Solutions of Fermat equations and covers of curves of genus 2
by
Gerhard Frey
Institute for Experimental Mathematics, University of Essen

It is well known that the existence of solutions of equations of the type

aXn + bYn = cZn

is closely related to properties of Galois representations of elliptic curves. In the talk we translate these properties into properties of the fundamental groups of curves of genus 2 and so we are led to new interesting diophantine questions about rational points on Hurwitz spaces (following the construction of Fried-Völklein).

In our case these Hurwitz spaces can be described explicitly by diagonal surfaces which are quotients of pairs of modular curves, and hence our questions about solutions of Fermat equations give rise to arithmetical questions about these surfaces.

Date received: February 26, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacf-08.