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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.

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 description of the completed cohomology 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., 142 pages, in French

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 $ζ_{p^n}$ of the $p^n$-th cyclotomic polynomial $Φ_{p^n}(T)$ in the $p$-adic Mal'cev-Neumann field $\mathbb{L}_p$. We establish a $\frac{2}{(p-1)p^{n-2}}$-truncated expansion of $ζ_{p^n}$ via a variant of the transfinite Newton algorithm, which gives the first $\aleph_0^2$ terms of the canonical expansion of $ζ_{p^n}$. The harmonic number identity simplifies the expression of this expansion., 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 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., 19pages, in French. arXiv admin note: substantial text overlap with arXiv:1211.3767, arXiv:1211.4256

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., 36 pages, in French; Improve the presentation of previous version and correct some computations

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., 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., 66pages. Submitted. Comments are welcome!