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

An Huang, Christian Jepsen

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 Néron-Tate local height function of the Tate curve.

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

Abstract

The -adic string worldsheet action on the quotient of the Bruhat-Tits tree of by a genus 1 Schottky group has a dual description on the asymptotic boundary, the Tate curve . We show that the two point function of the dual action coincides with the Néron-Tate local height function of the Tate curve.

Paper Structure

This paper contains 6 sections, 1 theorem, 39 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Properties of the dual action
  3. Green's function and the height function
  4. Spectrum of $D$
  5. Determinant of $D$
  6. The Néron local height function and holography

Key Result

Theorem 3.1

$G(x,y):=h(\frac{x}{y})$ when $m-1\geq v(x)\geq v(y)\geq 0$, extended by symmetry $G(x,y)=G(y,x)$ to the case when $v(x)< v(y)$, is the unique symmetric Green's function up to an additive constant, for the operator $D$, i.e. $DG(x,y)=\delta_{x,y}-\frac{1}{\text{Vol}}$. $\blacktriangleleft$$\blacktri

Figures (1)

  • Figure 1: The $p$-adic string worldsheet at one loop: the tree quotient $T_p/\Gamma$. For this example $p=2$ and $m=5$.

Theorems & Definitions (7)

  • Remark 1.1
  • Remark 2.1
  • Remark 2.2
  • Theorem 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 4.1