Category: absolute

Almost three decades ago, Yuri Manin submitted the paper “New dimensions in geometry” to the 25th Arbeitstagung, Bonn 1984. It is published in its proceedings, Springer Lecture Notes in Mathematics 1111, 59-101 and there’s a review of the paper available online in the Bulletin of the AMS written by Daniel Burns.

In the introduction Manin makes some highly speculative but inspiring conjectures. He considers the ring

$$\mathbb{Z}[x_1,\ldots,x_m;\xi_1,\ldots,\xi_n]$$

where $\mathbb{Z}$ are the integers, the $\xi_i$ are the “odd” variables anti-commuting among themselves and commuting with the “even” variables $x_j$. To this ring, Manin wants to associate a geometric object of dimension $1+m+n$ where $1$ refers to the “arithmetic dimension”, $m$ to the ordinary geometric dimensions $(x_1,\ldots,x_m)$ and $n$ to the new “odd dimensions” represented by the coordinates $(\xi_1,\ldots,\xi_n)$. Manin writes :

“Before the advent of ringed spaces in the fifties it would have been difficult to say precisely what me mean when we speak about this geometric object. Nowadays we simply define it as an “affine superscheme”, an object of the category of topological spaces locally ringed by a sheaf of $\mathbb{Z}_2$-graded supercommutative rings.”

Here’s my own image (based on Mumford’s depiction of $\mathsf{Spec}(\mathbb{Z}[x])$) of what Manin calls the three-space-2000, whose plain $x$-axis is supplemented by the set of primes and by the “black arrow”, corresponding to the odd dimension.

Manin speculates : “The message of the picture is intended to be the following metaphysics underlying certain recent developments in geometry: all three types of geometric dimensions are on an equal footing”.

Probably, by the addition “2000” Manin meant that by the year 2000 we would as easily switch between these three types of dimensions as we were able to draw arithmetic schemes in the mid-80ties. Quod non.

Twelve years into the new millenium we are only able to decode fragments of this. We know that symmetric algebras and exterior algebras (that is the “even” versus the “odd” dimensions) are related by Koszul duality, and that the precise relationship between the arithmetic axis and the geometric axis is the holy grail of geometry over the field with one element.

For aficionados of $\mathbb{F}_1$ there’s this gem by Manin to contemplate :

“Does there exist a group, mixing the arithmetic dimension with the (even) geometric ones?”

Way back in 1984 Manin conjectured : “There is no such group naively, but a ‘category of representations of this group’ may well exist. There may exist also certain correspondence rings (or their representations) between $\mathsf{Spec}(\mathbb{Z})$ and $x$.”

Here are the scans of my crude prep-notes for some of the later seminar-talks. These notes still contain mistakes, most of them were corrected during the talks. So, please, read these notes with both mercy are caution!

Hurwitz formula imples ABC : The proof of Smirnov’s argument, but modified so that one doesn’t require an $\epsilon$-term. This is known to be impossible in the number-theory case, but a possible explanation might be that not all of the Smirnov-maps $q~:~\mathsf{Spec}(\mathbb{Z}) \rightarrow \mathbb{P}^1_{\mathbb{F}_1}$ are actually covers.

Frobenius lifts and representation rings : Faithfully flat descent allows us to view torsion-free $\mathbb{Z}$-rings with a family of commuting Frobenius lifts (aka $\lambda$-rings) as algebras over the field with one element $\mathbb{F}_1$. We give several examples including the two structures on $\mathbb{Z}[x]$ and Adams operations as Frobenius lifts on representation rings $R(G)$ of finite groups. We give an example that this extra structure may separate groups having the same character table. In general this is not the case, the magic Google search term is ‘Brauer pairs’.

Big Witt vectors and Burnside rings : Because the big Witt vectors functor $W(-)$ is adjoint to the tensor-functor $- \otimes_{\mathbb{F}_1} \mathbb{Z}$ we can view the geometrical object associated to $W(A)$ as the $\mathbb{F}_1$-scheme determined by the arithmetical scheme with coordinate ring $A$. We describe the construction of $\Lambda(A)$ and describe the relation between $W(\mathbb{Z})$ and the (completion of the) Burnside ring of the infinite cyclic group.

Density theorems and the Galois-site of $\mathbb{F}_1$ : We recall standard density theorems (Frobenius, Chebotarev) in number theory and use them in combination with the Kronecker-Weber theorem to prove the result due to James Borger and Bart de Smit on the etale site of $\mathsf{Spec}(\mathbb{F}_1)$.

New geometry coming from $\mathbb{F}_1$ : This is a more speculative talk trying to determine what new features come up when we view an arithmetic scheme over $\mathbb{F}_1$. It touches on the geometric meaning of dual-coalgebras, the Habiro-structure sheaf and Habiro-topology associated to $\mathbb{P}^1_{\mathbb{Z}}$ and tries to extend these notions to more general settings. These scans are unintentionally made mysterious by the fact that the bottom part is blacked out (due to the fact they got really wet and dried horribly). In case you want more info, contact me.

October 21st : Dear Diary,

today’s seminar was fun, though a bit unconventional. The intention was to explain faithfully flat descent, but at the last moment i had the crazy idea to let students discover the main idea themselves (in the easiest of examples) by means of this thought experiment :

“I am an alien, and a very stubborn alien at that. To us, the only existing field is $\mathbb{C}$ and the only rings we accept are $\mathbb{C}$-algebras. We’ve heard rumours that you humans think there is some geometry hidden under $\mathbb{C}$, in particular we’ve heard that you consider something called real manifolds. Can you explain what an algebra over this non-existant field under $\mathbb{C}$ is in a way we can understand?”

The first hurdle was to explain the concept of complex conjugation, as the alien (me) was unwilling to decompose a number $c$ in two ‘ghost components’ $a+bi$. But, i had to concede that i knew about addition and multiplication and knew $1$ and that $-1$ had a square root which they called $i$.

‘Oh, but then you know what $\mathbb{Z}[i]$ is! You just add a number of times $1$’s and $i$’s.’

‘Why are you humans so focussed on counting? We do not count! We can’t! We have neither fingers nor toes!’

Admittedly a fairly drastic intervention, but i had to keep them on the path leading to Galois descent… After a while we agreed on a map, which they called conjugation, sending sums to sums and products to products and taking a root of unity to its inverse.

Next, they asked me to be a bit flexible and allow for ‘generalized’ fields such as consisting of all elements fixed under conjugation! Clearly, the alien refused : ‘We’re not going on the slippery road called generalization, we’ve seen the havock this has caused in human-mathematics.’

It took them a while to realize they would never be able to sell me an $\mathbb{R}$-algebra $A$, but could perhaps try to sell me the complex algebra $B= A \otimes_{\mathbb{R}} \mathbb{C}$.

Alien : ‘But, how do i recognize one of your algebras? Do they have a special property i can check?’

Humans : ‘Yes, they have some map (which we know to be the map $a \otimes c \mapsto a \otimes \overline{c}$, but you cannot see it) sending sums to sums, products to products and extending conjugation on $\mathbb{C}$.’

Alien : ‘But if i take a basis for any of my algebras and apply conjugation to all its coordinates, then all my algebras have this property, not?’

Humans : ‘No, such maps are good for sums, but not always for products. For example, take $\mathbb{C}[x]/(x^2-c)$ for $c$ a complex-number not fixed under conjugation.’

Alien : ‘Point taken. But then, your algebras are just a subclass of my algebras, right?’

Humans : ‘No! An algebra can have several of such additional maps. For example, take $B = \mathbb{C} \times \mathbb{C}$ then there is one sending $(a,b)$ to $(\overline{a},\overline{b})$ and another sending it to $(\overline{b},\overline{a})$. (because we know there are two distinct real algebras $\mathbb{R} \times \mathbb{R}$ and $\mathbb{C}$ of dimension two, tensoring both to $\mathbb{C} \times \mathbb{C}$.)’

By now, the alien and humans agreed on a dictionary : what to humans is the $\mathbb{R}$-algebra $A$ is to the alien the complex algebra $B=A \otimes \mathbb{C}$ together with a map $\gamma_B : B \rightarrow B$ sending sums to sums, products to products and extending conjugation on $\mathbb{C}$ (the extra structure, that is the map $\gamma_B$ is called the ‘descent data’).

Likewise, a human-observed $\mathbb{R}$-algebra morphism $\phi : A \rightarrow A’$ is to the alien the the $\mathbb{C}$-algebra morphism $\Phi = \phi \otimes id_{\mathbb{C}} : B \rightarrow B’$ which commutes with the extra structures, that is, $\Phi \circ \gamma_B = \gamma_{B’} \circ \Phi$.

Phrased differently (the alien didn’t want to hear any of this) : there is an equivalence of categories between the category $\mathbb{R}-\mathsf{algebras}$ of commutative $\mathbb{R}$-algebras and the category $\gamma-\mathsf{algebras}$ consisting of complex commutative algebras $B$ together with a ringmorphism $\gamma_B$ extending complex conjugation and with morphisms $\mathbb{C}$-algebra morphisms compatible with the $\gamma$-structure.

Further, what to humans is the base-extension (or tensor) functor

$- \otimes_{\mathbb{R}} \mathbb{C}~:~\mathbb{R}-\mathsf{algebras} \rightarrow \mathbb{C}-\mathsf{algebras}$

is (modulo the above equivalence) to the alien merely the forgetful functor

$\mathsf{Forget}~:~\gamma-\mathsf{algebras} \rightarrow \mathbb{C}-\mathsf{algebras}$

stripping off the descent-data.

After the break (yes, it took us that long to get here) we used this idea to invent rings living ‘under $\mathbb{Z}$’, or if you want, algebras over the field with one element $\mathbb{F}_1$.

Alien : ‘Ha-ha-ha a field with one element? Surely, you’re joking Mr. Human’

Note to self : Dare to waste more time like this in a seminar. It may very well be the only thing they will still remember next year.

We’ve had three seminar-sessions so far, and the seminar-blog ‘angs+’ contains already 20 posts and counting. As blogging is not a linear activity, I will try to post here at regular intervals to report on the ground we’ve covered in the seminar, providing links to the original angs+ posts.

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)

F_un Mathematics

Hardly a ‘new’ blog, but one that is getting a new life! On its old homepage you’ll find a diagonal banner stating ‘This site has moved’ and clicking on it will guide you to its new location : cage.ugent.be/~kthas/Fun.

From now on, this site will be hosted at the University of Ghent and maintained by Koen Thas. So, please update your bookmarks and point your RSS-aggregator to the new feed.

Everyone interested in contributing to this blog dedicated to the mathematics of the field with one element should contact Koen by email.

angst

Though I may occasionally (cross)post at F_un mathematics, my own blog-life will center round a new blog to accompany the master-course ‘seminar noncommutative geometry’ I’m running at Antwerp University this semester. Its URL is noncommutative.org and it is called :

Here, angs is short for Antwerp Noncommutative Geometry Seminar and the additions @t resp. + are there to indicate we will experiment a bit trying to find useful interactions between the IRL seminar, its blog and social media such as twitter and Google+.

The seminar (and blog) are scheduled to start in earnest september 30th, but I may post some prep-notes already. This semester the seminar will try to decode Smirnov’s old idea to prove the ABC-conjecture in number theory via geometry over the field with one element and connect it with new ideas such as Borger’s $\mathbb{F}_1$-geometry using $\lambda$-rings and noncommutative ideas proposed by Connes, Consani and Marcolli.

Again, anyone willing to contribute actively is invited to send me an email or to comment on ‘angst’, tweet about it using the hashtag #angs (all such tweets will appear on the frontpage) or share its posts on Google+.

Noncommutative Arithmetic Geometry Media Library

Via the noncommutative geometry blog a new initiative maintained by Alain Connes and Katia Consani was announced : the Noncommutative Arithmetic Geometry Media Library.

This site is dedicated to maintain articles, videos, and news about meetings and activities related to noncommutative arithmetic geometry. The website is still `under construction’ and the plan is to gradually add more videos (also from past conferences and meetings), as well as papers and slides.