On the existence of F-crystals

Let (N,F) be an F-isocrystal, with associated Newton vector νin (Q^n)_+. To any lattice M in N (an F-crystal) is associated its Hodge vector μ(M) in (Z^n)_+. By Mazur's inequality we have μ(M)>= ν. We show that, conversely, for any μin (Z^n)_+ with μ>= ν, there exists a lattice M in N such that μ=μ(M). We also give variants of this existence theorem for symplectic F-isocrystals, and for periodic lattice chains.

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