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Shanwen Wang

Publications and source records attributed to Shanwen Wang.

27 records · Page 2Linked to original sources
arXiv2024

Une factorisation de la cohomologie complétée et du système de Beilinson-Kato

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.

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arXiv2023

Uniformizer of the False Tate Curve Extension of $\mathbb{Q}_p$ (II)

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 ${\mathbb{K}}_p^{m,1}={\mathbb{Q}}_p(ζ_{p^m}, p^{1/p})$ 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.

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arXiv2023

Truncated expansion of $ζ_{p^n}$ in the $p$-adic Mal'cev-Neumann field

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.

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arXiv2021

Uniformizer of the False Tate Curve Extension of $\mathbb{Q}_p$

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 $ζ_{p^n}$ 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 $ζ_{p^n}$ and as an application, we use it to construct a uniformizer of $K_{2,m}=\mathbb{Q}_p\left(ζ_{p^2},p^{1/p^m}\right)$ with $m\geq 1$.

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arXiv2015

Le système d'Euler de Kato en famille (II)

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.

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arXiv2013

Une loi de réciprocité explicite pour le polylogarithme elliptique

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.

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arXiv2012

Le système d'Euler de Kato en famille (I)

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.

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arXiv2012

Le système d'Euler de Kato

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.

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