Hearing Tamagawa Factors modulo q-1

The Tamagawa factor w.r.t.\ the prime p of an abelian variety A over a non- archimedean local field K is found to be congruent modulo q−1 to the wavelet eigenvalues of a p-adic Laplacian integral operator on the OK-rational points of its Néron model, where q=p$^f$ is the cardinality of the residue field of K. The method is to express the volume of the K-rational points of A w.r.t.\ the canonical measure in terms of the local L-factor given by the Frobenius action on ℓ-adic cohomolgy, and the Tamagawa factor; and then observe that this coincides with the Serre invariant of that compact p-adic analytic manifold modulo q−1. A previous result by Á.M.\ Ledezma and the author on hearing Serre invariants then yields the asserted congruence.