Definition 2. AFrobenius structure on a manifold H is a Frobenius structure on each tangent space TtH such that (1) The metric <,> is flat (∇ = 0) (2) The vector field 1 is covariantly constant (∇1 = 0) (3) The system of PDE’s ~∇ws = w ◦ s is integrable ∀~ 6= 0, where w and s are vector fields and ◦ denotes the Frobenius multiplication. In ∇-flat coordinates {t}, this means that the family of c...

— Let X be a smooth projective curve of genus g ≥ 2 defined over an algebraically closed field k of characteristic p > 0. Given a semistable vector bundle E over X, we show that its direct image F∗E under the Frobenius map F of X is again semistable. We deduce a numerical characterization of the stable rank-p vector bundles F∗L, where L is a line bundle over X. Résumé (Semi-stabilité des images...

this article introduces a systematic study for computational aspects of classical wavelet transforms over finite fields using tools from computational harmonic analysis and also theoretical linear algebra. we present a concrete formulation for the frobenius norm of the classical wavelet transforms over finite fields. it is shown that each vector defined over a finite field can be represented as...

Let X be a smooth projective curve of genus g ≥ 2 defined over an algebraically closed field k of characteristic p > 0. For p sufficiently large (explicitly given in terms of r, g) we construct an atlas for the locus of all Frobenius-destabilized bundles (i.e. we construct all Frobenius-destabilized bundles of degree zero up to isomorphism). This is done by exhibiting a surjective morphism from...

Let X be a projective curve of genus 2 over an algebraically closed field of characteristic 2. The Frobenius map on X induces a rational map on the moduli space of rank-2 bundles. We show that up to isomorphism, there is only one (up to tensoring by an order two line bundle) semi-stable vector bundle of rank 2 with determinant equal to a theta characteristic whose Frobenius pull-back is not sta...

Let X be a projective curve of genus 2 over an algebraically closed field of characteristic 2. The Frobenius map on X induces a rational map on the moduli scheme of rank-2 bundles. We show that up to isomorphism, there is only one (up to tensoring by an order two line bundle) semi-stable vector bundle of rank 2 (with determinant equal to a theta characteristic) whose Frobenius pull-back is not ...

Let C be a smooth curve, and Mr(C) the coarse moduli space of vector bundles of rank r and trivial determinant on C. We discuss the generalized Verschiebung map Vr : Mr(C) 99K Mr(C) induced by pulling back under Frobenius. We begin with a survey of general background results on the Verschiebung, as well as certain more specialized work in fixed genus, characteristic, and rank. We then discuss t...

One reason for the universal interest in Frobenius algebras is that their characterisation can be formulated in arbitrary categories: a functor K : A → B between categories is Frobenius if there exists a functor G : B → A which is at the same time a right and left adjoint of K; a monad F on A is a Frobenius monad provided the forgetful functor AF → A is a Frobenius functor, where AF denotes the...