Abstract. We give necessary and sufficient conditions for the existence of pin ± and spin structures on Riemannian manifolds with holonomy group Z k 2 . For any n ≥ 4 (resp. n ≥ 6) we give examples of pairs of compact manifolds (resp. compact orientable manifolds) M1, M2, non homeomorphic to each other, that are Laplace isospectral on functions and on p-forms for any p and such that M1 admits a pin ± (resp. spin) structure whereas M2 does not.

For q = p m with p prime and k | q − 1, we consider the generalized Paley graphq }, and the irreducible p-ary cyclic code C(k, q) = {(Tr q/p (γω ik ) n−1 i=0 )} γ∈Fq , with ω a primitive element of Fq and n = q−1 k . We compute the spectra of Γ(k, q) in terms of Gaussian periods and give Spec(Γ(k, q)) explicitly in the semiprimitive case. We then show that the spectra of Γ(k, q) and C(k, q) are mutually determined by each other if further k | q−1 p−1 . Also, we use known characterizations of generalized Paley graphs which are cartesian decomposable to explicitly compute the spectra of the corresponding associated irreducible cyclic codes. As applications, we give reduction formulas for the number of rational points in Artin-Schreier curves and to the computation of Gaussian periods.

We consider a special class of generalized Paley graphs over finite fields, namely the Cayley graphs with vertex set Fqm and connection set the nonzero (q ℓ + 1)-th powers in Fqm , as well as their complements. We explicitly compute the spectrum of these graphs. As a consequence, the graphs turn out to be (with trivial exceptions) simple, connected, non-bipartite, integral and strongly regular (of Latin square type in half of the cases). As applications, on the one hand we solve Waring's problem over Fqm for the exponents q ℓ +1, for each q and for infinite values of ℓ and m. We obtain that the Waring's number g(q ℓ + 1, q m ) = 1 or 2, depending on m and ℓ, thus tackling some open cases. On the other hand, we construct infinite towers of Ramanujan graphs in all characteristics.

Abstract. Let M be an orientable compact flat Riemannian manifold endowed with a spin structure. In this paper we determine the spectrum of Dirac operators acting on smooth sections of twisted spinor bundles of M , and we derive a formula for the corresponding eta series. In the case of manifolds with holonomy group Z k 2 , we give a very simple expression for the multiplicities of eigenvalues that allows us to compute explicitly the η-series, in terms of values of Hurwitz zeta functions, and the η-invariant. We give the dimension of the space of harmonic spinors and characterize all Z k 2 -manifolds having asymmetric Dirac spectrum.Furthermore, we exhibit many examples of Dirac isospectral pairs of Z k 2 -manifolds which do not satisfy other types of isospectrality. In one of the main examples, we construct a large family of Dirac isospectral compact flat n-manifolds, pairwise nonhomeomorphic to each other of the order of a n .

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