Heat traces and spectral zeta functions for $p$-adic Laplacians

The study of the heat traces and spectral zeta functions for certain $p$-adic Laplacians is initiated. It is shown that the heat traces are given by $p$-adic integrals of Laplace type, and that the spectral zeta functions are $p$-adic integrals of Igusa type. Good estimates are found for the behaviour of the heat traces when the time tends to infinity, and for the asymptotics of the function counting the eigenvalues less than or equal to a given quantity.