The group algebra of all algebraic numbers

Some weeks ago, Robert Kucharczyk and Peter Scholze found a topological realisation of the ‘hopeless’ part of the absolute Galois group $\mathbf{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$. That is, they constructed a compact connected space $M_{cyc}$ such that etale covers of it correspond to Galois extensions of the cyclotomic field ${\mathbb{Q}}_{cyc}$. This gives, at least in theory, a handle on the hopeless part of the Galois group $\mathbf{Gal}(\bar{\mathbb{Q}}/{\mathbb{Q}}_{cyc})$, see the previous post in this series.

Here, we will get halfway into constructing $M_{cyc}$. We will try to understand the topology of the prime ideal spectrum $\mathbf{Spec}(\mathbb{C}[{\bar{\mathbb{Q}}}^{\times }])$ of the complex group algebra of the multiplicative group ${\bar{\mathbb{Q}}}^{\times }$ of all non-zero algebraic numbers.

[section_title text=”Pontryagin duals”]

Take an Abelian locally compact group $A$ (for example, an Abelian group equipped with the discrete topology), then its Pontryagin dual $A^{\vee }$ is the space of all continuous group morphisms $A\to {\mathbb{S}}^1$ to the unit circle ${\mathbb{S}}^1$ endowed with the compact open topology.

There are these topological properties of the locally compact group $A^{\vee }$:

– $A^{\vee }$ is compact if and only if $A$ has the discrete topology,

– $A^{\vee }$ is connected if and only if $A$ is a torsion free group,

– $A^{\vee }$ is totally disconnected if and only if $A$ is a torsion group.

If we take the additive group of rational numbers with the discrete topology, the dual space ${\mathbb{Q}}^{\vee }$ is the one-dimensional solenoid

It is a compact and connected group, but is not path connected. In fact, it path connected components can be identified with the finite adele classes ${\mathbb{A}}_f/\mathbb{Q}=\hat{\mathbb{Z}}/\mathbb{Z}$ where $\hat{\mathbb{Z}}$ is the ring of profinite integers.

Keith Conrad has an excellent readable paper on this fascinating object: The character group of $\mathbb{Q}$. Or you might have a look at this post.

[section_title text=”The multiplicative group of algebraic numbers”]

A torsion element $x$ in the multiplicative group ${\bar{\mathbb{Q}}}^{\times }$ of all algebraic numbers must satisfy $x^N=1$ for some $N$ so is a root of unity, so we have the exact sequence of Abelian groups

$0\to \mu {\mu }_{\mathrm{\infty }}\to {\bar{\mathbb{Q}}}^{\times }\to {\bar{\mathbb{Q}}}_{tf}^{\times }\to 0$

where the last term is the maximal torsion-free quotient of ${\bar{\mathbb{Q}}}^{\times }$. By Pontryagin duality this gives us an exact sequence of compact topological groups

$0\to ({\bar{\mathbb{Q}}}_{tf}^{\times }{)}^{\vee }\to ({\bar{\mathbb{Q}}}^{\times }{)}^{\vee }\to \mu {\mu }_{\mathrm{\infty }}^{\vee }\to 0$

Here, the left-most space is connected and $\mu {\mu }_{\mathrm{\infty }}^{\vee }$ is totally disconnected. That is, the connected components of $({\bar{\mathbb{Q}}}^{\times }{)}^{\vee }$ are precisely the translates of the connected subgroup $({\bar{\mathbb{Q}}}_{tf}^{\times }{)}^{\vee }$.

[section_title text=”Prime ideal spectra”]

The short exact sequence of Abelian groups gives a short exact sequence of the corresponding group schemes

$0\to \mathbf{Spec}(\mathbb{C}[{\bar{\mathbb{Q}}}_{tf}^{\times }])\to \mathbf{Spec}(\mathbb{C}[{\bar{\mathbb{Q}}}^{\times }]\to \mathbf{Spec}(\mathbb{C}[\mu {\mu }_{\mathrm{\infty }}])\to 0$

The torsion free abelian group ${\bar{\mathbb{Q}}}_{tf}^{\times }$ is the direct limit $\underrightarrow{lim}{M}_i$ of finitely generated abelian groups $M_i$ and as the corresponding group algebra $\mathbb{C}[M_i]=\mathbb{C}[x_1,x_1^{-1},\cdots ,x_k,x_k^{-1}]$, we have that $\mathbf{Spec}(\mathbb{C}[M_i])$ is connected. But then this also holds for

$\mathbf{Spec}(\mathbb{C}[{\bar{\mathbb{Q}}}_{tf}^{\times }])=\underleftarrow{lim}\mathbf{Spec}(\mathbb{C}[M_i])$

The underlying group of $\mathbb{C}$-points of $\mathbf{Spec}(\mathbb{C}[\mu {\mu }_{\mathrm{\infty }}])$ is $\mu {\mu }_{\mathrm{\infty }}^{\vee }$ and is therefore totally disconnected. But then we have

${\pi }_0(\mathbf{Spec}(\mathbb{C}[{\bar{\mathbb{Q}}}^{\times }])\simeq {\pi }_0(\mathbf{Spec}(\mathbb{C}[\mu {\mu }_{\mathrm{\infty }}])\simeq \mu {\mu }_{\mathrm{\infty }}^{\vee }$

and, more importantly, for the etale fundamental group

${\pi }_1^{et}(\mathbf{Spec}(\mathbb{C}[{\bar{\mathbb{Q}}}^{\times }],x)\simeq {\pi }_1^{et}(\mathbf{Spec}(\mathbb{C}[{\bar{\mathbb{Q}}}_{tf}^{\times }],y)$

So, we have to compute the latter one. Again, write the torsion-free quotient as a direct limit of finitely generated torsion-free Abelian groups and recall that connected etale covers of $\mathbf{Spec}(\mathbb{C}[M_i])=\mathbf{Spec}(\mathbb{C}[x_1,x_1^{-1},\cdots ,x_k,x_k^{-1}])$ are all of the form $\mathbf{Spec}(\mathbb{C}[N])$, where $N$ is a subgroup of $M_i\otimes \mathbb{Q}$ that contains $M_i$ with finite index (that is, adjoining roots of the $x_i$).

Again, this goes through the limit and so a connected etale cover of $\mathbf{Spec}(\mathbb{C}[{\bar{\mathbb{Q}}}_{tf}^{\times }])$ would be determined by a subgroup of the $\mathbb{Q}$-vectorspace ${\bar{\mathbb{Q}}}_{tf}^{\times }\otimes \mathbb{Q}$ containing ${\bar{\mathbb{Q}}}_{tf}^{\times }$ with finite index.

But, ${\bar{\mathbb{Q}}}_{tf}^{\times }$ is already a $\mathbb{Q}$-vectorspace as we can take arbitrary roots in it (remember we’re using the multiplicative structure). That is, $\mathbf{Spec}(\mathbb{C}[{\bar{\mathbb{Q}}}^{\times }])$ is simply connected!

[section_title text=”Bringing in the Galois group”]

Now, we’re closing in on the mysterious space $M_{cyc}$. Clearly, it cannot be the complex points of $\mathbf{Spec}(\mathbb{C}[{\bar{\mathbb{Q}}}^{\times }])$ as this has no proper etale covers, but we still have to bring the Galois group $\mathbf{Gal}(\bar{\mathbb{Q}}/{\mathbb{Q}}_{cyc})$ into the game.

The group algebra $\mathbb{C}[{\bar{\mathbb{Q}}}^{\times }]$ is a commutative and cocommutative Hopf algebra, and all the elements of the Galois group act on it as Hopf-automorphisms, so it is natural to consider the fixed Hopf algebra

$H_{cyc}=\mathbb{C}[{\bar{\mathbb{Q}}}^{\times }{]}^{\mathbf{Gal}(\bar{\mathbb{Q}}/{\mathbb{Q}}_{cyc})}$

This Hopf algebra has an interesting alternative description as a subalgebra of the Witt ring $W({\mathbb{Q}}_{cyc})$, bringing it into the realm of ${\mathbb{F}}_1$-geometry.

This ring of Witt vectors has as its underlying set of elements $1+{\mathbb{Q}}_{cyc}[[t]]$ of formal power series in ${\mathbb{Q}}_{cyc}[[t]]$. Addition on this set is defined by multiplication of power series. The surprising fact is that we can then put a ring structure on it by demanding that the product $\odot$ should obey the rule that for all $a,b\in {\mathbb{Q}}_{cyc}$ we have

$(1-at)\odot (1-bt)=1–abt$

In this mind-boggling ring the Hopf algebra $H_{cyc}$ is the subring consisting of all power series having a rational expression of the form

$\frac{1+a_{1}t+a_2{t}^2+\cdots +a_n{t}^n}{1+b_{1}t+b_2{t}^2+\cdots +b_m{t}^m}$

with all $a_i,b_j\in {\mathbb{Q}}_{cyc}$.

We can embed $\mu {\mu }_{\mathrm{\infty }}$ by sending a root of unity $\zeta$ to $1–\zeta t$, and then the desired space $M_{cyc}$ will be close to

$\mathbf{Spec}(H_{cyc}{\otimes }_{\mathbb{Z}[\mu {\mu }_{\mathrm{\infty }}]}\mathbb{C})$

but I’ll spare the details for another time.

In case you want to know more about the title-picture, quoting from John Baez’ post The Beauty of Roots:

“Sam Derbyshire decided to to make a high resolution plot of some roots of polynomials. After some experimentation, he decided that his favorite were polynomials whose coefficients were all 1 or -1 (not 0). He made a high-resolution plot by computing all the roots of all polynomials of this sort having degree ≤ 24. That’s $2^{24}$ polynomials, and about $24\times 2^{24}$ roots — or about 400 million roots! It took Mathematica 4 days to generate the coordinates of the roots, producing about 5 gigabytes of data.”