(Both the published version and the arXiv version are available online.)

Let q  be a power of a prime. For each positive integer i ≤ 3,  we give an explicit description of the points on the curve u^q-1+v^q-1+ w^q-1=0  which have coordinates in the field F of cardinality qⁱ. As a consequence, we obtain much quicker proofs of the formulas for the number of such points which were obtained previously by Moisio. We also derive the following unexpected corollary: for any i ≤ 3,  and any point (u : v : w) as above, it always happens that the product uvw  is a cube in F. In case q  is even and i=3, this corollary was proved by a different argument in a previous paper by Scherr and Zieve, as a step in the proof of a conjecture related to finite projective planes.

We also describe the points on the surface u^q-1+v^q-1+ w^q-1+x^q-1=0  which have coordinates in the field of cardinality q². It turns out that, for any such point, the product uvwx  is a square. We do not know whether there is a common generalization of this result and the corollary mentioned above.

Michael Zieve:  home page   publication list