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Slope filtrations in families. (English) Zbl 1285.14023
This paper is of interest to those studying number theory. It studies how certain invariants (HN-polygons) of \(\phi\)-modules over the Robba ring behave in families.
Let \(K\) be a field equipped with a discrete valuation. Suppose that \(K\) is complete of characteristic zero and the residue field of the ring of integers is of characteristic \(p >0\). Let \(R\) be a ring equipped with an endomorphism \(\phi\). A \(\phi\)-module over \(R\) is a pair of a finite free \(R\)-module \(M\) and an \(R\)-linear isomorphism \(\theta: \phi^*M \rightarrow M\). An example of interest to this paper is to take \(R=\mathcal{R}_{A_K}\) to be the Robba ring of an affinoid algebra \(A_K\) and \(\phi\) a lift of the relative Frobenius.
The structure morphism \(\theta\) of a \(\phi\)-module \(M\) and the valuation on \(K\) can be used to define the degree of \(M\). The slope of \(M\) is defined as \(\mu(M) = \deg(M)/\mathrm{rank}(M)\). K. S. Kedlaya [Astérisque 319, 259–301 (2008; Zbl 1168.11053)] has shown that \(M\) has a unique Harder-Narasimhan (HN) filtration \(0 = M_0 \subset M_1 \subset \dots \subset M_l =M\) by saturated \(\phi\)-submodules. The slopes of this filtration determine a Newton polygon called the HN-polygon.
The paper under review studies how these HN-filtrations and their associated HN-polygons behave in families. It studies the fiberwise behavior of the HN-filtration/HN-polygons of \(\phi\)-modules over \(\mathcal{R}_{A_K}\). The paper has four main results/applications. For brevity, only one will be fully stated. For any \(\phi\)-module \(M_A\) over \(\mathcal{R}_{A_K}\) and for any \(x \in M(A)\) there is a Weierstrass subdomain \(M(B)\) of \(M(A)\) such that the subset of points of \(M(B)\) where the HN-polygons agree with the HN-polygon at \(x\) is a Zariski closed set \(M(C)\). Moreover, the HN-filtration of the fiber \((M_A)_x\) at \(x\) lifts to a slope filtration on the module \(\widetilde{M_C} = M_{A} \otimes_{\mathcal{R}_{A_K}} (C \widehat{\otimes}_{\mathbb{Q}_p} \widetilde{\mathcal{R}}_L)\) where \(L\) is an admissible extension of \(K\) with strongly difference-closed residue field. The other main theorems are variations on this theorem when the hypothesis is that the fiber \((M_A)_x\) is pure of slope \(s\), when \(K\) is assumed to be a finite unramified extension of \(\mathbb{Q}_p\), or where the conclusion is that the HN-polygon at \(y\) lies above the HN-polygon at \(x\).

14F30 \(p\)-adic cohomology, crystalline cohomology
11S20 Galois theory
Full Text: DOI arXiv
[1] Astérisque 319 pp 303– (2008)
[2] Spectral theory and analytic geometry over non-Archimedean fields Volume 33 pp 169– (1990) · Zbl 0715.14013
[3] DOI: 10.2140/ant.2008.2.91 · Zbl 1219.11078
[4] Publ. Math. IHÉS 14 pp 47– (1962) · Zbl 0119.03701
[5] Int. Math. Res. Not. 2008 3
[6] DOI: 10.2140/ant.2010.4.943 · Zbl 1278.11060
[7] p-adic differential equations Volume 125 (2010)
[8] Astérisque 319 pp 259– (2008)
[9] Doc. Math. 10 pp 447– (2005)
[10] DOI: 10.4007/annals.2004.160.93 · Zbl 1088.14005
[11] DOI: 10.1090/S0002-9939-01-06001-4 · Zbl 1012.12007
[12] DOI: 10.1007/s002220050266 · Zbl 0929.14029
[13] Astérisque 330 pp 13– (2010)
[14] Astérisque 319 pp 213– (2008)
[15] Astérisque 319 pp 117– (2008)
[16] Non-Archimedean analysis Volume 261 (1984)
[17] Non-Archimedean functional analysis (2002)
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