标题： $p$-进分布代数由F级数的逐点乘积诱导 显示英文标题

标题： Algebras of $p$-Adic Distributions Induced by Pointwise Products of F-Series

摘要： 设$p$为一个整数$\geq2$，并设$K$为一个全局域。 一个叶状的$p$-adic F级数是一个函数$X$，它是一个$p$-adic整数变量$\mathfrak{z}$的函数，满足泛函方程$X\left(p\mathfrak{z}+j\right)=a_{j}X\left(\mathfrak{z}\right)+b_{j}$对于所有$\mathfrak{z}\in\mathbb{Z}_{p}$和所有$j\in\left\{ 0,\ldots,p-1\right\} $，其中$a_{j}$和$b_{j}$是不定元。 将$X$视为取值于$K$上的某个形式幂级数环，本文为F级数建立了一个普遍/函子性的傅里叶理论：我们证明$X$具有傅里叶变换，并且对于几乎任何理想$I\subseteq R$，其中$R=\mathcal{O}_{K}\left[a_{0},\ldots,a_{p-1},b_{0},\ldots,b_{p-1}\right]$，这个傅里叶变换通过模$I$的商下降，这在$X$上施加了由$I$编码的关系。 此外，我们证明了$X$与自身相乘$n$次的结果也具有与下降相容的傅里叶变换。 这些结果推广到任何$d$个不同的 F-级数$X_{1},\ldots,X_{d}$的乘积$X_{1}^{e_{1}}\cdots X_{d}^{e_{d}}$，其指数为整数$e_{1},\ldots,e_{d}\geq0$。 使用这些傅里叶变换，F级数及其乘积可以与在$\mathbb{Z}_{p}$上的分布相对应，这种对应方式与模$I$的下降相容，从而在逐点乘法下形成代数。 此外，对于任何给定的 F 级数或其乘积，可以关联一个在$K$上的仿射代数簇，我称之为分解簇。 在模$I$下，由 F 级数乘积引起的分布表现出对$I$中包含对应于分布分解簇的理想的相关性。 这提供了一种通过分布以与逐点乘积、卷积和张量积相容的方式对给定仿射代数簇进行编码的新方法。 显示英文摘要

摘要： Let $p$ be an integer $\geq2$ and let $K$ be a global field. A foliated $p$-adic F-series is a function $X$ of a $p$-adic integer variable $\mathfrak{z}$ satisfying the functional equations $X\left(p\mathfrak{z}+j\right)=a_{j}X\left(\mathfrak{z}\right)+b_{j}$ for all $\mathfrak{z}\in\mathbb{Z}_{p}$ and all $j\in\left\{ 0,\ldots,p-1\right\} $, where the $a_{j}$s and $b_{j}$s are indeterminates. Treating $X$ as taking value in a certain ring of formal power series over $K$, this paper establishes a universal/functorial Fourier theory for F-series: we show that $X$ has a Fourier transform, and that, for nearly any ideal $I\subseteq R$, where of $R=\mathcal{O}_{K}\left[a_{0},\ldots,a_{p-1},b_{0},\ldots,b_{p-1}\right]$, this Fourier transform descends through the quotient mod $I$ which imposes on $X$ the relations encoded by $I$. Furthermore, we show that the pointwise product of $X$ with itself $n$ times also has a Fourier transform compatible with descent. These results generalize to products $X_{1}^{e_{1}}\cdots X_{d}^{e_{d}}$ of any $d$ distinct F-series $X_{1},\ldots,X_{d}$ with integer exponents $e_{1},\ldots,e_{d}\geq0$. Using these Fourier transforms, F-series and their products can be identified with distributions on $\mathbb{Z}_{p}$ in a manner compatible with descent mod $I$, forming algebras under pointwise multiplication. Also, to any given F-series or product thereof, one can associate an affine algebraic variety over $K$ which I call the breakdown variety. The distributions induced by a product of F-series under descent mod $I$ exhibit sensitivity to $I$'s containment of the ideal corresponding to the distributions' breakdown varieties. This yields a novel method of encoding given affine algebraic varieties through distributions in a way compatible with pointwise products, convolutions, and tensor products.