Rank- $$2$$ 2 syzygy bundles on Fermat curves and an application to Hilbert–Kunz functions

Abstract

Abstract. In this paper we describe the Frobenius pull-backs of the syzygy bundles Syz C (X a , Y a , Z a ), a ≥ 1, on the projective Fermat curve C of degree n in characteristics coprime to n, either by giving their strong HarderNarasimhan filtration if Syz C (X a , Y a , Z a ) is not strongly semistable or in the strongly semistable case by their periodicity behavior. Moreover, we apply these results to Hilbert-Kunz functions, to find Frobenius periodicities of the restricted cotangent bundle Ω P 2 | C of arbitrary length and a problem of Brenner regarding primes with strongly semistable reduction.