Conway’s Big Picture

In this note we briefly describe the combinatorial/diagrammatic “big picture” framework introduced by J.H. Conway in his paper Understanding groups like \(\Gamma_0(N)\). Conway’s big picture can be used to understand the structure of sets of commensurable lattices in the plane (amongst other things).

Background

In what follows we always assume \(\omega_1, \omega_2 \in \mathbb{C}\) are linearly indepenent over \(\mathbb{R}\).

Definition 1. The \(\mathbb{Z}\)-module \(\langle \omega_1, \omega_2\rangle\) defined by

\begin{equation*} \langle \omega_1, \omega_2 \rangle = \mathbb{Z}\omega_1 \oplus \mathbb{Z}\omega_2 \end{equation*}

is called the lattice (in \(\mathbb{C}\)) generated by \(\omega_1, \omega_2\). We shall write \(\mathrm{Lat}(\mathbb{C})\) to denote the set of all lattices in \(\mathbb{C}\). ∎

The set of all lattices in \(\mathbb{C}\) is slightly too large for us to say anything useful about here. To make things more managable we will consider two lattices to be equivalent if one is a rational multiple of the other. More precisely:

Definition 2. Let us write \(L_1 \sim L_2\) if \(L_1, L_2\) are two lattices satisfying

\begin{equation*} L_1 = \langle \omega_1, \omega_2 \rangle = \langle \lambda \omega_1 \lambda \omega_2 \rangle = L_2 \end{equation*}

for some \(\lambda \in \mathbb{Q}^\times\). It is easy to see that \(\sim\) is an equivalence relation on \(\mathrm{Lat}(\mathbb{C})\). If \(L_1 \sim L_2\) we say \(L_1\) and \(L_2\) are projectively equivalent. Equivalence classes under \(\sim\) are also called projective classes. We shall write \(\mathrm{PLat}(\mathbb{C})\) to denote the set of projective classes of lattices in \(\mathbb{C}\). ∎

Definition 3. We say that two lattices are commensurable if the subgroup \(L_1 \cap L_2\) has finite index in both \(L_1\) and \(L_2\). Two projective classes of lattices are commensurable if we can choose commensurable representative elements. The set of lattices (resp. projective lattices) commensurable to \(L_1\) is denoted \(\mathrm{Lat} _ {L_1}(\mathbb{C})\) (resp. \(\mathrm{PLat} _ {L_1}(\mathbb{C})\)). ∎

Notice that a lattice in \(\mathbb{C}\) is the same thing as a discrete additive subgroup of \(\mathbb{C}\).

Lattice Normal Form

Let \(e_1 = 1\) and \(e_2 = i\) be the standard basis vectors for \(\mathbb{C}\) as an \(\mathbb{R}\)-vector space. The lattice \(\langle e_1, e_2\rangle\) is called the standard lattice. In this section we show that any projective class of lattice commensurable to \(\langle e_1, e_2\rangle\) has a representative of the form \(\langle M e_1 + \frac{g}{h} e_2, e_2\rangle\) where \(M > 0\) and \(0 \leq \frac{g}{h} <1\) is a fraction in lowest terms.

To this end, let us first observe that a generic lattice commensurable with \(\langle e_1, e_2\rangle\) can be written in the form

\begin{equation} \label{eq:fracs} \left\langle \frac{a_0}{a_1} e_1 + \frac{b_0}{b_1} e_2, \frac{c_0}{c_1} e_1, \frac{d_0}{d_1} e_2 \right\rangle \end{equation}

where each of the coefficients on the \(e_i\) are elements of \(\mathbb{Q}\) in lowest terms. Multiplying through by \(\frac{a_1 c_1}{\mathrm{gcd}(a_1 c_0, a_0 c_1)}\) in \ref{eq:fracs} yields a projectively equivalent lattice

\begin{equation} \label{eq:ints} \langle Ae_1 + Be_2 , Ce_1 + De_2 \rangle \end{equation}

where \(A\) and \(C\) are co-prime integers. If we choose a matrix \(\left(\begin{smallmatrix}\alpha & \beta \\ \gamma & \delta\end{smallmatrix}\right) \in \mathrm{SL}(2;\mathbb{Z})\) then the lattice

\begin{equation} \label{eq:big} \langle \alpha(Ae_1 + Be_2) + \beta(Ce_1 + De_2), \gamma(Ae_1 + Be_2) + \delta(Ce_1 + De_2)\rangle \end{equation}

is the same as the one in equation \ref{eq:ints}. We want this expression to reduce to something of the form \(\langle Me_1 + \frac{g}{h} e_2, e_2\rangle\), so it will be necessary to choose \(\left(\begin{smallmatrix}\alpha & \beta \\ \gamma & \delta\end{smallmatrix}\right)\) such that \(\gamma A + \delta C = 0\). One obvious choice is \(\gamma = -C\) and \(\delta = A\). Then \(\gamma\) and \(\delta\) are co-prime since \(A\) and \(C\) are chosen to be co-prime. As such, there exist \(\alpha, \beta \in \mathbb{Z}\) such that

\begin{equation*} \alpha\delta - \beta\gamma = 1 \end{equation*}

In other words, it is possible to choose \(\left(\begin{smallmatrix}\alpha & \beta \\ \gamma & \delta\end{smallmatrix}\right) \in \mathrm{SL}(2;\mathbb{Z})\) such that equation \ref{eq:big} reduces to

\begin{equation} \label{eq:e2clear} \langle (\alpha A + \beta C)e_1 + (\alpha B + \beta D)e_2, (AD - BC)e_2\rangle \end{equation}

Finally, dividing by \(AD - BC\) yields a lattice projectively equivalent to \ref{eq:e2clear}:

\begin{equation} \label{eq:almost} \left\langle \frac{\alpha A + \beta C}{AD - BC} e_1 + \frac{\alpha B + \beta D}{AD - BC} e_2, e_2 \right\rangle \end{equation}

Setting \(M = \frac{\alpha A + \beta C}{AD - BC}\), \(g = \alpha B + \beta D\), and \(h = AD - BC\) we arrive at a representative for our projective class of lattices in the desired form. The \(0\leq \frac{g}{h} < 1\) condition follows from the fact that changing base by \(\left(\begin{smallmatrix}1 & n \\ 0 & 1\end{smallmatrix}\right)\) in \ref{eq:almost} allows us to add/subtract multiples of \(e_2\) from \(\frac{\alpha B + \beta D}{AD - BC}\), so its value only matters modulo 1. Positivity of \(M\) follows from the fact we have made no orientation reversing change of base.

Hyperdistance

We can define a metric on the set \(\mathrm{PLat} _ L (\mathbb{C})\) of all lattices commensurable to some given lattice \(L\). In order to do so, it is convenient to first adopt a matrix-based perspective on lattices in \(\mathbb{C}\). With this in mind, notice that if we fix a basis \({ \omega_1, \omega_2}\) for \(L\) we obtain identifications

\begin{equation} \mathrm{SL}(2;\mathbb{Z}) \backslash \mathrm{GL}^+(2;\mathbb{Q}) \longrightarrow \mathrm{Lat} _ L(\mathbb{C}) \end{equation}

and

\begin{equation} \mathrm{PSL}(2;\mathbb{Z}) \backslash \mathrm{PGL}^+(2;\mathbb{Q}) \longrightarrow \mathrm{PLat} _ L(\mathbb{C}) \end{equation}

via the map

\begin{equation} \begin{pmatrix} a & b \\ c & d \end{pmatrix} \longmapsto (a\omega_1 + b\omega_2)\mathbb{Z} + (c\omega_1 + b\omega_2)\mathbb{Z} \end{equation}

once we recall the \(\mathrm{SL}(2;\mathbb{R})\)-action on lattices only serves to change bases. Under this identification the lattice \(\langle Me_1 + \frac{g}{h}e_2 , e_2\rangle\) corresponds to the class of matrices represented by \(\left(\begin{smallmatrix} M & \frac{g}{h} \\ 0 & 1 \end{smallmatrix}\right)\).

For concreteness, let us again restrict to the case \(L = \langle e_1, e_2 \rangle\). Suppose we are given two projective lattices \(L_1 = \langle Me_1 + \frac{g}{h}e_2, e_2\rangle\) and \(L = \langle Ne_1 + \frac{i}{j}e_2, e_2\rangle\) both commensurable to \(L\). Define \(\widetilde{d} : \mathrm{PLat} _ L ^ 2 \to \mathbb{Z} _ {>0}\) by

\begin{equation} \widetilde{d}(L_1, L_2) = \mathrm{Pdet}\left( \begin{pmatrix} M & \frac{g}{h} \\ 0 & 1 \end{pmatrix} \begin{pmatrix} N & \frac{i}{j} \\ 0 & 1 \end{pmatrix}^{-1} \right) \end{equation}

where \(\mathrm{Pdet}\) is the rational projective determinant. Well-definedness of \(\widetilde{d}\) follows from the fact that \(\mathrm{Pdet}\) is a well-defined map \(\mathrm{PSL}(2;\mathbb{Z}) \backslash \mathrm{PGL}^+(2;\mathbb{Q}) \to \mathbb{Z} _ {>0}\).

Definition 4. The map \(\widetilde{d}\) above is called the hyperdistance function. ∎

It follows from the elementary properties of the rational projective determinant that \(\log(\widetilde{d})\) is a distance function. In particular, \(\widetilde{d}\) is symmetric.