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

Abstract

Let $p$ be an integer $\ge 2$ 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 $j\in \left\{0,\dots ,p-1\right\}$, 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 $R={\mathcal{O}}_K\left[a_0,\dots ,a_{p-1},b_0,\dots ,b_{p-1}\right]$, 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 $X_1,\dots ,X_d$ with integer exponents $e_1,\dots ,e_d\ge 0$. 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.

154 pages