**A working seminar on the
topics of Expander Graphs and on Point counting (MATH 667), Winter 2010**

**
Announcement**

**Instructors**: Profs. Henri Darmon and
Eyal Goren.

**Time**: Fridays from 8:30 to 10:00 in BH 920.

This working seminar has as one of its
goals to
provide students with preparation

for some of the workshops in the scope of the
thematic semester at the CRM

"Number Theory as Experimental and Applied
Science"

(see
www.crm.umontreal.ca/NT2010/
for
details.)

Goren will take responsibility
of the
seminar
during the months of January and February.

This part of the seminar will be
devoted to topics in expander graphs, using the

paper by Hoory,
Linial and Wigderson
(Bull.
AMS, 43 (2006)) as our main text.

The requirement from
the students is to present a 2 hour lecture on a topic selected

together with
the instructor and to participate in the weekly meetings.

Darmon will take responsibility
of the
seminar
during the months of March and April.

This part of the seminar will be
devoted to point counting on algebraic varieties

in positive characteristic and
the machinery that goes into it.

**Part 2: Point counting and p-adic cohomology**

The goal of this part of the course will be to go over at least part of the
following references, with the goal of preparing ourselves for the CRM
workshop
on point counting that will be held at the end
of April.

1) R. Schoof,
Counting points on elliptic curves over finite fields.

Particularly the last section, which focuses on Schoof's polynomial
time algorithm based on l-adic cohomology.

2) B. Edixhoven,
Point counting after Kedlaya.

This is a nice introduction
to the circle of ideas related to point counting on curves using
p-adic cohomology.

3) A. Chambert-Loir,
Compter (rapidement) le nombre de solutions d'\'equations dans les corps finis.

A beautifully written survey of the subject. Highly recommended!

Anyone who is interested in giving a
presentation related to the material above
is
encouraged to get in touch with me (H. Darmon).

## Schedule of lectures

**Friday, March 5.** I will give an overview of the seminar and
start to line up some volunteers.

**Friday, March 12. ** There will be no seminar because
of the workshop on Graph Theory at the CRM.

**Friday, March 19, at 8:30 AM. **
*Note on the time*: Please note that from now on the seminar will meet at
8:30 AM. I apologise for the confusion about the starting
time last week!!
Please note that there is always a small chance (if the metros
are running more slowly than usual) that I could only make
it a short time after 8:30, but certainly we will not start
after 8:40.

*The program*: I will discuss p-adic point counting algorithms for
elliptic curves based on the theory of the canonical lift,
following the article

*An extension of Satoh's algorithm and its
implementation*, by Mireille Fouquet, Pierrick Gaudry, and Robert
Harley.

This article can be downloaded
from here.

Any seminar participant who is interested in covering this article
with me
should contact me.

*An exercise based on my first lecture* At the end of the first lecture,
I gave a cohomological proof (by calculating the trace of Frobenius on deRham cohomology)
of the fact the number of points of an elliptic curve *E* over a finite field
**F**_{q} of odd characteristic
is equal to the *q-1*st coefficient in the polynomial
*f(x)*^{(q-1/2)}, where *y*^{2} =f(x) is a defining equation for the curve.
To test your understanding of this theorem, try extending the statement (and the cohomological proof, as well as the lowbrow proof!) to the case of a hyperelliptic curve *y*^{2}=f(x) of genus *g*,
where *f* is a polynomial of degree *2g+1*.

**Friday, March 26, at 8:30 AM. **

*Note that this lecture will be at the CRM, in room 4336.*

**Darmon**.
Satoh's algorithm, cont'd.

**Friday, April 2, at 8:30 AM. **

**Aurel Page**, Schoof's algorithm.

**Friday, April 9, at 8:30 AM. **

**Francesc Castella**, Kedlaya's algorithm.

**Friday, April 16, at 8:30 AM. **

*Note that this lecture will be at the CRM, in room 4336.*

**Adam Logan**, Kedlaya's algorithm, cont'd.