All posts by lievenlb

Also blogs at NeverEndingBooks.

Prep-notes dump

Here are the scans of my rough 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 and 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.

$\mathbb{F}_1$ and noncommutative geometry

why noncommutative geometry?

Some motivate noncommutative geometry as follows : assume you have a space (or variety) $X$ on which a group $G$ acts wildly so that the ‘orbit-space’ $X/G$ does not exists or has bad topological properties. Let $A$ be the ring of continuous functions on $X$ (or the coordinate ring $\mathcal{O}(X)$), then every $g \in G$ acts as an automorphism $\alpha_g$ on $A$.

Traditionally one associates the orbit-space (when possible) to the commutative fixed-point algebra $A^G$. However, when this algebra is too small to give information on the $G$-orbits in $X$ one can still associate a noncommutative algebra to the situation, the crossed product algebra $A \ast G$ which as a vectorspace is merely $A \otimes \mathbb{C} G$ but with multiplication induced by $(a \otimes g) (b \otimes h) = a \alpha_g(b) \otimes g h$. Some argue that ringtheoretical invariants of $A \ast G$ give some insight into the horrible orbit-space $X/G$.

relevant to $\mathbb{F}_1$-geometry?

We’ve defined an algebra $A$ over $\mathbb{F}_1$ to be a torsion-free $\mathbb{Z}$-ring having a commuting family of endomorphisms $\psi^n~:~A \rightarrow A$ having the property that for every prime number $p$ the endomorphism $\Psi^p$ is a lift of the Frobenius map on $A/pA$. This gives an action by endomorphisms of the multiplicative monoid $\mathbb{N}_{\times}$ on $A$.

We’ve interpreted this additional structure as descent-data from $\mathbb{Z}$ to $\mathbb{F}_1$. Now, in the case of Galois-descent between two fields $k \subset K$ with $Gal(K/k)=G$, the $k$-algebra corresponding to a $K$-algebra $A$ with descent-data $G \rightarrow Aut(A)$ is, of course, the fixed-point algebra $A^G$.

Of course, in the $\mathbb{F}_1$-setting it makes no sense to look at the fixed-point ring $A^{\mathbb{N}_{\times}}$, but we can still consider the corresponding noncommutative ring

$A \ast \mathbb{N}_{\times}$

which as a $\mathbb{Z}$-module is the tensor-product $A \otimes_{\mathbb{Z}} \mathbb{Z} [\mathbb{N}_{\times}]$ where $\mathbb{Z} [\mathbb{N}_{\times}]$ is the monoid-algebra of the commutative monoid $\mathbb{N}_{\times}$. As above, the multiplication is induced by the rule (using the variables $X_n = 1 \otimes n$)

$(a X_n) (b X_m) = a \Psi^n(b) X_{mn}$

If you are a lowly ringtheorist this is already daunting enough because the fact that the crossing is made with endos rather than autos kills most of the desired properties of your noncommutative ring (for example Noetherianness). But, if your a $C^{\ast}$-algebraist then you want to complicate matters even more as you need variables $X_n^{\ast}$ corresponding to the $X_n$ satisfying suitable properties. If this is possible, we will denote the noncommutative algebra generated by $A$, the $X_n$ and the $X_n^*$ by $A \circ \mathbb{N}_{\times}$.

the giant mashup-algebra $\mathbb{Z}[\mathbb{Q}/\mathbb{Z}] \circ \mathbb{N}_{\times}$, aka BC

Lots of papers are written trying to get novel insights into the BC-algebra by looking at its adelic-, motivic-, semi-hemi-demi-, p-adic-, $\mathbb{F}_1$-gadgety or whatever-comes-next interpretation. It is the archetypical example of the above construction.

Let’s define it by generators and relations using its ‘integral’ incarnation. Generators are $e(r)$, one for each $r \in \mathbb{Q}/\mathbb{Z}$ and elements $\tilde{\mu}_n$ and $\mu_n^*$ for $n \in \mathbb{N}_+$. The relations are

$e(r) e(s) = e(r+s)~\forall r,s \in \mathbb{Q}/\mathbb{Z}$

$\tilde{\mu}_n \tilde{\mu}_m = \tilde{\mu}_{nm}~\forall n,m \in \mathbb{N}_+$

$\mu_n^* \mu_m^* = \mu^*_{nm}~\forall n,m \in \mathbb{N}_+$

$\mu_n^* \tilde{\mu}_n = n~\quad \text{and} \quad \tilde{\mu}_n \mu^*_m = \mu^*_m \tilde{\mu}_n~\quad~\text{whenever} \quad (m,n)=1$

$\mu^*_n e(r) = e(nr) \mu^*_n~\forall r \in \mathbb{Q}/\mathbb{Z}, n \in \mathbb{N}_+$

$e(r) \tilde{\mu}_n = \tilde{\mu}_n e(nr)~\forall r \in \mathbb{Q}/\mathbb{Z}, n \in \mathbb{N}_+$

$\tilde{\mu}_n e(r) \mu^*_n = \sum_{ns=r} e(s)~\forall r \in \mathbb{Q}/\mathbb{Z}, n \in \mathbb{N}_+$

The first relations imply that the $\mathbb{Z}$-ring generated by the $e(r)$ is the integral group-ring $\mathbb{Z}[\mathbb{Q}/\mathbb{Z}]$. Taking $e(r) \mapsto e^{2 \pi i r}$ we see that this ring is isomorphic to the integral group-ring $\mathbb{Z}[\pmb{\mu}_{\infty}]$ of the multiplicative group of all roots of unity.

$\mathbb{Z}[\pmb{\mu}_{\infty}]$ is a $\lambda$-ring (actually, our best shot at the algebraic closure $\overline{\mathbb{F}}_1$) with endomorphisms $\Psi^n(e^{2 \pi i r}) = e^{2 \pi i nr}$ (which correspond to the endomorphisms $e(r) \mapsto e(nr)$ in $\mathbb{Z}[\mathbb{Q}/\mathbb{Z}]$).

Hence, we see that the subring generated by the $e(r)$ and the $\mu_n^*$ is actually isomorphic to the noncommutative crossed product $\mathbb{Z}[\pmb{\mu}_{\infty}] \ast \mathbb{N}_{\times}$ constructed before. The full BC-algebra is then what we have denoted $\mathbb{Z}[\pmb{\mu}_{\infty}] \circ \mathbb{N}_{\times}$.

More information on the (classical) BC-algebra can be found in these neverendingbook-posts : as a giant mash-up of arithmetical information and its relation to the Riemann zeta-function.

In view of the Borger-de Smit result characterizing the etale site of $\mathsf{Spec}(\mathbb{F}_1)$ it is perhaps interesting to consider the multi-variate BC-algebras $\mathbb{Z}[\pmb{\mu}_{\infty}] \otimes \cdots \otimes \mathbb{Z}[\pmb{\mu}_{\infty}] \circ \mathbb{N}_{\times}$ defined in the now obvious way.

More food-for-thought : take your favorite torsion free $\mathbb{Z}$-ring $A$ and construct your own BC-lookalike algebra $W(A) \circ \mathbb{N}_{\times}$ making clever use of the Adams operations $\Psi^n$ and the ‘Verschiebung’-operations on the ring of big Witt vectors $W(A)$.

Aliens and reality

October 21st : Dear Diary,

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

“I am an alien (laughter…), and a very stubborn alien at that. There’s just one field, the complex numbers $\mathbb{C}$, and all rings are $\mathbb{C}$-algebras. I’ve heard strange rumours that you humans believe in a geometry ‘hidden under $\mathbb{C}$’, something called real manifolds. What then is an algebra over this obviously virtual ‘real’ field?”

Their first hurdle was to convey the concept of complex conjugation as the alien(me) was unwilling to decompose a complex number $c$ into two ‘ghost components’ $a+bi$. Still i had to concede that i knew about addition and multiplication, i had a $1$ and a square root of $-1$, which for some reason they preferred to call $i$.

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

‘Why are you humans so obsessed with 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 (they called it conjugation) sending sums to sums, 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 the one consisting of all elements fixed under conjugation! Clearly, the alien refused : ‘We’re not going on that slippery road called generalization, we’ve seen the havock caused by it in human-mathematics.’

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

Alien : ‘But, how do i recognize one of your algebras among mine? Is there a test to detect them?’

Humans : ‘Yes, they have a 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 that extends the conjugation on $\mathbb{C}$.’

Alien : ‘But if i take a basis for any of my algebras and apply conjugation to all its coordinates, then surely 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 $\mathbb{C}$ (this extra structure, the map $\gamma_B$, is called the ‘descent data’).

A human-observed $\mathbb{R}$-algebra morphism $\phi : A \rightarrow A’$ is to the alien 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 properly define obviously non-existing rings living ‘under $\mathbb{Z}$’, or if you like silly terminology, 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 time like this in a seminar.

$\mathbb{F}_1$ and the ABC-conjecture

Finally, we’re closing in on Smirnov’s approach to the ABC-conjecture via geometry over the field with one element.

The geometric defect

Let $\phi : C_1 \mapsto C_2$ be a cover of curves defined over $k$, then the scheme-version of the Riemann-Hurwitz inequality is

$$2 g_{C_1} – 2 \geq deg(\phi) (2 g_{C_1} -2) + \sum^{scheme}_{P \in C_1} (e_{\phi}(P)-1) deg(P)$$

In the special case, when $f$ is a non-constant rational function in $k(C)$ and $f~:~C \mapsto \mathbb{P}^1_k$ is the corresponding cover, this reads

$$2g_C-2 \geq -2 deg(f) + \sum^{scheme}_{P \in C} (e_f(P)-1) deg(P)$$

which can be turned into the inequality

$$\sum^{scheme}_{P \in C} \frac{(e_f(P)-1) deg(P)}{deg(f)} \leq 2 – \frac{2-2g_C}{deg(f)}$$

We call the expression $\delta(P) = \tfrac{(e_f(P)-1) deg(P)}{deg(f)}$ the defect of $P$. Observe that $\delta(P) \geq 0$ and so this inequality only improves it we restrict the summation to some subset of schematic $C$-points.

The arithmetic defect

Take a positive rational number $q = \frac{m}{n}$ with $1 \leq n < m$ and $(m,n)=1$ and consider the cover

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

Recall that the fiber over the point $[d] \in \mathbb{P}^1 / \mathbb{F}_1$ consists of all prime divisors of $m^d-n^d$ not dividing any $m^e-n^e$ for $e < d$. The fiber of $[0]$ (resp. of $[\infty]$) consists of all prime divisors of $m$ (resp. of $n$ together with $\infty$). Here's part of the cover for $q=\frac{104348}{33215}$ (a good rational approximation for $\pi$).

pi-map

It is tempting to define the ramification index $e_q(p)$ for the map $q$ in the prime $p$ lying in the fiber $q^{-1}([d])$ to be the largest power of $p$ dividing $m^d-n^d$. Likewise, for $p \in q^{-1}([0])$ (resp. in $q^{-1}([\infty])$) take for $e_q(p)$ the largest power of $p$ dividing $m$ (resp. dividing $n$). Finally, take $e_q(\infty) = log(q)$.

Combine this with our previous definitions for the degree of $p$ to be $log(p)$ and of the degree of the map $q$ to be $log(m)$, to define the arithmetic defect of $q$ in the prime $p$ to be

$$\delta(p) = \frac{(e_q(p)-1) log(p)}{log(m)}$$

We can now define the total defect of the cover $q$ over the point $[d] \in \mathbb{P}^1 / \mathbb{F}_1$ to be

$$\delta_{[d]} = \sum_{p \in q^{-1}([d])} \delta(p)$$

It is easy to work out these total defects for the four $\mathbb{F}_1$-rational points of $\mathbb{P}^1 / \mathbb{F}_1$ : $\{ [0],[1],[2],[\infty] \}$ (the primes lying on the blue lines in the graph).

For a natural number $a$ let $a_0$ be its square-free part and $a_1 = \tfrac{a}{a_0}$ the remaining part. Then

  • $\delta_{[0]} = \frac{log(m_1)}{log(m)}$
  • $\delta_{[\infty]} = \frac{log(n_1)+log(q)-1}{log(m)}$
  • $\delta_{[1]} = \frac{log((m-n)_1)}{log(m)}$
  • $\delta_{[2]}= \frac{log(k_1)}{log(m)}$

where $k$ is $m+n$ divided by the largest $2$-power it may contain.

Hurwitz-conjecture for $\mathbb{Q}$

If we sum the defects of $q$ in all primes over the points $\{ [0],[1],[\infty] \}$ we would get, in analogy with the Hurwitz-inequality in the function field case

$$\delta_{[0]}+\delta_{[1]}+\delta_{[\infty]} \leq 2 – \frac{2 – 2g_{\mathsf{Spec}(\mathbb{Z})}}{log(m)}$$

We do not know what the genus of the arithmetic curve $\mathsf{Spec}(\mathbb{Z})$ might be, but is sure is a constant not depending on the map $q$. If we could develop a geometry over $\mathbb{F}_1$ such that all wild guesses we made before would turn out to be the correct ones for an $\mathbb{F}_1$-version of the Hurwitz inequality, we would have the statement below :

For every $\epsilon > 0$ there exists a constant $C(\epsilon)$ such that the following inequality holds for every pair $1 \leq m < n$ with $(m,n)=1$

$$\frac{log(m_1) + log((m-n)_1) + log(n_1) + log(m)-log(n)-1}{log(m)} \leq 2 + \epsilon + \frac{C(\epsilon)}{log(m)}$$

‘Proof’ of the ABC-conjecture

The ABC-conjecture requires for every $\epsilon > 0$ a constant $D(\epsilon)$ such that for all coprime natural numbers $A$ and $B$ we have with $A+B=C$

$$C \leq D(\epsilon) (A_0B_0C_0)^{1+\epsilon}$$

Well, take $m=C$ and $n=min(A,B)$ then in the conjectural Hurwitz inequality for the cover corresponding to $q=\frac{m}{n}$ above we have that

  • $\frac{log(m_1)}{log(m)} = 1 – \frac{log(m_0)}{log(m)}$
  • $\frac{log(n_1)+log(m)-log(n)-1}{log(m)} = 1 – \frac{log(n_0)}{log(m)} – \frac{1}{log(m)}$
  • $\frac{log((m-n)_1)}{log(m)}=\frac{log(m-n)}{log(m)}-\frac{log((m-n)_0)}{log(m)} \geq 1 – \frac{log((m-n)_0)}{log(m)} – \frac{log(2)}{log(m)}$

(the latter inequality because $m-n \geq \frac{m}{2}$ and so $log(m-n) \geq log(m)-log(2)$). Plug this into the inequality above and get

$$3-\frac{log(n_0m_0(m-n)_0)}{log(m)} \leq 2 + \epsilon + \frac{C(\epsilon) + 1 + log(2)}{log(m)}$$

Take $log(C'(\epsilon))=C(\epsilon)+1+log(2)$ and reshuffle in order to get the inequality $m^{1-\epsilon} \leq C'(\epsilon)(n_0m_0(m-n)_0)$. But then, finally (finally!) with $D(\epsilon)=C'(\epsilon)^{1+\epsilon}$

$$C=m \leq D(\epsilon)(n_0m_0(m-n)_0)^{1+\epsilon} = D(\epsilon)(A_oB_0C_0)^{1+\epsilon}$$

The exceptional map and Mersenne primes

Last time we’ve seen that almost all rational numbers $q \in \mathbb{Q}$ determine a finite cover $q~:~\mathsf{Spec}(\mathbb{Z}) \mapsto \mathbb{P}^1 / \mathbb{F}_1$, the exceptional cases controlled by Zsigmondy’s theorem.

The prime exceptional case corresponds to $q=2$. Below,we sketch a small portion of the graph of this non-cover (the image does not contain the points $[1]$ and $[6]$, the red lines) where we use a logarithmic scale on $\mathsf{Spec}(\mathbb{Z})$.

Clearly, such images should be taken with a bucket of salt. The linear depiction of $\mathsf{Spec}(\mathbb{Z})$ suggests for example that the prime $3$ has more affinity with $2$ and $5$ than with say $2147483647$ or $524287$. However, the relevant topology on $\mathsf{Spec}(\mathbb{Z})$ is the cofinite topology, so one is always allowed to reshuffle finitely many primes in order to get a smoother covering map!

The fiber over $[n]$ consists of the primitive prime factors of $2^n-1$ (that is, those primes not dividing $2^d-1$ for $d$ a proper divisor of $n$). For small values of $n$, most fibers consist of just one prime (for $n \leq 35$ only $n=11,23,23,28$ and $35$ have two principal primes and only $n=29$ has three primes in its fiber).

A Mersenne prime is a prime number of the form $M_p = 2^p-1$ (from which it follows that $p$ must be prime, too). Only $47$ Mersenne primes are known, the smallest being

$M_2,M_3,M_5,M_7,M_{13},M_{17},M_{19}$ and $M_{31}$

corresponding to the points on ‘the diagonal’ in the graph of the exceptional map. Naturally, if $M_p$ is a Mersenne prime, the fiber $2^{-1}([p])$ has only one element.

If we believe a geometry over $\mathbb{F}_1$ can be developed such that the morphisms $q$ make sense (and hence their graphs define divisors in the Smirnov plane $\mathsf{Spec}(\mathbb{Z}) \times \mathbb{P}^1 / \mathbb{F}_1$) one might expect that the subsets of points $[n] \in \mathbb{P}^1 / \mathbb{F}_1$ with a fiber containing at least $k$ points, should be cofinite.

In particular, in view of Schinzel’s result (stating that all maps $q$ have infinitely many points with a fiber containing at least two elements) this would imply that the set of Mersenne primes has to be finite , contradicting the Lenstra-Pomerance-Wagstaff conjecture.

On the positive side, it would imply that infinitely many numbers of the form $2^p-1$ (with $p$ a prime number) are highly composite (as they must have at least two principal prime factors) which is another big open problem…