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Theta-Induced Diffusion on Tate Elliptic Curves over Non-Archimedean Local Fields

Patrick Erik Bradley

TL;DR

This work develops diffusion operators on Tate elliptic curves over non-archimedean local fields, constructing an $L^2$-type operator $\mathcal{H}_\theta$ from a theta-function kernel using the invariant measure $|\omega|$ and proving a Berkovich-analytic analogue. The spectrum of the diffusion is purely discrete, comprising a finite part from a $v(q)$-dimensional circle-structure and an infinite Kozyrev-wavelet part, and a corresponding heat equation yields a Markov process on the rational points. Extending to the Berkovich space via the Monge-Ampère equation and Chambert-Loir measures, the diffusion $\mathcal{H}_{\theta,\sigma}$ retains the same spectrum and induces a diffusion on $E_q^{an}$. Importantly, spectral data encode arithmetic features of the Tate curve, such as 2-torsion and the parity of $v(q)$, establishing a bridge between ultrametric diffusion and Tate-curve arithmetic with potential applications ranging from geometric reconstruction to data analysis in topological settings.

Abstract

A diffusion operator on the $K$-rational points of a Tate elliptic curve $E_q$ is constructed, where $K$ is a non-archimedean local field, as well as an operator on the Berkovich-analytification $E_q^{an}$ of $E_q$. These are integral operators for measures coming from a regular $1$-form, and kernel functions constructed via theta functions. The second operator can be described via certain non-archimedan curvature forms on $E_q^{an}$. The spectra of these self-adjoint bounded operators on the Hilbert spaces of $L^2$-functions are identical and found to consist of finitely many eigenvalues. A study of the corresponding heat equations yields a positive answer to the Cauchy problem, and induced Markov processes on the curve. Finally, some geometric information about the $K$-rational points of $E_q$ is retrieved from the spectrum.

Theta-Induced Diffusion on Tate Elliptic Curves over Non-Archimedean Local Fields

TL;DR

This work develops diffusion operators on Tate elliptic curves over non-archimedean local fields, constructing an -type operator from a theta-function kernel using the invariant measure and proving a Berkovich-analytic analogue. The spectrum of the diffusion is purely discrete, comprising a finite part from a -dimensional circle-structure and an infinite Kozyrev-wavelet part, and a corresponding heat equation yields a Markov process on the rational points. Extending to the Berkovich space via the Monge-Ampère equation and Chambert-Loir measures, the diffusion retains the same spectrum and induces a diffusion on . Importantly, spectral data encode arithmetic features of the Tate curve, such as 2-torsion and the parity of , establishing a bridge between ultrametric diffusion and Tate-curve arithmetic with potential applications ranging from geometric reconstruction to data analysis in topological settings.

Abstract

A diffusion operator on the -rational points of a Tate elliptic curve is constructed, where is a non-archimedean local field, as well as an operator on the Berkovich-analytification of . These are integral operators for measures coming from a regular -form, and kernel functions constructed via theta functions. The second operator can be described via certain non-archimedan curvature forms on . The spectra of these self-adjoint bounded operators on the Hilbert spaces of -functions are identical and found to consist of finitely many eigenvalues. A study of the corresponding heat equations yields a positive answer to the Cauchy problem, and induced Markov processes on the curve. Finally, some geometric information about the -rational points of is retrieved from the spectrum.
Paper Structure (9 sections, 26 theorems, 160 equations)

This paper contains 9 sections, 26 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Diffusion on the Rational Points of a Tate Elliptic Curve
  3. Measure on $E_q(K)$
  4. A Kernel Function on a Tate Curve
  5. Spectrum of the Diffusion Operator on $E_q(K)$
  6. The heat equation
  7. Heat equation on $K$-analytic manifolds
  8. Diffusion on the Berkovich-Analytification
  9. On Hearing the Shape of a Tate Curve

Key Result

Theorem 1

The space $L^2(E_q(K),\left\lvert\omega\right\rvert)$ has an orthogonal decomposition into $\mathcal{H}_{\theta}$-invariant subspaces. The subspace $L^2(E_q(K))_\sigma$ is of finite dimension $v(q)$ and spanned by the indicator functions supported on the circles $S_\ell(0)$ of radius $p^{-f\ell}$ centred in zero, with $\ell = 0,\dots,v(q) - 1$, and the subspace $L^2(E_q(K))_0$ is spa

Theorems & Definitions (54)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 2.1: Tate's Uniformisation of Tate Curves
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3: Tate
  • proof
  • Lemma 2.4
  • ...and 44 more