I'm reading a paper On the Torsion Jacquet-Langlands correspondence by Akshay Venkatesh and Frank Calegari.

Truthfully speaking I have no idea what Jacquet-Landlands *is*. I'm just trying to understand why there are knots in a paper on algebraic number theory and some of the players involved in that paper.

To keep matters simple, how do we pass between number fields and 3-manifolds and why is this beneficial? Poking around this paper, I find a version of the congruence groups: $$\Gamma_0(\mathfrak{n}) = \left\{ \left( \begin{array}{cc} a & b \\ c & d\end{array} \right) : \mathfrak{n}\big|\, c\right\} \subset PGL_2(\mathbf{O}_F)$$

where $\mathbf{O}_F$ is an order of a number field. We get a 3-manifold by quotienting hyperbolic 3-space: $\mathbb{H}^3/\Gamma_0(\mathfrak{n})$. Apparently, there's also another similar way to do it with *quaternions*.

They then proceed to look at look at some group-cohomology invariants of the group and then they use some spectral theory and the rest of paper mostly goes over my head. Well... we do get this:

However, along the way, we took many detours to explore related phenomena, some of which was inspired by the data computed for the ﬁrst author by Nathan Dunﬁeld. In view of the almost complete lack of rigorous understanding of torsion for (nonHermitian) locally symmetric spaces, we have included many of these results, even when what we can prove is rather modest.

The take-home message seems to be that we have constructed a large collection of infinite groups and actions on low-dimensional spaces of interest number theorists. Arithmetic lattices look like they play an important role.

I guess I'm trying to understand better **how this connection between knots and number fields works** and how Calegari and Venkatesh are using it to get a handle of the many invariants (which I may save for later questions).

thebook on this topic. I have to find it in a library or buy it... I don't know if my question is clear or specific enough to get an answer other than "goto Machlachlan-Reid" $\endgroup$2more comments