Let K be a complete discrete valuation field of mixed characteristic (0, p) and GK the absolute Galois group of K. In this paper, we will prove the p-adic monodromy theorem for p-adic representations of GK without any assumption on the residue field of K, for example the finiteness of a p-basis of the residue field of K. The main point of the proof is a construction of (ϕ, GK )-module N ∇+ rig (V ) for a de Rham representation V , which is a generalization of Pierre Colmez' N
We study the asymptotic behavior of solutions of Frobenius equations defined over the ring of overconvergent series. As an application, we prove Chiarellotto-Tsuzuki's conjecture on the rationality and right continuity of Dwork's logarithmic growth filtrations associated to ordinary linear p-adic differential equations with Frobenius structures.
In this paper, we answer a question due to Y. André related to B. Dwork's conjecture on a specialization of the logarithmic growth of solutions of p-adic linear differential equations. Precisely speaking, we explicitly construct a ∇-module M over Qp [[X]]0 of rank 2 such that the left endpoint of the special log-growth Newton polygon of M is strictly above the left endpoint of the generic log-growth Newton polygon of M .
In the 1970s, Dwork defined the logarithmic growth (log-growth for short) filtrations for
$p$
-adic differential equations
$Dx=0$
on the
$p$
-adic open unit disc
$|t|<1$
, which measure the asymptotic behavior of solutions
$x$
as
$|t|\to 1^{-}$
. Then, Dwork calculated the log-growth filtration for
$p$
-adic Gaussian hypergeometric differential equation. In the late 2000s, Chiarellotto and Tsuzuki proposed a fundamental conjecture on the log-growth filtrations for
$(\varphi ,\nabla )$
-modules over
$K[\![t]\!]_0$
, which can be regarded as a generalization of Dwork's calculation. In this paper, we prove a generalization of the conjecture to
$(\varphi ,\nabla )$
-modules over the bounded Robba ring. As an application, we prove a generalization of Dwork's conjecture proposed by Chiarellotto and Tsuzuki on the specialization property for log-growth Newton polygons.
Let K be a complete discrete valuation field of mixed characteristic (0, p), whose residue fields may not be perfect, and GK the absolute Galois group of K. In the first part of this paper, we prove that Scholl's generalization of fields of norms over K is compatible with Abbes-Saito's ramification theory. In the second part, we construct a functor N dR associating a de Rham representation V with a (ϕ, ∇)-module in the sense of Kedlaya. Finally, we prove a compatibility between Kedlaya's differential Swan conductor of N dR (V ) and Swan
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