In this directory: http://www.maths.ox.ac.uk/~flynn/genus2/durham/
are the transparencies which I used for my talk
at the LMS Durham Symposium, held 24 July - 3 August 2000.
Many thanks to Judi Young for scanning them.
The abstract of the talk was as follows:
FERMAT QUARTICS AND A CHALLENGE CURVE OF SERRE
E.V. Flynn.
A study of rational points on Fermat quartics of the form X^4 + Y^4 = c
immediately reveals many values of c which can be dismissed by
congruence considerations. Many other values of c can be dismissed
if one of two associated elliptic curves has rank 0. What remains
are the stubborn values of c which cannot be trivially dismissed:
c = 17, 82, 97 and 257 being the only such less than 300. A solution
of the case c = 17 (posed as a challenge by Serre) is presented,
representing the first success with a nontrivial value of c, and
we discuss the extent to which the method might hope to solve other
difficult values of c. The talk is based on joint work with Joe Wetherell.
There is an associated manuscript (to appear in Acta Arith) at:
http://www.maths.ox.ac.uk/~flynn/genus2/manuscripts/serrecurve.ps
The Home Page of the LMS Durham Symposium is at:
http://www.maths.nott.ac.uk/personal/jec/durham/index.html
and there is an Online Proceedings at:
http://www.maths.nott.ac.uk/personal/jec/durham/proc.html