-adic cohomology (l≠p)

· Ramanujan conjecture (comment by Manin, (16) is the Ramanujan Conjecture)

· Classical Gauss sums and Jacobi sums, Fermat varieties

· Katz-Messing

· Rankin’s trick

· Etale cohomology of abelian varieties

· Algebraic cycles, K-theory

· Flat cohomology

· Arithmetic duality theorems

Tentative plan of the course:

Week ending	Topic	Reference
8/31	Introduction and Overview	A. Weil in PNAS A. Weil in Bulletin of AMS Dieudonne in Math Intelligencer (also reprinted in ECW) P. Roquette’s articles on history of RH MEC 1
9/7	Etale morphisms	MEC 2
9/14	Etale fundamental group	MEC 3
9/21	Local ring for the etale topology	MEC 4
9/28	Sites
10/5	Sheaves for the etale topology
10/12	Operations on sheaves
10/19	Cohomology: Definitions
10/26	Cech cohomology
11/2	Torsors and H^1
11/9
11/16
11/23
11/30
12/7
12/14

General References:

• Deligne’s “La conjecture de Weil I” (just 34 pages!)

(B. Mazur’s Zentralblatt review)

• Deligne’s follow-up “La conjecture de Weil II”
• N. Katz’s Lectures on Weil II
• J. Milne’s Etale Cohomology (the book)
• SGA
• Grothendieck Circle
• Grothendieck’s Séminaire Bourbaki “Formule de Lefschetz et rationalité des fonctions L”
• Grothendieck’s Standard conjectures
• Notices of AMS issue about André Weil
• Grothendieck-Serre correspondence
• Leila Schneps’s summary (especially pages 14-17) of Grothendieck-Serre correspondence
• The Rising Sea:Grothendieck on simplicity and generality by Colin McClarty
• Comme Appelé du Néant -- As If Summoned From the Void (Part I) (Part II)

by Allyn Jackson, Notices of the AMS (Part I is in Vol 51, No. 4, Part II is in Vol 51, No. 10)

• CARTIER Pierre, Alain HERREMAN on Grothendieck and Motives (French) (PDF)
• Kahlerian analogue of the Weil conjectures due to J.-P. Serre
• Serre on Weil
• Serre’s report on Deligne’s proof of the Weil conjectures at Séminaire Bourbaki (only French names allowed!)
• Verdier’s report on Deligne’s proof of the Weil conjectures at Séminaire Bourbaki
• Deligne’s report on Weil implies Ramanujan at Séminaire Bourbaki
• Kowalski’s A tale in two parts on Weil I
• Hartshorne’s Algebraic Geometry has an appendix on the Weil conjectures
• Algebraic Geometry (Arcata 1974), AMS has many articles on the Weil conjectures