On the existence of F-crystals

By R. Kottwitz, Michael Rapoport. 2002

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

Subfields: Geometry and Topology, Materials Chemistry, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b87a353fafa67b93467bed8e3b1b6a1e013a378d

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Citations in the corpus

On the Hodge-Newton decomposition for split groups (2003, methodology): “ Non-emptiness of aﬃne Deligne-Lusztig subsets of G(L)/G(oL). We will now determine when the aﬃne Deligne-Lusztig set XG µ (b) (see (1.1)) is nonempty. The split case is treated in Proposition 4.6 of [KR], and we follow closely the method used there. As in [KR], for µ ∈ X∗(T) we put (4.7) Pµ := {ν ∈”
Dimensions of some affine Deligne–Lusztig varieties (2005, methodology): “Moreover the strategy used to prove the proposition is simply a refinement of the method used to prove nonemptiness in [16].”
Connected components of affine Deligne–Lusztig varieties in mixed characteristic (2013, background): “erage of the Γ-conjugates of µ.Then the set Xµ(b) is non-empty if and only if κG(b) = [µ] in π1(G)Γ,and ¯µ−νdom is a linear combination of positive coroots with non-negative rational coeﬃcients - see [KR03], [Win05], [GHKR06], Prop. 5.6.1, and [Gas10]. We denote by B(G,µ) the set of 1In fact in this”
Kottwitz-Rapoport conjecture on unions of affine Deligne-Lusztig varieties (2014, methodology): “The main purpose of this paper is to prove the following result, conjectured by Kottwitz and Rapoport in [17] and [22].”
The \mu-ordinary locus for Shimura varieties of Hodge type (2013, unknown): “jection X∗(T) →π1(G)hσi = (X∗(T)/hα ∨ |α∨ ∈Φ∨i) hσi. Deﬁnition 5.6. Let B(G,µ) := b∈B(G) |κG(b) = µ♮, νG(b) µ¯ . We endow B(G,µ) ⊂B(G) with the induced partial order . By work of Kottwitz-Rapoport ([KR]), Lucarelli ([Lu]) and Gashi ([Ga]) we knowthat B(G,µ) isexactlythe imageofthedouble cosetG(W(k))”
Stratifications in the reduction of Shimura varieties (2015, background): “As a preliminary, we mention the following result (conjectured in [30] and [41]).”
The Newton stratification on deformations of local G-shtukas (2008, background): “There is a criterion for XmðbÞ to be non-empty, see [34] and [14].”
GEOMETRIC AND HOMOLOGICAL PROPERTIES OF AFFINE DELIGNE-LUSZTIG VARIETIES (2012, background): “As a consequence, we solve the emptiness/nonemptiness question for aﬃne Deligne-Lusztig varieties in the aﬃne Grassmannian for tamely ramiﬁed groups in Theorem 7.1, generalizing previous results of Rapoport-Richartz [28], Kott-witz-Rapoport [23], Lucarelli [24] and Gashi [5] for unramiﬁed groups.”