An explicit example of Frobenius periodicity

Abstract

In this note we show that the restriction of the cotangent bundle ${\Omega }_{\mathbb{P}}^2$ of the projective plane to a Fermat curve $C$ of degree $d$ in characteristic $p\equiv -1\mathrm{mod}2d$ is, up to tensoration with a certain line bundle, isomorphic to its Frobenius pull-back. This leads to a Frobenius periodicity $F^{\ast }(\mathcal{E})\cong \mathcal{E}$ on the Fermat curve of degree 2d, where $\mathcal{E}=\mathrm{Syz}(U^2,V^2,W^2)(3)$.