Counting points and acquiring flesh

TL;DR

This paper explores the combinatorial and scheme-theoretic foundations of finite geometry over the field with one element, including zeta functions of Deitmar schemes, highlighting recent developments.

Contribution

It introduces ideas based on $\

Findings

Connections between combinatorial theory and $\

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contribution

Abstract

This set of notes is based on a lecture I gave at "50 years of Finite Geometry | A conference on the occasion of Jef Thas's 70th birthday," in November 2014. It consists essentially of three parts: in a first part, I introduce some ideas which are based in the combinatorial theory underlying $\mathbb{F}_1$, the field with one element. In a second part, I describe, in a nutshell, the fundamental scheme theory over $\mathbb{F}_1$ which was designed by Deitmar. The last part focuses on zeta functions of Deitmar schemes, and also presents more recent work done in this area.