One-loop $p$-adic string theory and the N\'eron local height function

TL;DR

This paper explores the duality between $p$-adic string theory on a Bruhat-Tits tree quotient and the Tate curve, demonstrating that the two-point function matches the Néron-Tate local height function.

Contribution

It establishes a connection between $p$-adic string worldsheet actions and arithmetic geometry via the Néron-Tate height function on Tate curves.

Findings

Two-point function of the dual action equals the Néron-Tate local height function.

Provides a geometric interpretation of $p$-adic string correlators.

Links string theory on Bruhat-Tits trees to arithmetic invariants.

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 $\mathbb{Q}_p^\ast/q^\mathbb{Z}$. We show that the two point function of the dual action coincides with the N\'eron-Tate local height function of the Tate curve.