Counting points and acquiring flesh

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.