This year’s goal is to obtain a somewhat definite verdict on the field-with-one-element hype.

In short, the plan is to outline Smirnov’s approach to the ABC-conjecture using geometry over $\mathbb{F}_1$, to describe Borger’s idea for such an $\mathbb{F}_1$-geometry and to test it on elusive objects such as $\mathbb{P}^1_{\mathbb{F}_1} \times_{\mathbb{F}_1} \mathsf{Spec}(\mathbb{Z})$ (relevant in Smirnov’s paper) and $\mathsf{Spec}(\mathbb{Z}) \times_{\mathbb{F}_1} \mathsf{Spec}(\mathbb{Z})$ (relevant to the Riemann hypothesis).

We did start with an historic overview, using recently surfaced material such as the Smirnov letters. Next, we did recall some standard material on the geometry of smooth projective curves over finite fields, their genus leading up to the Hurwitz formula relating the genera in a cover of curves.

Using this formula, a version of the classical ABC-conjecture in number theory can be proved quite easily for curves.

By analogy, Smirnov tried to prove the original ABC-conjecture by viewing $\mathsf{Spec}(\mathbb{Z})$ as a ‘curve’ over $\mathbb{F}_1$. Using the connection between the geometric points of the projective line over the finite field $\mathbb{F}_p$ and roots of unity of order coprime to $p$, we identify $\mathbb{P}^1_{\mathbb{F}_1}$ with the set of all roots of unity together with $\{ [0],[\infty] \}$. Next, we describe the schematic points of the ‘curve’ $\mathsf{Spec}(\mathbb{Z})$ and explain why one should take as the degree of the ‘point’ $(p)$ (for a prime number $p$) the non-sensical value $log(p)$.

To me, the fun starts with Smirnov’s proposal to associate to any rational number $q = \tfrac{a}{b} \in \mathbb{Q} – \{ \pm 1 \}$ a cover of curves

$q~:~\mathsf{Spec}(\mathbb{Z}) \rightarrow \mathbb{P}^1_{\mathbb{F}_1}$

by mapping primes dividing $a$ to $[0]$, primes dividing $b$ to $[\infty]$, sending the real valuation to $[0]$ or $[\infty]$ depending onw whether or not $b > a$ and finally sending a prime $p$ not involved in $a$ or $b$ to $[n]$ where $n$ is the order of the unit $\overline{a}.\overline{b}^{-1}$ in the finite cyclic group $\mathbb{F}_p^*$. Somewhat surprisingly, it does follow from Zsigmondy’s theorem that this is indeed a finite cover for most values of $q$. A noteworthy exception being the map for $q=2$ (which fails to be a cover at $[6]$) and of which Pieter Belmans did draw this beautiful graph

True believers in $\mathbb{F}_1$ might conclude from this graph that there should only be finitely many Mersenne primes… Further, the full ABC-conjecture would follow from a natural version of the Hurwitz formula for such covers.

(to be continued)