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

2025/11/25 by Bradley, Patrick Erik, Ledezma, Ángel Morán
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analysis of PDEs (math.AP) #FOS: Mathematics #Number Theory (math.NT) #advanced mathematical theories

paper · doi:10.48550/arxiv.2511.20631

openalex publication_date 2025/11/25 · openalex created_date 2025/11/28 · openalex updated_date 2026/07/28

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 Δ₀^s with s∈\mathdsR 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^s₀, 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.

Citations

• Boundary Value Problems for p-Adic Elliptic Parisi-Zúñiga Diffusion
• Time-Varying Energy Landscapes and Temperature paths: Dynamic Transition Rates in locally Ultrametric Complex Systems
• Approximating Diffusion on Finite Multi-Topology Systems Using Ultrametrics
• A non-autonomous p-Adic diffusion equation on time changing graphs
• Schottky-Invariant p-Adic Diffusion Operators
• Brownian Motion in a Vector Space over a Local Field is a Scaling Limit
• Heat Equations and Hearing the Genus on p-adic Mumford Curves via Automorphic Forms
• Components and Exit Times of Brownian Motion in two or more p-Adic Dimensions
• Reaction-diffusion Equations on Complex Networks and Turing Patterns, via p-Adic Analysis
• Linear and Nonlinear Heat Equations on a p-Adic Ball

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