In the paper The notion of dimension in geometry and algebra, Yuri I. Manin writes :

“This guess involves the conjectural existence of a geometrical world defined over “an absolute point” $\mathsf{Spec}~\mathbb{F}_1$ where $\mathbb{F}_1$ is a mythical field with one element. For some insights about this world, see [J. Tits, Sur les analogues algebriques des groupes semi–simples complexes], [A. Smirnov, Hurwitz inequalities for number fields], [A. Smirnov, Letters to Yu. Manin of Sept. 29 and Nov. 29, 2003], [M. Kapranov, A. Smirnov, Cohomology determinants and reciprocity laws: number field case], [Yu. Manin, Three–dimensional hyperbolic geometry as ∞–adic Arakelov geometry], [C. Soule, Les varietes sur le corps a un element].”

The first of two letters from Alexandr Smirnov to Manin also appears in the paper Lambda-rings and the field with one element by James Borger :

“The second purpose is to prove the Riemann hypothesis. With the analogy between integers and polynomials in mind, we might hope that $\mathsf{Spec}~\mathbb{Z}$ would be a kind of curve over $\mathsf{Spec}~\mathbb{F}_1$ , that $\mathsf{Spec}~\mathbb{Z} \times \mathsf{Spec}~\mathbb{Z}$ would not only make sense but be a surface bearing some kind of intersection theory, and that we could then mimic over $\mathbb{Z}$ Weil’s proof of the Riemann hypothesis over function fields. The origins of this idea are unknown to me. Manin [Yuri Manin, Lectures on zeta functions and motives (according to Deninger and Kurokawa)] mentions it explicitly. According to Smirnov [Alexandr L. Smirnov, Letter to Y. Manin. September 29, 2003], the idea occurred to him in 1985 and he mentioned it explicitly in a talk in Shafarevich’s seminar in 1990. It may well be that a number of people have had the idea independently since the appearance of Weil’s proof.”

I thank Yuri I. Manin and James Borger for providing me with copies of these letters. Some of their content is crucial to understand the genesis of Smirnov’s paper Hurwitz inequalities for number fields, which is the first paper we will study in the seminar.

From Smirnov’s letter to Yu. I. Manin, dated september 29th 2003 (links added):

“As to my investigations, I work on the topic since 1985. Till then the subject has been paid next to no attention, with the exception of a paper by J. Tits (1957) and the idea (due to D. Quillen?) to interpret the Barrat-Priddy-Quillen Theorem as the equality $K(\mathbb{F}_1) = \pi^{st}(S^0)$.

My initial goal was (and still is) to construct a “world” which contains algebraic geometry as well as arithmetic, and where all constructions from algebraic geometry (including $\mathsf{Spec}~\mathbb{Z} \times \mathsf{Spec}~\mathbb{Z}$) would be available. From the beginning I believed that this could give an approach to the Riemann hypothesis (similar to Weil’s approach).

I started with the idea, known for me from a seminar (early 80-s), that sets can be considered as vectorspaces over $\mathbb{F}_1$. I believed that this idea was promising in view of the mentioned interpretation of the Barrat-Priddy-Quillen Theorem.

Since I couldn’t invent $\mathsf{Spec}~\mathbb{Z} \times \mathsf{Spec}~\mathbb{Z}$, I worked out a strategy which I have adhered:

- “If we can’t develop the whole desired theory, we should invent as many objects over $\mathbb{F}_1$ as possible and establish connections between them.”
- “Since the situation is extremely rigid, any flexibility of constructions would lead to essential progress.”

The idea to construct finite extensions of $\mathbb{F}_1$ (thus getting the missing flexibility) and the suggestion to consider the monoids $0 \cap \pmb{\mu}$ as a naive technical approximation to these extensions arose precisely from this strategy.

Having on hand the extensions, I discovered I could effectively work with a number of new objects over $\mathbb{F}_1$, for instance with $\mathbb{P}^n$. Thus I decided to handle intersection theory (which is part of Weil’s approach to the Riemann hypothesis) on the surface

$$\mathbb{P}^1/\mathbb{F}_1 \times \mathsf{Spec}~\mathbb{Z}$$

instead of the more complicated $\mathsf{Spec}~\mathbb{Z} \times \mathsf{Spec}~\mathbb{Z}$. The Hurwitz genus formula for a map of curves $f~:~X \rightarrow Y$ can be viewed as an example of using intersection theory on the surface $X \times Y$, and I started with it. I succeeded in stating a certain approximation to the Hurwitz formula for the “map”

$$f~:~\mathsf{Spec}~\mathbb{Z} \rightarrow \mathbb{P}^1/\mathbb{F}_1 \qquad \text{where $f \in \mathbb{Q}$}$$

It was somewhat surprising (and confirming the importance of the approach) that this approximation gave very profound assertions like the ABC-conjecture and others. An impulse to publish these results was given to me by M. Kapranov after he and V. Voevodsky learned about them (1988 or 1989).”