Green's function on the Tate curve

TL;DR

The paper defines a $p$-adic flat Laplacian $D$ on the Tate curve $E_q=\mathbb{Q}_p^{\times}/q^{\mathbb{Z}}$ with $|q|<1$ and proves the Green's function $G$ exists as a limit of finite-quotient Green's functions. It establishes a precise decomposition $G(x,y)=B(x,y)+C(x,y)$ where $C$ depends only on $|x|,|y|$ and $B$ on $|x|,|y|,|x-y|$, and provides an explicit infinite-series form for $B$ built from a log singularity plus corrections, with absolute convergence. The cross-term $DB_{p,m}(x,y)$ is given in closed form in terms of valuations via $U(y)$, and $C$ is characterized through a linear system that admits a symmetrization and recurrence-based computation, with extensions to finite extensions $K$ of $\mathbb{Q}_p$ outlined. These results yield a concrete non-Archimedean analogue of the Archimedean torus Green's function and support further study of $p$-adic string worldsheet theories on genus-one curves and their connections to Tate's thesis and related arithmetic structures.

Abstract

Motivated by the question of defining a $p$-adic string worldsheet action in genus one, we define a Laplacian operator on the Tate curve, and study its Green's function. We show that the Green's function exists. We provide an explicit formula for the Green's function, which turns out to be a non-Archimedean counterpart of the Archimedean Green's function on a flat torus.

Theorems & Definitions (33)

• Remark 1.1
• Remark 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Corollary 2.3
• proof
• Corollary 2.4
• proof