The Langlands program and non-commutative geometry

The Bulletin of the AMS just made this paper by Julia Mueller available online: “On the genesis of Robert P. Langlands’ conjectures and his letter to Andre Weil” (hat tip +ChandanDalawat and +DavidRoberts on Google+).

It recounts the story of the early years of Langlands and the first years of his mathematical career (1960-1966)leading up to his letter to Andre Weil in which he outlines his conjectures, which would become known as the Langlands program.

Langlands letter to Weil is available from the IAS.

The Langlands program is a vast net of conjectures. For example, it conjectures that there is a correspondence between

– $n$-dimensional representations of the absolute Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$, and

– specific data coming from an adelic quotient-space $GL_n(\mathbb{A}_{\mathbb{Q}})/GL_n(\mathbb{Q})$.

For $n=1$ this is essentially class field theory with the correspondence given by Artin’s reciprocity law.

Here we have on the one hand the characters of the abelianised absolute Galois group

\[
Gal(\overline{\mathbb{Q}}/\mathbb{Q})^{ab} \simeq Gal(\mathbb{Q}(\pmb{\mu}_{\infty})/\mathbb{Q}) \simeq \widehat{\mathbb{Z}}^{\ast} \]

and on the other hand the connected components of the idele class space

\[
GL_1(\mathbb{A}_{\mathbb{Q}})/GL_1(\mathbb{Q}) = \mathbb{A}_{\mathbb{Q}}^{\ast} / \mathbb{Q}^{\ast} = \mathbb{R}_+^{\ast} \times \widehat{\mathbb{Z}}^{\ast} \]

For $n=2$ it involves the study of Galois representations coming from elliptic curves. A gentle introduction to the general case is Mark Kisin’s paper What is … a Galois representation?.

One way to look at some of the quantum statistical systems studied via non-commutative geometry is that they try to understand the “bad” boundary of the Langlands space $GL_n(\mathbb{A}_{\mathbb{Q}})/GL_n(\mathbb{Q})$.

Here, the Bost-Connes system corresponds to the $n=1$ case, the Connes-Marcolli system to the $n=2$ case.

If $\mathbb{A}’_{\mathbb{Q}}$ is the subset of all adeles having almost all of its terms in $\widehat{\mathbb{Z}}_p^{\ast}$, then there is a well-defined map

\[
\pi~:~\mathbb{A}’_{\mathbb{Q}}/\mathbb{Q}^{\ast} \rightarrow \mathbb{R}_+ \qquad (x_{\infty},x_2,x_2,\dots) \mapsto | x_{\infty} | \prod_p | x_p |_p \]

The inverse image of $\pi$ over $\mathbb{R}_+^{\ast}$ are exactly the idele classes $\mathbb{A}_{\mathbb{Q}}^{\ast}/\mathbb{Q}^{\ast}$, so we can view them as the nice locus of the horrible complicated quotient of adele-classes $\mathbb{A}_{\mathbb{Q}}/\mathbb{Q}^*$. And we can view the adele-classes as a ‘closure’ of the idele classes.

But, the fiber $\pi^{-1}(0)$ has horrible topological properties because $\mathbb{Q}^*$ acts ergodically on it due to the fact that $log(p)/log(q)$ is irrational for distinct primes $p$ and $q$.

This is why it is better to view the adele-classes not as an ordinary space (one with bad topological properties), but rather as a ‘non-commutative’ space because it is controlled by a non-commutative algebra, the Bost-Connes algebra.

For $n=2$ there’s a similar story with a ‘bad’ quotient $M_2(\mathbb{A}_{\mathbb{Q}})/GL_2(\mathbb{Q})$, being the closure of an ‘open’ nice piece which is the Langlands quotient space $GL_2(\mathbb{A}_{\mathbb{Q}})/GL_2(\mathbb{Q})$.

A forgotten type and roots of unity (again)

The monstrous moonshine picture is the finite piece of Conway’s Big Picture needed to understand the 171 moonshine groups associated to conjugacy classes of the monster.

Last time I claimed that there were exactly 7 types of local behaviour, but I missed one. The forgotten type is centered at the number lattice $84$.

Locally around it the moonshine picture looks like this
\[
\xymatrix{42 \ar@{-}[dr] & 28 \frac{1}{3} \ar@[red]@{-}[d] & 41 \frac{1}{2} \ar@{-}[ld] \\ 28 \ar@[red]@{-}[r] & \color{grey}{84} \ar@[red]@{-}[r] \ar@[red]@{-}[d] \ar@{-}[rd] & 28 \frac{2}{3} \\ & 252 & 168} \]

and it involves all square roots of unity ($42$, $42 \frac{1}{2}$ and $168$) and $3$-rd roots of unity ($28$, $28 \frac{1}{3}$, $28 \frac{2}{3}$ and $252$) centered at $84$.

No, I’m not hallucinating, there are indeed $3$ square roots of unity and $4$ third roots of unity as they come in two families, depending on which of the two canonical forms to express a lattice is chosen.

In the ‘normal’ expression $M \frac{g}{h}$ the two square roots are $42$ and $42 \frac{1}{2}$ and the three third roots are $28, 28 \frac{1}{3}$ and $28 \frac{2}{3}$. But in the ‘other’ expression
\[
M \frac{g}{h} = (\frac{g’}{h},\frac{1}{h^2M}) \]
(with $g.g’ \equiv 1~mod~h$) the families of $2$-nd and $3$-rd roots of unity are
\[
\{ 42 \frac{1}{2} = (\frac{1}{2},\frac{1}{168}), 168 = (0,\frac{1}{168}) \} \]
and
\[
\{ 28 \frac{1}{3} = (\frac{1}{3},\frac{1}{252}), 28 \frac{2}{3} = (\frac{2}{3},\frac{1}{252}), 252 = (0 , \frac{1}{252}) \} \]
As in the tetrahedral snake post, it is best to view the four $3$-rd roots of unity centered at $84$ as the vertices of a tetrahedron with center of gravity at $84$. Power maps in the first family correspond to rotations along the axis through $252$ and power maps in the second family are rotations along the axis through $28$.

In the ‘normal’ expression of lattices there’s then a total of 8 different local types, but two of them consist of just one number lattice: in $8$ the local picture contains all square, $4$-th and $8$-th roots of unity centered at $8$, and in $84$ the square and $3$-rd roots.

Perhaps surprisingly, if we redo everything in the ‘other’ expression (and use the other families of roots of unity), then the moonshine picture has only 7 types of local behaviour. The forgotten type $84$ appears to split into two occurrences of other types (one with only square roots of unity, and one with only $3$-rd roots).

I wonder what all this has to do with the action of the Bost-Connes algebra on the big picture or with Plazas’ approach to moonshine via non-commutative geometry.

Penrose tiles in Helsinki


(image credit: Steve’s travels & stuff)

A central street in Helsinki has been paved with Penrose tiles.


(image credit: Sattuman soittoa)

From a Finnish paper:

“The street could also be an object to mathematical awe. The stone under one’s feet is embroidered with some profound geometry, namely, Penrose tiling.

In 1974, a British mathematician Roger Penrose realised a plane could be fully covered with a few simple rules such that the pattern constantly changes. These kind of discontinuous patterns are interesting to mathematicians since the patterns can be used to solve other geometrical problems. Together, the tiles can randomly form patterns reminding a star or the Sun but they do not regularly recur in the tiling.

Similar features are found in the old Arabic ornaments. The tiling of the Central Street prom was selected by Yrjö Rossi.

If your kid stays put to stare at the tiling, they might have what they need in order to become a mathematician.”

(via Reddit/m)

nc-geometry and moonshine?

A well-known link between Conway’s Big Picture and non-commutative geometry is given by the Bost-Connes system.

This quantum statistical mechanical system encodes the arithmetic properties of cyclotomic extensions of $\mathbb{Q}$.

The corresponding Bost-Connes algebra encodes the action by the power-maps on the roots of unity.

It has generators $e_n$ and $e_n^*$ for every natural number $n$ and additional generators $e(\frac{g}{h})$ for every element in the additive group $\mathbb{Q}/\mathbb{Z}$ (which is of course isomorphic to the multiplicative group of roots of unity).

The defining equations are
\[
\begin{cases}
e_n.e(\frac{g}{h}).e_n^* = \rho_n(e(\frac{g}{h})) \\
e_n^*.e(\frac{g}{h}) = \Psi^n(e(\frac{g}{h}).e_n^* \\
e(\frac{g}{h}).e_n = e_n.\Psi^n(e(\frac{g}{h})) \\
e_n.e_m=e_{nm} \\
e_n^*.e_m^* = e_{nm}^* \\
e_n.e_m^* = e_m^*.e_n~\quad~\text{if $(m,n)=1$}
\end{cases}
\]

Here $\Psi^n$ are the power-maps, that is $\Psi^n(e(\frac{g}{h})) = e(\frac{ng}{h}~mod~1)$, and the maps $\rho_n$ are given by
\[
\rho_n(e(\frac{g}{h})) = \sum e(\frac{i}{j}) \]
where the sum is taken over all $\frac{i}{j} \in \mathbb{Q}/\mathbb{Z}$ such that $n.\frac{i}{j}=\frac{g}{h}$.

Conway’s Big Picture has as its vertices the (equivalence classes of) lattices $M,\frac{g}{h}$ with $M \in \mathbb{Q}_+$ and $\frac{g}{h} \in \mathbb{Q}/\mathbb{Z}$.

The Bost-Connes algebra acts on the vector-space with basis the vertices of the Big Picture. The action is given by:
\[
\begin{cases}
e_n \ast \frac{c}{d},\frac{g}{h} = \frac{nc}{d},\rho^m(\frac{g}{h})~\quad~\text{with $m=(n,d)$} \\
e_n^* \ast \frac{c}{d},\frac{g}{h} = (n,c) \times \frac{c}{nd},\Psi^{\frac{n}{m}}(\frac{g}{h})~\quad~\text{with $m=(n,c)$} \\
e(\frac{a}{b}) \ast \frac{c}{d},\frac{g}{h} = \frac{c}{d},\Psi^c(\frac{a}{b}) \frac{g}{h}
\end{cases}
\]

This connection makes one wonder whether non-commutative geometry can shed a new light on monstrous moonshine?

This question is taken up by Jorge Plazas in his paper Non-commutative geometry of groups like $\Gamma_0(N)$

Plazas shows that the bigger Connes-Marcolli $GL_2$-system also acts on the Big Picture. An intriguing quote:

“Our interest in the $GL_2$-system comes from the fact that its thermodynamic properties encode the arithmetic theory of modular functions to an extend which makes it possible for us to capture aspects of moonshine theory.”

Looks like the right kind of paper to take along when I disappear next week for some time in the French mountains…

Everything’s wrappable to a sphere

One of the better opening quotes of a paper:

“Even quite ungainly objects, like chairs and tables, will become almost spherical if you wrap them in enough newspaper.”

The paper in question is The orbifold notation for surface groups by John Conway.

Here’s Conway talking leisurely about Thurston’s idea to capture the acting group via the topology of the orbifold space and his own notation for such orbifolds.



Here’s another version of the paper, with illustrations: The orbifold notation for two-dimensional groups, by Conway and Daniel H. Huson.

A very accessible account are these lecture notes:

A field guide to the orbifolds, notes from class on “Geometry and the Imagination” in Minneapolis, with John Conway, Peter Doyle, Jane Gilman and Bill Thurston, on June 17–28, 1991.

And, here are notes by Thurston on The Geometry and Topology of Three-Manifolds, including stuff about orbifolds.

I came across these papers struggling my way through On the discrete groups of moonshine by Conway, McKay and Sebbar.

On the genus $0$ property of moonshine groups they have this to say:

“As for groups of the form $(n|h)+e,f,\dots$, the genus can be determined from the fundamental regions using the Riemann-Hurwitz formula. Since most of the groups are not subgroups of the modular group, the calculations of the genus, which cannot be produced here because of their length, are carried out by finding the elliptic fixed points and the cone points in the orbifolds attached to the fundamental regions. The Euler characteristic of the orbifold determines the genus of the group. See [paper] for more details on orbifold techniques.”