Positive Characteristic
Linear differential equations over differential fields of characteristic p > 0 have been studied for a long time ([139, 152, 153, 8],…). Grothendieck’s conjecture on p-curvatures is one of the motivations for this. Another motivation is the observation
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Linear differential equations over differential fields of characteristic p > 0 have been studied for a long time ([139, 152, 153, 8],...). Grothendieck’s conjecture on p-curvatures is one of the motivations for this. Another motivation is the observation that for the factorization of differential operators over, say, the differential field Q(z) the reductions modulo prime numbers yield useful information. In this chapter we first develop the classification of differential modules over differential fields K with [K : K p ] = p. It turns out that this classification is rather explicit and easy. It might be compared with Turrittin’s classification of differential modules over C((z)). Algorithms are developed to construct and obtain standard forms for differential modules. From the viewpoint of differential Galois theory, these linear differential equations in characteristic p do not behave well. A completely different class of equations, namely the “linear iterative differential equations”, is introduced. These equations have many features in common with linear differential equations in characteristic 0. We will give a survey and explain the connection with p-adic differential equations.
13.1 Classification of Differential Modules In this chapter, K denotes a field of characteristic p > 0 satisfying [K : K p ] = p. The universal differential module Ω K of K has dimension 1 over K . Indeed, choose an element z ∈ K \ K p . Then K has basis 1, z, . . . , z p−1 over K p and this implies that Ω K = Kdz. Let f → f = ddzf denote the derivation given by (a0 + a1 z + · · · + a p−1 z p−1 ) = a1 + 2a2 z + · · · + ( p − 1)a p−1 z p−2 , for any a0 , . . . , a p−1 ∈ K p . Every derivation on K has the form g dzd for a unique g ∈ K. Examples of fields K with [K : K p ] = p are k(z) and k((z)) with k a perfect field of charactersitic p. We note that any separable algebraic extension L ⊃ K of a field with [K : K p ] = p again has the property [L : L p ] = p. A connection over the field K is a pair (∇, M) where M is a finite dimensional vector space over K and where ∇ : M → Ω K ⊗ M is an additive map satisfying the M. van der Put. et al., Galois Theory of Linear Differential Equations © Springer-Verlag Berlin Heidelberg 2003
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13 Positive Characteristic
usual rule ∇( fm) = d f ⊗m + f ∇(m) for f ∈ K and m ∈ M. One observes that ∇ is determined by ∂ := ∇ d : M → M given by ∂ = ⊗ id ◦ ∇, where : Ω K → K is dz the K -linear map defined by (dz) = 1. In what follows we will fix z, the derivation f → f = ddzf and consider instead of connections differential modules (M, ∂) defined by: M is a finite dimensional vector space over K and ∂ : M → M is an additive map satisfying ∂( fm) = f m + f∂m for f ∈ K and m ∈ M. Our aim is to show that, under some condition on K , the tannakian category Diff K of all differential modules over K is equivalent to the tannakian category Mod K p [T ] of all K p [T ]-modules, which are finite dimensional as vector spaces over the field K p . The objects of the latter category can be given as pairs (N, t N ) consist
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