Saturday January 27, 2001
Harbour Centre, 1325 Axa Pacific Lecture Room

515 West Hastings Street
Vancouver, BC V6B 5K3

(see www.harbour.sfu.ca/maes/findpark.htm for a map and parking
locations)

**Three talks from 11am to 4pm
**

**
**

**11AM-12:00, Nils Bruin (PIMS, SFU, UBC)
will lecture on
**

**"Generalised Fermat equations."
**

** Abstract:**
The famous equation x^n+y^n=z^n can de be seen as a special case of a more
general equation x^r+y^s=z^t. The latter has become known as the
"Generalised Fermat equation". For given exponents r,s,t, the structure of
the set of primitive solutions (integral solution with coprime x,y,z)
depends heavily on the sum of the inverses of the exponents, 1/r+1/s+1/t.
Especially, if r,s,t are large enough, then there are only finitely many
primitive solutions. If we assume the ABC-conjecture, then there should be
only trivial solutions for large r,s,t.

However, some surprisingly large solutions are known. For instance

In general, the structure of the solution set has close ties to the structure of the set of rational points on an algebraic curve. In this talk, I will explain this relation and I will sketch how one can try to describe the solution sets for specific exponent triples.

The first part of the talk is accessible to a general mathematical audience. At the end, we will hit on current areas of research.

**
12-13:30, LUNCH
**

**
**

**1:30-2:30, Adrian Iovita (University of Washington, Seattle)
will lecture on
**

**"Explicit description of the local Galois representations
attached to modular forms"
**

** Abstract:**
(joint work with R.Coleman) We give an explicit description, using p-adic
integration, of the log-crystalline cohomology of semistable curves
over p-adic fields with values in F-isocrystals. This provides, via
Fontaine's theory, an explicit description of the Galois representations
arising as etale cohomology on the respective semistable curves.
We have nice applications of this to p-adic L-functions of modular forms.

**
**

**3:00 - 4:00, Stephen Choi (Simon Fraser University)
will lecture on
**

**"A Problem of Cohn on Classifying Characters"
**

** Abstract:**
Let $p$ be a prime and $F$ be a finite field with $q=p^s$
elements. It is well-known that for any nontrivial
multiplicative character $f$ of $F$,
\[
\sum_{b\in F}f(b)\overline{f(b+a)}=
\begin{cases}
q-1 & \text{if $a=0$;} \\
-1 & \text{if $a\neq 0.$}
\end{cases}
\]
H. Cohn asked whether the converse is true. For the case $p$
is odd and $s=1$, A. Bir\'{o} gives a partial answer to
Cohn's problem. In this talk, we give a negative answer to
Cohn's problem when $q 4$ and $s 1$.

We welcome all those with an interest in Number Theory to attend as this will be a good opportunity to meet the Number Theory community from the three universities (Simon Fraser University, University of British Columbia, and University of Washington). Please feel free to pass this announcement on. We hope to see you there!

Imin Chen

Adrian Iovita

Nike Vatsal