One-loop $p$-adic string theory and the Néron local height function

The $p$-adic string worldsheet action on the quotient of the Bruhat-Tits tree of $PGL(2,{\mathbb{Q}}_p)$ by a genus 1 Schottky group has a dual description on the asymptotic boundary, the Tate curve ${\mathbb{Q}}_p^{\ast }\mathrm{/}{q}^{\mathbb{Z}}$. We show that the two point function of the dual action coincides with the Néron-Tate local height function of the Tate curve.