| Let k be an algebraically closed field of characteristic p > 0 and W(k) be its ring of Witt vectors. An F-crystal is a pair (M, ϕ) such that M is a free W(k)-module of finite rank and ϕ : M → M is a Frobenius-linear monomorphism. The classical Dieudonne theory says that the category of p-divisible groups D over k is anti-equivalent to the category of Dieudonne modules (M, ϕ) over k. A Dieudonne module is a special type of F-crystal such that pM ⊂ ϕ(M). Manin proved that for each p-divisible group D, there exists a smallest integer n such that the isomorphism type of D is determined by its finite truncation D[pn]. The isomorphism number nM of an arbitrary F-crystal (M, ϕ) is a generalization of such an n for p-divisible groups. It is the smallest non-negative integer such that for each g ∈ GL M(W(k)) with the property that g ≡ 1 mod pnM, we have (M, ϕ) ≅ (M, gϕ). Lau, Nicole and Vasiu have found an optimal upper bound in the case of Dieudonne modules in 2009 which states n M ≤ [2nu(c)] where nu is the Newton polygon and c is the codimension of the corresponding p-divisible group. In this manuscript, we discuss how to compute the isomorphism number for general F-crystals, in particular when the F-crystal are isoclinic, i.e. all Newton slopes are equal. More precisely, we have the following estimate of the isomorphism number for isoclinic F-crystals: nM≤&fll0;eri> lhi+&parl0;i< lhi-i>l hi&parr0;l&flr0;. &parl0;*&parr0; Here er is the maximal Hodge slope, hi are the Hodge numbers and lambda is the unique Newton slope. This estimate recovers the result of Lau, Nicole and Vasiu in the isoclinic case. But in general this upper bound is not optimal.;By using minimal F-crystals, i.e. F-crystals with isomorphism number less than or equal to 1, we define the minimal height qM of (M, ϕ) which is the smallest non-negative integer such that there exists an isogeny (M', ϕ') → (M, ϕ) such that p qM annihilates M/M'. Here ( M', ϕ) is a minimal F-crystal. Since nM ≤ 2qM + 1, we have another estimate for isosimple F-crystals: nM≤2&fll0;gr,s,re r-sr&flr0;+3. &parl0;* *&parr0; Here s is the sum of all Hodge slopes and g(r, s, rer -- s) is the Frobenius number, namely the greatest positive integer that cannot be expressed as a linear combination of r, s and rer -- s with coefficients in Z≥0 . In general, (*) and (**) are not comparable. |
