The structure of Deitmar Schemes, II. Zeta functions and automorphism groups

TL;DR

This paper explores the structure of Deitmar schemes over the field with one element, focusing on zeta functions for graphs and automorphism groups, advancing the understanding of schemes in this novel setting.

Contribution

It introduces a new zeta function for graphs via Deitmar schemes and analyzes their automorphism groups and base extensions, expanding the theory of schemes over the field with one element.

Findings

Defined a new zeta function for graphs using Deitmar schemes

Determined automorphism groups of Deitmar schemes

Analyzed base extensions to fields

Abstract

We provide a coherent overview of a number of recent results obtained by the authors in the theory of schemes defined over the field with one element. Essentially, this theory encompasses the study of a functor which maps certain geometries including graphs to Deitmar schemes with additional structure, as such introducing a new zeta function for graphs. The functor is then used to determine automorphism groups of the Deitmar schemes and base extensions to fields.