Joint Number Theory Event

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

43^8+96222^3=30042907^2.

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

(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