Abstract
Define a $p$-adic analogue to the Laplacian by taking the Fourier conjugate of the $p$-adic norm.This gives an integral over $\mathbb{Q}_p^\times$ which has significance in $p$-adic string theory as it can be used to write down the $p$-adic string action.
We can extend the definition of the $p$-adic Laplacian from the genus 0 case to the genus 1 case, so the integral is now over sections of the quotient $\mathbb{Q}_p^\times/\Gamma$ for some discrete subgroup $\Gamma$ of $\text{GL}(2,\mathbb{Q}_p)$.
In this paper, we restrict our attention to the genus 1 $p$-adic Laplacian where $\Gamma$ is a cyclic subgroup of the form $\Gamma_m = \bigg\langle\left[\begin{array}{cc}
p^m & 0 \\
0 & 1
\end{array}\right]\bigg\rangle$, where $m$ is an integer $\geq 1$.
We diagonalize this operator and find its corresponding Green's fucntion.