Related papers: Sur le th\'eor\`eme de l'indice des \'equations di…

For an odd prime number $p$, we consider degree $p$ extensions $L/K$ of $p$-adic fields with normal closure $\widetilde{L}$ such that the Galois group of $\widetilde{L}/K$ is the dihedral group of order $2p$. We shall prove a complete…

This article discusses variants of Weber's class number problem in the spirit of arithmetic topology to connect the results of Sinnott--Kisilevsky and Kionke. Let $p$ be a prime number. We first prove the $p$-adic convergence of class…

We survey our recent work on an extension of the theory of motivic integration, called arithmetic motivic integration. We developed this theory to understand how p-adic integrals of a very general type depend on p.

Motivated by applications in point counting algorithms using p-adic cohomology, we give an explicit description of integral lattices in rigid cohomology spaces that p-adically approximate logarithmic crystalline cohomology modules. These…

One of the phenomena peculiar in the theory of $p$-adic differential equations is that solutions $f$ of $p$-adic differential equations defined on open discs may satisfy growth conditions at the boundaries. This phenomenon is first studied…

The notion of a $p$-adic superspace is introduced and used to give a transparent construction of the Frobenius map on $p$-adic cohomology of a smooth projective variety over $\zp$ (the ring of $p$-adic integers), as well as an alternative…

Ardakov-Wadsley defined the sheaf D-cap of $p$-adic analytic differential operators on a smooth rigid analytic variety $X$ by restricting to the case where $X$ is affinoid and the tangent sheaf admits a smooth Lie lattice. We generalize…

We prove $p$-complete arc-descent results for finite projective modules and perfect complexes over integral perfectoid rings. Using our results, we clarify a reduction argument in the proof of the classification of $p$-divisible groups over…

The p-cohomology of an algebraic variety in characteristic p lies naturally in the category $D_{c}^{b}(R)$ of coherent complexes of graded modules over the Raynaud ring (Ekedahl-Illusie-Raynaud). We study homological algebra in this…

For a finite group scheme G over an algebraically closed field k of characteristic p>0 we study G-modules M, which are defined in terms of properties of their pull-backs along p-points of G. We show that the corresponding subcategories…

The polyadic integer numbers, which form a polyadic ring, are representatives of a fixed congruence class. The basics of polyadic arithmetic are presented: prime polyadic numbers, the polyadic Euler function, polyadic division with a…

In this paper we investigate the following related problems: (A) the separation of $p$-adic roots of integer polynomials of a fixed degree and bounded height; and (B) counting integer polynomials of a fixed degree and bounded height with…

This is the second paper of a series. It extends the results of the first paper from number fields to finitely generated fields, based on the recent theory of adelic line bundles of the same authors. We prove an arithmetic Hodge index…

This note outlines an approach to defining $p$-adic Shimura classes and $p$-adic derived Hecke operators on the completed cohomology of modular curves from upcoming work by the author. After reviewing the modulo-$p$ constructions of Harris…

In this paper, we construct stable distributions on the set of elliptic elements of an odd orthogonal groups (over a p-adic field with p large). Theses distributions are of zero level. They are parametrised by Langlands like parameters. In…

We construct $p$-adic $L$-functions associated with $p$-refined cohomological cuspidal Hilbert modular forms over any totally real field under a mild hypothesis. Our construction is canonical, varies naturally in $p$-adic families, and does…

An analogue of the Gauss-Lucas theorem for polynomials over the algebraic closure $\mathbb C_p$ of the field of $p$-adic numbers is considered.

We prove \cite[Conjecture~5.17]{Clausen} on the local light--profinite structure of smooth $p$-adic analytic Artin stacks. The argument proceeds in several reductions. First, by proving a generalization of van~Dantzig theorem for groupoids,…

This paper is a continuation of the paper F. A. Arias and M. Malakhaltsev "A generalization of the Gauss-Bonnet and Hopf-Poincar\'e theorems", ArXiv:1510.01395 [MathDG] 5 Oct 2015. Let $\pi : E \to M$ be a locally trivial fiber bundle over…

Let $K$ be a $p$-adic field. We continue to develop the theory of rigid analytic $p$-divisible groups over $K$. For example, we explain how to find back the category of Banach-Colmez spaces from rigid analytic $p$-divisible groups "in…