Hearing the Serre invariant of a compact $p$-adic analytic manifold
Patrick Erik Bradley, Ángel Morán Ledezma
TL;DR
The paper develops a $p$-adic diffusion framework on compact $p$-adic analytic manifolds using a finite equitising atlas to define a Vladimirov-Zúñiga-type operator $Δ_0^s$. It proves that the Kozyrev wavelet eigenvalues $λ_ψ$ of $Δ_0^s$ satisfy $λ_ψ ≡ i(X) \pmod{q-1}$, enabling the Serre invariant $i(X)$ to be heard from the spectrum. By relating $i(X)$ to measures arising from Néron models, it shows that for Tate elliptic curves $E/K$ the eigenvalues are congruent to zero modulo $q-1$, i.e. $λ_ψ ≡ 0 \pmod{q-1}$. This builds a bridge between $p$-adic diffusion spectra and arithmetic invariants, offering spectral means to infer geometric and arithmetic structure of elliptic curves over $K$. Overall, the work links diffusion, Serre invariants, and elliptic curve reduction in the $p$-adic analytic setting, with potential applications to arithmetic geometry and spectral number theory.
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 $Δ_0^s$ 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 Néron model of an elliptic curve defined over $K$ relates to the wavelet spectrum of $Δ^s_0$, 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 $q-1$.