Hearing the Serre invariant of a compact $p$-adic analytic manifold

TL;DR

This paper introduces a novel method to determine the Serre invariant of a compact p-adic analytic manifold using Laplacian operators and wavelet spectra, linking geometric invariants to spectral properties.

Contribution

It develops a new approach to 'hear' the Serre invariant via Laplacian eigenvalues and relates it to the spectrum of a wavelet operator on p-adic manifolds, especially elliptic curves.

Findings

Wavelet eigenvalues are congruent to the Serre invariant modulo q-1.

The number of rational points relates to the wavelet spectrum.

The Serre invariant vanishes modulo q-1 for elliptic curves with split multiplicative reduction.

Abstract

Using a previous novel way of defining kernel functions for Laplacian integral operators on a compact $p$-adic analytic manifold $X$, one such operator $\Delta_0^s$ with $s\in\mathds{R}$ is applied to hearing the Serre invariant $i(X)$ by showing that a wavelet eigenvalue is always congruent to $i(X)$ modulo $q-1$, where $q$ is the cardinality of the residue field $k$ attached to a $p$-adic number field $K$. It is shown how the number of $k$-rational points of the special fibre of the N\'eron model of an elliptic curve defined over $K$ relates to the wavelet spectrum of $\Delta^s_0$, and this then leads to the realisation that the Serre invariant $i(E(X))$ in the case of an elliptic curve $E$ with split multiplicative reduction vanishes modulo $q-1$.