Sen attached to each $p$-adic Galois representation of a $p$-adic field a multiset of numbers called generalized Hodge–Tate weights. In this paper, we discuss a rigidity of these numbers in a geometric family. More precisely, we consider a $p$-adic local system on a rigid analytic variety over a $p$-adic field and show that the multiset of generalized Hodge–Tate weights of the local system is constant. The proof uses the $p$-adic Riemann–Hilbert correspondence by Liu and Zhu, a Sen–Fontaine decompletion theory in the relative setting, and the theory of formal connections. We also discuss basic properties of Hodge–Tate sheaves on a rigid analytic variety.
We study horizontal semistable and horizontal de Rham representations of the absolute Galois group of a certain smooth affinoid over a
$p$
-adic field. In particular, we prove that a horizontal de Rham representation becomes horizontal semistable after a finite extension of the base field. As an application, we show that every de Rham local system on a smooth rigid analytic variety becomes horizontal semistable étale locally around every classical point. We also discuss potentially crystalline loci of de Rham local systems and cohomologically potentially good reduction loci of smooth proper morphisms.
For an abelian variety A over an algebraically closed non-archimedean field of residue characteristic p, we show that there exists a perfectoid space which is the tilde-limit of lim ← −[p] A.Our proof also works for the larger class of abeloid varieties.
For a lisse $\ell $-adic sheaf on a scheme flat and of finite type over $\mathbb{Z}$, we consider the field generated over $ \mathbb{Q}$ by Frobenius traces of the sheaf at closed points. Assuming conjectural properties of geometric Galois representations of number fields and the Generalized Riemann Hypothesis, we prove that the field is finite over $\mathbb{Q}$ when the sheaf is de Rham at $\ell $ pointwise. This is a number field analog of Deligne’s finiteness result about Frobenius traces in the function field case.
Deligne conjectured that a single l-adic lisse sheaf on a normal variety over
a finite field can be embedded into a compatible system of l'-adic lisse
sheaves with various l'. Drinfeld used Lafforgue's result as an input and
proved this conjecture when the variety is smooth. We consider an analogous
existence problem for a regular flat scheme over Z and prove some cases using
Lafforgue's result and the work of Barnet-Lamb, Gee, Geraghty, and Taylor.Comment: Some arguments are simplified and corrected. Typos are fixed. To
appear in Algebra and Number Theor
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