Regularized Products over arithmetic Schemes

TL;DR

This paper introduces a novel regularization method for the product of residue field cardinalities in arithmetic schemes, leading to a new invariant that is transcendental and confirms the infinitude of closed points.

Contribution

It develops a regularization technique generalizing zeta-regularization to define a new arithmetic invariant for schemes, with explicit computations in several cases.

Findings

The invariant is always transcendental.

The set of closed points in an arithmetic scheme is infinite.

Explicit calculations of the invariant in specific cases.

Abstract

In this paper, we study the closed points of arithmetic schemes. We accomplish this by showing that the product of the cardinals of residue fields of closed points in an arithmetic scheme can be regularized. This regularization yields a new arithmetic invariant attached to the scheme. We compute it explicitly in several cases and find that it is always a transcendental number. This result provides a proof that the set of closed points is infinite. Our main tool is a regularization technique, which generalizes the zeta-regularization method introduced by Mu\~noz and P\'erez.