On variation of Hodge-Tate structures

We construct a functor from the category of p-adic étale local systems on a smooth rigid analytic variety X over a p-adic field to the category of vector bundles with an integrable connection on its "base change to B dR ", which can be regarded as a first step towards the sought-after p-adic RiemannHilbert correspondence. As a consequence, we obtain the following rigidity theorem for p-adic local systems on a connected rigid analytic variety: if the stalk of such a local system at one point, regarded as a p-adic Galois representation, is de Rham in the sense of Fontaine, then the stalk at every point is de Rham. Along the way, we also establish some basic properties of the p-adic Simpson correspondence. Finally, we give an application of our results to Shimura varieties. R.

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“…Note that the local version of OC first appeared in the work of Hyodo [14] (under the notation S ∞ ).…”

Section: Statement Of the Theoremmentioning

confidence: 99%

Rigidity and a Riemann–Hilbert correspondence for p-adic local systems

2016

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“…We obtain it as a Hodge-Tate comparison map except that on the one hand thé etale cohomology group is global (on X), while the differential object only lives on the affinoid X(w). Moreover, to make things worse theétale sheaf associated with the Γ -representation D U is not a Hodge-Tate sheaf, i.e., it does not define (locally on X) Hodge-Tate representations in the sense of [12]. Its cohomology is not a Hodge-Tate G K -representation!…”

Section: Theorem 12 ([9]mentioning

confidence: 99%

Overconvergent Eichler–Shimura isomorphisms

Andreatta

Iovită

Stevens

2014

J. Inst. Math. Jussieu

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Given a prime $p\gt 2$, an integer $h\geq 0$, and a wide open disk $U$ in the weight space $ \mathcal{W} $ of ${\mathbf{GL} }_{2} $, we construct a Hecke–Galois-equivariant morphism ${ \Psi }_{U}^{(h)} $ from the space of analytic families of overconvergent modular symbols over $U$ with bounded slope $\leq h$, to the corresponding space of analytic families of overconvergent modular forms, all with ${ \mathbb{C} }_{p} $-coefficients. We show that there is a finite subset $Z$ of $U$ for which this morphism induces a $p$-adic analytic family of isomorphisms relating overconvergent modular symbols of weight $k$ and slope $\leq h$ to overconvergent modular forms of weight $k+ 2$ and slope $\leq h$.

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“…[14]). For each prime ideal p of A of height 1 containing p, let O K p denote the completion of the local ring A p , which is a complete discrete valuation ring, and let K p denote its field of fractions.…”

Section: T Tsujimentioning

confidence: 99%

“…In this section, we review Hodge-Tate representations of G A defined by Hyodo [14]. See also [5,7,10,12,13] and [25].…”

Section: Hodge-tate Representationsmentioning

confidence: 99%

Purity for Hodge-Tate representations

Tsuji

2010

Math. Ann.

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We prove that a kind of purity holds for Hodge-Tate representations of the fundamental group of the generic fiber of a semi-stable scheme over a complete discrete valuation ring of mixed characteristic with perfect residue field. As an application, we see that the relative p-adic étale cohomology with proper support of a scheme separated of finite type over the generic fiber is Hodge-Tate if it is locally constant.

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On variation of Hodge-Tate structures

Cited by 16 publications

References 7 publications

Rigidity and a Riemann–Hilbert correspondence for p-adic local systems

Rigidity and a Riemann–Hilbert correspondence for p-adic local systems

Overconvergent Eichler–Shimura isomorphisms

Purity for Hodge-Tate representations

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