We compute the Hilbert-Kunz functions of two-dimensional rings of type ADE by using representations of their indecomposable, maximal Cohen-Macaulay modules in terms of matrix factorizations, and as first syzygy modules of homogeneous ideals.

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.

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We generalize a theorem of Monsky to compute the Hilbert-Kunz multiplicity of -graded rings k X Y Z / F , where F is a quasi-homogeneous, regular trinomial. This result will be used to study a limit of the Hilbert-Kunz multiplicity in families of such rings, where one exponent of F is dominating the others.

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