Probing holography in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>p</mml:mi></mml:math> -adic CFT

We holographically calculate the partition functions of certain types isotropic sectors CFTs dual to Bruhat-Tits trees and $p$-adic BTZ black holes. Along way, we propose new spectral decompositions Laplacian operator other than plane-wave basis on these two background, with both analytical numerical evidence. extract density states hence entropy from function via inverse Laplace transform. Then one-loop Witten diagram is computed in hole yielding constraints heavy-heavy-light averaged three-point coefficient its boundary CFT. Finally, for general (not necessarily holographic), analyze representation theory their global conformal group $PGL\left(2,\mathbb{Q}_p\right)$, discuss suitability different representations as Hilbert spaces