The mantra recited by $\mathbb{F}_1$-followers is that $\mathsf{Spec}(\mathbb{Z})$ is far too large to serve as the terminal object in the category of schemes, and, one should view it as a ‘geometric’ object over ‘something’ living ‘under $\mathbb{Z}$’ called $\mathbb{F}_1$ : the field with one element.
In this seminar we will encounter a fair number of proposals as to what this elusive object $\mathsf{Spec}(\mathbb{Z})$ viewed over $\mathsf{Spec}(\mathbb{F}_1)$ might be. Let’s start with the simplest and earliest proposal.
Smirnov’s proposal is that the smooth projective curve $\mathsf{Spec}(\mathbb{Z})$ should have as its schematic points the set $\{ 2,3,5,7,11,13,17,\cdots \} \cup \{ \infty \}$, that is, the set of all prime numbers together with $\infty$, and, that the degree of the ‘point’ $p$ should be equal to $log(p)$ whereas the degree of $\infty$ is equal to $1$.
Attempted explanation : We have seen before that a schematic point $P$ of a curve $C$ defined over $k$ corresponds to a discrete valuation ring in the function field $k(C)$ and that its degree $deg(P)$ equals $[\mathcal{O}_P/\mathfrak{m}_P : k]$.
By analogy, the schematic points of the ‘projective curve’ $\mathsf{Spec}(\mathbb{Z})$ should correspond to all discrete valuations on $\mathbb{Q}$, which by Ostrovski’s theorem are either the $p$-adic valuations $v_p(q)=n$ if $q=p^n \frac{r}{s}$ and $(r,p)=(s,p)=1$ or the real valuation $v_{\infty}(q) = -log |q|$ (minus sign because of the convention that the value of $0$ should be $\infty$).
To motivate the non-sensical definition of the degrees, recall that the degree of the divisor $div(f) = \sum_{P \in C} ord_P(f) [P]$ equals zero for all $f \in \overline{k}(C)$.
Now, if $f$ is in the function field $k(C)$, then its divisor must be invariant under the action of the Galois group $Gal(\overline{k}/k)$ (that is, $ord_{\sigma(P)}(f) = ord_P(f)$ for all Galois-automorphisms $\sigma$). But then, we can write $div(f)$ as a sum over the schematic points (which are the orbits of the geometric points under the action of the Galois group) and hence its degree is
$deg(div(P)) = \sum^{scheme}_{P \in C}~ord_P(f) deg(P) = 0$
where now the sum is taken over all schematic points of $C$. Once again, by analogy, if $f = \pm \frac{p_1^{e_1} \cdots p_r^{e_r}}{q_1^{f_1} \cdots q_s^{f_s}} \in \mathbb{Q}$, then its ‘divisor’ is
$div(f) = \sum_i e_i [p_i] – \sum_j f_j [q_j] – log |f| [\infty]$
and Smirnov’s proposal for the degrees of the scheme points of $\mathsf{Spec}(\mathbb{Z})$ is (up to a common multiple) the only one assuring that the degree of all such divisors is zero.
What is the field of constants?
In this proposal $\mathsf{Spec}(\mathbb{Z})$ is a smooth projective curve with function field $\mathbb{Q}$. To determine the ‘field’ over which it is defined we have (in analogy with the functionfield case where the field of constants is $K \cap \overline{k}$) to determine
$\mathbb{Q} \cap \overline{\mathbb{F}_1} = \mathbb{Q} \cap \pmb{\mu} = \{ +1,-1 \}$
So, $\mathsf{Spec}(\mathbb{Z})$ is not really a curve over $\mathbb{F}_1$, but rather over $\mathbb{F}_{1^2}$.
Smirnov’s surface
In a letter to Manin Smirnov explained that, as a first step towards the intersection theory on $\mathsf{Spec}(\mathbb{Z}) \times \mathsf{Spec}(\mathbb{Z})$ (which might lead to a proof of the Riemann hypothesis by mimicking Weil’s proof in the function field case) he embarked on the intersection theory in the somewhat easier product of two curves
$\mathbb{P}^1~/~\mathbb{F}_1~\times~\mathsf{Spec}(\mathbb{Z})$
Combining the above with the description of the projective line over $\mathbb{F}_1$, we can now depict this Smirnov surface
Recall that the schematic points of $\mathbb{P}^1~/~\mathbb{F}_1$ are $\{ 0,\infty \} \cup \{ [1],[2],[3],\cdots \}$ where the point $[n]$ represents all primitive $n$-th roots of unity and so has degree $\phi(n)$.
