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Let $p\geq 3$ be a prime. The hyper-algebraic elements in the $p$-adic Mal'cev-Neumann field $\mathbb{L}_p$ form an algebraically closed subfield $\mathbb{L}_p^{\operatorname{ha}}$. In this article, we clarify the relations among the fields $\mathbb{L}_p^{\operatorname{ha}}$, $\overline{\mathbb{Q}}_p$ and $\mathbb{C}_p$. We introduce two arithmetic invariants (hyper-tame index and hyper-inertia index) of hyper-algebraic elements and study the relation between these invariants and classical arithmetic invariants of $p$-adic algebraic numbers. Finally, we give a criterion for hyper-algebraic elements to be tamely ramified over $\mathbb{Q}_p$., Comment: Comments are welcomed!

In the era of Big Data, prompt analysis and processing of data sets is critical. Meanwhile, statistical methods provide key tools and techniques to extract valuable insights and knowledge from complex data sets. This paper creatively applies statistical methods to the field of traffic, particularly focusing on the preprocessing of multi-frame signals obtained by optical fiber-based Distributed Acoustic Sensing (DAS) system. An online non-parametric regression model based on Local Polynomial Regression (LPR) and variable bandwidth selection is employed to dynamically update the estimation of mean function as signals flow in. This mean estimation method can derive average information of multi-frame fiber signals, thus providing the basis for the subsequent vehicle trajectory extraction algorithms. To further evaluate the effectiveness of the proposed method, comparison experiments were conducted under real highway scenarios, showing that our approach not only deals with multi-frame signals more accurately than the classical filter-based Kalman and Wavelet methods, but also meets the needs better under the condition of saving memory and rapid responses. It provides a new reliable means for signal processing which can be integrated with other existing methods., Comment: 10 pages, 11figures

In the realm of intelligent transportation systems, accurate and reliable traffic monitoring is crucial. Traditional devices, such as cameras and lidars, face limitations in adverse weather conditions and complex traffic scenarios, prompting the need for more resilient technologies. This paper presents traffic flow monitoring method using optical fiber-based Distributed Acoustic Sensors (DAS). An innovative vehicle trajectory extraction algorithm is proposed to derive traffic flow statistics. In the processing of optical fiber waterfall diagrams, Butterworth low-pass filter and peaks location search method are employed to determine the entry position of vehicles. Subsequently, line-by-line matching algorithm is proposed to effectively track the trajectories. Experiments were conducted under both real highway and tunnel scenarios, showing that our approach not only extracts vehicle trajectories more accurately than the classical Hough and Radon transform-based methods, but also facilitates the calculation of traffic flow information using the low-cost acoustic sensors. It provides a new reliable means for traffic flow monitoring which can be integrated with existing methods., Comment: 10 pages,23 figures, references added

We extend the dictionary between Fontaine rings and $p$-adic functionnal analysis, and we give a refinement of the $p$-adic local Langlands correspondence for principal series representations of ${\rm GL}_2(\mathbf{Q}_p)$., Comment: 20 pages, in French language, accepted by JNT

Let $\mathscr{O}_K$ be a henselian DVR with field of fractions $K$ and residue field of characteristic $p>0$. Let $S$ denote $\mathop{\mathrm{Spec}} \mathscr{O}_K$ endowed with the canonical log structure. We show that the generic fiber functor $\mathbf{BT}_{S, {\mathrm{d}}}^{\log}\to \mathbf{BT}^{\mathrm{st}}_K$ between the category of dual representable log $p$-divisible groups over $S$ and the category of $p$-divisible groups with semistable reduction over $K$ is an equivalence. If $\mathscr{O}_K$ is further complete with perfect residue field and of mixed characteristic, we show that $\mathbf{BT}_{S, {\mathrm{d}}}^{\log}$ is also equivalent to the category of semistable Galois $\mathbb{Z}_p$-representations with Hodge-Tate weights in $\{0,1\}$. Finally, we show that the above equivalences respect monodromies., Comment: 41 pages. We have rewritten part of the paper, in particular the introduction in order to make more evident the novelty of our results and added a corollary on the antiequivalence between the category of log $p$-divisible groups and the category of strongly divisible modules of weight less or equal to 1

In this article, we investigate the explicit formula for the uniformizers of the false-Tate curve extension of $\mathbb{Q}_p$. More precisely, we establish the formula for the fields ${\mathbb{K}}_p^{m,1}={\mathbb{Q}}_p(\zeta_{p^m}, p^{1/p})$ with $m\geq 1$ and for general $n\geq 2$, we prove the existence of the recurrence polynomials ${\mathcal{R}}_p^{m,n}$ for general field extensions ${\mathbb{K}}_p^{m, n}$ of ${\mathbb{Q}}_p$, which shows the possibility to construct the uniformizers systematically., Comment: Accepted by IJNT

In this note, we show that the metaplectic theta correspondence is compatible with the tempered condition by directly estimating the matrix coefficients, without using the classification theorem.

Fix an odd prime $p$. In this article, we provide a $\mathrm{mod}\ p$ harmonic number identity, which appears naturally in the canonical expansion of a root $\zeta_{p^n}$ of the $p^n$-th cyclotomic polynomial $\Phi_{p^n}(T)$ in the $p$-adic Mal'cev-Neumann field $\mathbb{L}_p$. We establish a $\frac{2}{(p-1)p^{n-2}}$-truncated expansion of $\zeta_{p^n}$ via a variant of the transfinite Newton algorithm, which gives the first $\aleph_0^2$ terms of the canonical expansion of $\zeta_{p^n}$. The harmonic number identity simplifies the expression of this expansion., Comment: This version is seperated from the last version (v4). We move the uniformizer related part into a new article. arXiv admin note: text overlap with arXiv:2009.09807

We show that the modular symbol $(0,\infty)$, considered as an element of the dual of Emerton's completed cohomology, interpolates Kato's Euler system at classical points, and we deduce from this a factorisation of Beilinson-Kato's system as a product of two symbols $(0,\infty)$ (an algebraic analog of Rankin's method). The proof uses the $p$-adic local Langlands correspondence for ${\mathbf GL}_2({\mathbf Q}_p)$ and Emerton's factorization of the completed cohomology of the tower of modular curves for which we provide a new proof resting upon the construction of a Kirillov model for the completed cohomology, and which we refine by imposing conditions at classical points; the existence of such a refinement is a manifestation of an analyticity property for $p$-adic periods of modular forms., Comment: 187pages, in French. This new version includes a proof of Emerton's factorization of completed cohomology (with some local conditions removed) which makes the factorization of Beilinson-Kato system more streamlined. The paper is dedicated to the memory of Jan Nekov\'a\v{r} and Jo\"el Bella\"{\i}che

Let $p\geq 3$ be a prime number. In this article, we study the canonical expansion of the primitive $p^n$-th root of unity $\zeta_{p^n}$ in $p$-adic Mal'cev-Neumann field $\mathbb{L}_p$ for $n\geq 1$. More precisely, we give the explicit formula for the first $\aleph_0$ terms of the expansion of $\zeta_{p^n}$ and as an application, we use it to construct a uniformizer of $K_{2,m}=\mathbb{Q}_p\left(\zeta_{p^2},p^{1/p^m}\right)$ with $m\geq 1$., Comment: 37 pages, accepted by The Ramanujan Journal

This article is the second article on the generalization of Kato's Euler system. The main subject of this article is to construct a family of Kato's Euler systems over the cuspidal eigencurve, which interpolate the Kato's Euler systems associated to the modular forms parametrized by the cuspidal eigencurve. We also explain how to use this family of Kato's Euler system to construct a family of distributions on $Z_p$ over the cuspidal eigencurve; this distribution gives us a two variable $p$-adic L function which interpolate the $p$-adic L function of modular forms., Comment: 36 pages, in French; Improve the presentation of previous version and correct some computations

We construct a dual exponential map which relates the $p$-adic Eisenstein classes to Eisenstein series. From this map, we deduce a compatibility between the $p$-adic realization and the de Rham realization of the torsion sections of the elliptic polylogarithm prosheaf., Comment: 19pages, in French. arXiv admin note: substantial text overlap with arXiv:1211.3767, arXiv:1211.4256

This article is the first article of a serie of articles on the generalization of Kato's Euler system. The main subject of this article is to construct a family of Kato's Euler systems and a family of Kato's explicit reciprocity laws over the weight space, which interpolate the corresponding classical objects., Comment: 40pages. Submitted. arXiv admin note: text overlap with arXiv:1211.3767

This article is devoted to Kato's Euler system, which is constructed from modular unites, and to its image by the dual exponential map (so called Kato's reciprocity law). The presentation in this article is different form Kato's original one, and dual exponential map in this article is a modification of Colmez's construction in his Bourbaki talk., Comment: 66pages. Submitted. Comments are welcome!