Green's Functions for Vladimirov Derivatives and Tate's Thesis

TL;DR

This paper explores the connection between Vladimirov derivatives, Green's functions, and Tate's thesis within the context of $p$-adic field theories, revealing a deep link between local and global zeta integrals.

Contribution

It establishes a formula for Vladimirov derivatives via Fourier transforms and links Green's functions to the local and global functional equations of zeta integrals, highlighting Tate's thesis in physics.

Findings

Green's functions correspond to local zeta integral equations

Two-point functions satisfy an adelic product formula

Vladimirov derivatives are expressed through Fourier conjugates

Abstract

Given a number field $K$ with a Hecke character $\chi$, for each place $\nu$ we study the free scalar field theory whose kinetic term is given by the regularized Vladimirov derivative associated to the local component of $\chi$. These theories appear in the study of $p$-adic string theory and $p$-adic AdS/CFT correspondence. We prove a formula for the regularized Vladimirov derivative in terms of the Fourier conjugate of the local component of $\chi$. We find that the Green's function is given by the local functional equation for Zeta integrals. Furthermore, considering all places $\nu$, the field theory two-point functions corresponding to the Green's functions satisfy an adelic product formula, which is equivalent to the global functional equation for Zeta integrals. In particular, this points out a role of Tate's thesis in adelic physics.