One-loop $p$-adic string theory and the Néron local height function

Abstract: The $p$-adic string worldsheet action on the quotient of the Bruhat-Tits tree of $PGL(2,\mathbb{Q}_p)$ by a genus 1 Schottky group has a dual description on the asymptotic boundary, the Tate curve $\mathbb{Q}_p<sup>\ast/q<sup>\mathbb{Z}$. We show that the two point function of the dual action coincides with the Néron-Tate local height function of the Tate curve.

• The paper demonstrates that the one-loop p-adic string worldsheet action matches the Néron-Tate local height function on the Tate curve up to an additive constant.
• It employs explicit spectral analysis and zeta-function regularization to compute Green's functions and determinants associated with p-adic moduli spaces.
• The results establish a holographic correspondence that bridges non-Archimedean string amplitudes with classical arithmetic invariants, impacting p-adic AdS/CFT.

One-loop $p$-adic String Theory and the Néron Local Height Function

Introduction and Motivation

This paper rigorously establishes a correspondence between the one-loop worldsheet action in $p$-adic string theory and the Néron-Tate local height function on the Tate elliptic curve. While $p$-adic string theory has a rich history since the pioneering work of Freund and Olson, providing non-Archimedean analogues of tree-level and loop-level string amplitudes, the genus one (one-loop) case remains less developed due to the more intricate geometry of the associated Bruhat-Tits trees. Recent advances clarified this picture, culminating in the dual worldsheet/boundary description on the Tate curve $\mathbb{Q}_p^*/q^\mathbb{Z}$, with $q$ parametrizing the modulus.

The main result is that the two-point function for the boundary dual of the $p$-adic one-loop action coincides with the local Néron-Tate height function on the Tate curve, up to an explicit additive constant. This deepens the interplay between non-Archimedean physics and arithmetic geometry, highlighting the significance of Green's functions and spectral properties of associated pseudo-differential operators acting on $p$-adic moduli spaces.

The $p$-adic One-loop Worldsheet and Boundary Dual

The $p$-adic string one-loop amplitude is constructed from the quotient of the Bruhat-Tits tree $T_p$ by a rank-1 Schottky subgroup $p$0, resulting in a genus 1 graph. The asymptotic boundary is the Tate curve $p$1, modeled as the disjoint union of fundamental domains on which Haar measures are naturally defined.

Figure 1: The $p$2-adic string worldsheet at one loop: the tree quotient $p$3 for $p$4, $p$5.

The bulk (tree) Laplacian action reduces, via holography, to a non-local boundary action,

$p$6

where $p$7 is a singular, self-adjoint, positive semi-definite operator constructed from a weight function $p$8 encoding the geometric interaction kernel of the worldsheet. $p$9 is characterized by explicit formulae depending on $p$0, $p$1, and the valuations of its arguments, ensuring invariance under dilations and inversions.

Symmetries of the Action and Operator Structure

The specific form of $p$2 guarantees the action's symmetry under dilatations and inversions, reflecting the modular properties of the underlying Tate curve. Under a change of variable or group action, the kernel $p$3 transforms covariantly, preserving the self-adjoint structure of $p$4 and ensuring its spectral properties align with expectations for two-dimensional Laplacians on quotients of trees.

This symmetry extends to the spectrum of $p$5, as it acts diagonally on the abelian multiplicative character basis of $p$6. This yields a decomposition into radial (valuation-dependent) and angular (unit-dependent) eigenfunctions, with eigenvalues governed by conductors and roots of unity, respectively.

Green's Functions and the Néron Local Height

A central technical result is the explicit calculation of the Green's function $p$7 for $p$8. For $p$9, this coincides with the Néron-Tate local height function

$\mathbb{Q}_p^*/q^\mathbb{Z}$0

where $\mathbb{Q}_p^*/q^\mathbb{Z}$1 denotes $\mathbb{Q}_p^*/q^\mathbb{Z}$2-adic valuation and $\mathbb{Q}_p^*/q^\mathbb{Z}$3 reflects the Tate module.

The calculation proceeds by decomposing the integration domain according to valuation strata, evaluating finite sums over cosets in the multiplicative group, and leveraging group-theoretical relations to show that $\mathbb{Q}_p^*/q^\mathbb{Z}$4 for $\mathbb{Q}_p^*/q^\mathbb{Z}$5, with $\mathbb{Q}_p^*/q^\mathbb{Z}$6 the measure of $\mathbb{Q}_p^*/q^\mathbb{Z}$7. By leveraging covariance, this extends to all $\mathbb{Q}_p^*/q^\mathbb{Z}$8 (up to an additive constant and symmetrization).

This is a strong claim: up to a constant, the Green's function for a non-local Laplacian on a $\mathbb{Q}_p^*/q^\mathbb{Z}$9-adic stringy worldsheet is identical to a key geometric invariant from arithmetic geometry.

Spectrum and Weyl Law for $q$0

The operator $q$1 is shown to have discrete spectrum, with eigenvalues corresponding to multiplicative characters on $q$2. Radial eigenvalues depend solely on the conductor $q$3 and are of the form $q$4. The degeneracies follow from the group structure, yielding explicit multiplicities in analogy with Laplace spectra on Riemann surfaces.

Angular eigenvalues are parameterized by roots of unity $q$5, with corresponding eigenvalues

$q$6

The smallest eigenvalue is always among the angular sector, governing the spectral gap and determining the asymptotics.

A Weyl-type law is established for eigenvalue counting, mirroring classical spectral asymptotics for two-dimensional Laplacians.

Regularized Determinant and Amplitude Interpretation

The determinant of $q$7 is evaluated via a zeta-function regularization, splitting contributions into angular and radial parts. Explicit closed-form expressions are derived:

• The angular determinant is a finite product over eigenvalues at roots of unity.
• The radial part gets a resummed representation via zeta regularization, with contributions depending only on $q$8 and $q$9.

The total determinant,

$p$0

serves as the one-loop vacuum partition function for the $p$1-adic string on the genus one worldsheet. This makes the calculation directly relevant for physical amplitude computations in $p$2-adic string and AdS/CFT contexts.

Holographic Interpretation and Arithmetic Implications

Within the holographic $p$3-adic AdS/CFT framework, the operator $p$4 emerges as the boundary limit of a bulk Laplacian. The dimensionless limit ($p$5) of the two-point function of the dual operator $p$6 yields the Néron local height, including normalization constants. This shows that arithmetic invariants computed from intersection theory have a natural appearance as correlation functions in the $p$7-adic holographic boundary theory.

This result has several implications:

• It provides a direct analytic realization of arithmetic heights as physical correlators.
• The machinery developed can potentially generalize to higher genus, other $p$8-adic moduli spaces, and to the study of quantum gravity models on graphs.
• It points toward interactions between $p$9-adic string theory and the arithmetic geometry of elliptic curves, with consequences for partition functions, quantum amplitudes, and possibly non-Archimedean entanglement entropy.

Conclusion

This work demonstrates that the one-loop, genus-one $p$0-adic string worldsheet action is holographically dual to a boundary theory encoding the Néron-Tate local height function on the Tate curve. The identification of the two-point function with the local height, the explicit spectral analysis, and the determinant computation together suggest a robust and calculable correspondence between non-Archimedean physics and classical invariants in arithmetic geometry. The results pave the way for deeper exploration of $p$1-adic AdS/CFT, connections to arithmetic quantum field theory, and explicit calculations in non-Archimedean string theory, with potential ramifications for number theory and theoretical physics alike.