Green's function on the Tate curve

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. In particular, it turns out that this Green's function recovers the N\'eron local height function for the Tate curve in the $p\to\infty$ limit, when the $j$-invariant has odd valuation. So this non-Archimedean height function now acquires a physics meaning in terms of the large $p$ limit of a non-Archimedean conformal field theory two point function on the Tate curve, as well as a direct analytic interpretation as a Green's function, on the same footing as in the Archimedean place.