An Explicit Computation of a Family of Trivializing Étale Covers

Abstract

We explicitly compute étale covers of the smooth Fermat curves Y _p+1 = Proj k[u, v, w]/(u ^p+1 + v ^p+1 − w ^p+1) which trivialize the vector bundles Syz(u ², v ², w ²)(3), where k is a field of characteristic p ≥ 3.

Key Words:

• Étale cover
• Frobenius periodicity
• Vector bundle

2010 Mathematics Subject Classification:

• Primary: 14H30
• Secondary: 14H60

ACKNOWLEDGMENTS

I thank Holger Brenner for useful discussions and for pointing out several simplifications to arguments in an earlier version of this article. Furthermore, the computer algebra system CoCoA ([5]) has been helpful in the preparation of this article.

Notes

In other words, any generator of .

Communicated by U. Walther.