CHARACTERISTIC p ANALOGUE OF MODULES WITH FINITE CRYSTALLINE HEIGHT

In the case of local fields of positive characteristic we introduce an analogue of Fontaine’s concept of Galois modules with crystalline height h ∈ N. If h = 1 these modules appear as geometric points of Faltings’s strict modules. We obtain upper estimates for the largest upper ramification numbers of these modules and prove (under an additional assumption) that these estimates are sharp. 0. Introduction. Let p be a prime number. Let K be a complete discrete valuation field with perfect residue field k of characteristic p. Choose a separable closure Ksep of K and set ΓK = Gal(Ksep/K). Denote by R the valuation ring of K and for any v > 0, by Γ K the ramification subgroup of ΓK with the upper number v. Suppose, first, that K is of characteristic 0, i.e. K contains Qp, and consider e = e(K) — the ramification index of K over Qp. In this situation for h ∈ N, Fontaine [Fo3] introduced the category of finite Zp[ΓK ]-modules with crystalline height h. Examples of such modules are given by subquotients of crystalline representations of ΓK with Hodge-Tate filtration of length h or, more specifically, of Galois modules of h-th etale coomology of projective schemes over K with good reduction. If h = 1 then the corresponding Galois modules appear as points G(Ksep) of finite flat pgroup schemes (i.e. killed by a power of the endomorphism p idG) G over R . In this case Fontaine [Fo1] proved very important ramification estimate: if H ∈ MGK , pH = 0 and v > e ( N + 1 p− 1 ) − 1 then Γ K acts trivially on H. This result was generalised in [Ab1] (cf. also [Fo2], [Ab2]): if H is a subquotient of crystalline representation of ΓK with the Hodge-Tate filtration of length h < p− 1, pH = 0 and e = 1 then for v > ( N + h p− 1 ) − 1, Γ K acts trivially on H. Now suppose that K is of characteristic p and k ⊃ Fq, where q is a power of p. Introduce an analogue of Zp. This will be a subring O = Fq[[π]] of R, where π ∈ R is not invertible in R. If E is the fraction field of O inK then denote by e = e(K/E) 1991 Mathematics Subject Classification. 11S15, 11S20.