From: Dave Rusin <rusin@math.niu.edu>
Date: Thu, 6 Aug 1998 11:28:18 -0500 (CDT)
To: [Permission pending]
Subject: Curves in Euler surface

When Elkies proved there were so many rational points in the surface
x^4+y^4+z^4=1 by finding embedded curves with genus 1, it must have
occurred to you to ask whether there exist maps from P^1 to the surface,
defined over  Q. Is the existence or nonexistence of such a curve known?

dave

==============================================================================
Date: Thu, 6 Aug 1998 12:30:23 -0400
From: [Permission pending]
To: rusin@math.niu.edu
Subject: Re:  Curves in Euler surface

> Subject: Curves in Euler surface

[Usually, if perhaps illogically, called a "Fermat surface"...]

I do not know whether there are rational curves on x^4+y^4+z^4=1
defined over Q.  It might be known that there are no *smooth* such
curves, because once they are smooth one can determine their self-
intersection from the adjunction formula, use the Galois structure
of the Neron-Severi group, yada^3.  But proving that there are
no rational curves at all seems much harder.

Sincerely,
[Permission pending]
==============================================================================

From: Dave Rusin <rusin@math.niu.edu>
Date: Wed, 16 Sep 1998 11:38:57 -0500 (CDT)
To: [Permission pending]
Subject: Re:  Curves in Euler surface

I had recently written you asking if there were rational curves in the
surface x^4+y^4+z^4=1. I just came across a paper by Bremner,
"On Euler's quartic surface", Math. Scand. 61 (1987), no. 2, 165--180
which addressed this very question. MathRev states:

"... He shows there are no curves on $V$ defined over
   $\bold Q$, of arithmetic genus $0$. The question of whether there are
   such curves of geometric genus $0$ remains undecided. "

In Bremner style, he goes on to calculate some small possibilities, 
finding none.

dave