Hearing the Serre Invariant of a Compact <i>p</i> ‐Adic Analytic Manifold

ABSTRACT Using a previous novel way of defining kernel functions for Laplacian integral operators on a compact ‐adic analytic manifold , one such operator with is applied to hearing the Serre invariant by showing that a wavelet eigenvalue (with the wavelet having small support) is always congruent to modulo , where is the cardinality of the residue field attached to a ‐adic number field . It is shown how the number of ‐rational points of the special fiber of the Néron model of an elliptic curve defined over relates to the wavelet spectrum of , and this then leads to the realization that the Serre invariant in the case of an elliptic curve with split multiplicative reduction vanishes modulo .