Projective spaces over

Abstract

In this essay, we study various notions of projective space (and other schemes) over F 1 ℓ, with F 1 denoting the field with one element. Our leading motivation is the “Hidden Points Principle,” which shows a huge deviation between the set of rational points as closed points defined over F 1 ℓ and the set of rational points defined as morphisms monospace𝚂𝚙𝚎𝚌 ( double-struckF 1 ℓ ) ↦ scriptX. We also introduce, in the same vein as Kurokawa [Proc. Jpn. Acad. Ser. A Math. Sci. 81 (2005), pp. 180–184], schemes of F 1 ℓ‐type and consider their zeta functions.

A taste of Weil theory in characteristic one

“…For every prime p, there is a closed point, and the Kurokawa ({F 1 , F 1 2 }-)zeta function (see [12] and the author's second chapter) should involve a factor of the form i∈{0}∪P (s − ϑ(i)) −1 , where ϑ(•) is a function which arises because we work over F 1 2 . I will come back to this matter in [19].…”

Absolute Quantum Theory (after Chang, Lewis, Minic and Takeuchi), and a Road to Quantum Deletion

“…We call such points simple points. As we showed in [13], and as was conjectured by others (see the details in [13]), we need to include the extra points to set up a natural connection with the "functor-of-points viewpoint. "…”

“…Let F 1 be the algebraic closure of F 1 ; it consists of all complex roots of unity plus an element 0, endowed with the natural multiplication; see [13,12]. For every positive integer ℓ, we have that F 1 ℓ ≤ F 1 .…”

“…So 0 • 1 = 1 • 0 = 0 • 0 = 0, and 1 • 1 = 1. We adopt the traditional F 1 -Mantra that there is no addition [13].…”

“…In this section, our state space will be the vector space V = V (m, F 1 ℓ ). We will take on the viewpoint of our recent paper [13], and also, we will make no distinction at this point between vector spaces over F 1 and its extensions, and affine spaces.…”