Math 608R: Etale Cohomology and the Weil conjectures
Fall 2007
MWF 1pm 1:50pm, PHY4208
Professor
Niranjan Ramachandran, 4115, x55080
Textbooks:
(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
Deligne’s
“Cohomologie étale: les points de départ”
([Arcata] pp. 475 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 inclass
presentation. Some possible topics
are:
·
DeligneLusztig theory (notes of Yoshida, wiki)
·
Estimates
for exponential sums (notes
of Kowalski)
·
·
Ramanujan conjecture (comment
by Manin, (16) is the Ramanujan
Conjecture)
·
Classical
Gauss sums and Jacobi sums, Fermat varieties
·
Rankin’s
trick
·
Etale cohomology of abelian varieties
·
Algebraic
cycles, Ktheory
·
Flat
cohomology
·
Arithmetic
duality theorems
Tentative plan of the course:
Week
ending 
Topic 
Reference 
8/31 
Introduction and
Overview 
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:
(B.
Mazur’s Zentralblatt review)
by Allyn Jackson, Notices of the AMS (Part I is in Vol 51, No. 4, Part II is in Vol 51, No. 10)