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

Abstract

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.