Here are some details about my preprint [1]. These notes supplement a short presentation I gave at a Wework session in Pune organized by Anish.
Let \(G = \mathrm{SL}_{3}(\mathbb{R})\) and let \(\Gamma = \mathrm{SL}_{3}(\mathbb{Z})\). Let \(A \subseteq G\) be the subgroup of positive diagonal matrices. Let \(\mathfrak{a} \simeq \mathbb{R}^{2}\) be the Lie algebra of \(A\). From now on, it means \[\begin{equation} \mathfrak{a} =\{(x_{1},x_{2},x_{3}) \in \mathbb{R}^{3} \mid x_{1}+x_{2}+x_{3}=0 \}. \end{equation}\]
A periodic torus \(F\) is a compact diagonal orbit of \(A\) on \(G/ \Gamma\).
One is classically interested in the counting problems \[\begin{equation} \#\{F \text{ periodic tori } \mid \mathrm{vol}(F) \leq T \} \ll ? \end{equation}\] \[\begin{equation} \#\{F \text{ periodic tori } \mid \mathrm{sys}(F) \leq T \} \ll ? \end{equation}\] It is clear that small volume implies small systole but the reverse direction is not clear. To understand this we define a shape.
Fix an isomorphism \(\mathfrak{a} \simeq \mathbb{R}^{2}\), . Given a periodic tori \(F\), the point \(sh(F) \in \mathrm{SL}_{2}(\mathbb{R})/\mathrm{SL}_{2}(\mathbb{Z})\) is defined as follows. Consider a point \(x \in F\) and take the periods \[\begin{equation} P (F) = \{ a \in \mathfrak{a} \mid \exp(a) x = x\}. \end{equation}\] Then, \(P(F) \subseteq \mathfrak{a}\) defines a lattice \(c \cdot h \cdot \mathbb{Z}^{2} \subseteq \mathbb{R}^{2}\) where \(h \in \mathrm{SL}_{2}(\mathbb{R})\) and \(c > 0\) is some constant. We define \(sh(F) = h \mathrm{SL}_{2}(\mathbb{Z})\).
What are the shape points \(\{sh(F)\}_{F \text{ p.t. }}\)?
This is a good question because
(Dang, G., Li)
They are a dense subset of \(\mathrm{SL}_{2}(\mathbb{R})/\mathrm{SL}_{2}(\mathbb{Z})\).
There is a technicality about a finite group \(M=\mathrm{diag}\{\pm 1, \pm 1, \pm 1\} \cap \mathrm{SL}_{3}(\mathbb{R})\) which I will ignore. What we really prove is the density of shapes of compact \(A\)-orbits of \(M \backslash G / \Gamma\). One then takes a small step to go from \(M \backslash G / \Gamma\) to its \(4\)-fold cover \(G/\Gamma\).
See [1] for the gory details.
Given a totally real cubic number field \(K\), three embeddings \(\sigma_{1},\sigma_{2},\sigma_{3}: K \rightarrow \mathbb{R}\) and an order \(\mathcal{O} \subseteq K\), one can define \(sh(\mathcal{O})\) as the following.
Map \(\mathcal{O}^{\times} \in \mathfrak{a}\) by sending \(x \mapsto (\log| \sigma_{1}(a) | , \log |\sigma_{2}(a)|, \log |\sigma_{3}(a)|)\). By Dirichlet unit theorem, it is a lattice.
It is enough to show that \(sh(\mathcal{O})\) are dense in \(\mathrm{SL}_{2}(\mathbb{R})/\mathrm{SL}_{2}(\mathbb{Z})\). Let us show this. If \(\mathcal{O}\) is an order, we observe that \(\tilde{\sigma} = (\sigma_{1},\sigma_{2},\sigma_{3}) : K \otimes \mathbb{R} \rightarrow \mathbb{R}^{3}\) is an isomorphism in which \(|\mathrm{disc}\mathcal{O}|^{\frac{1}{3}}\mathcal{O}\) is a lattice of unit covolume, and hence a point in \(G / \Gamma\). The lattice is preseved under \(\mathcal{O}^{\times}\). Let \(h_{\mathcal{O}} \in \mathrm{SL}_{3}(\mathbb{R})\) be such that \(h_{\mathcal{O}} \mathbb{Z}^{3} \simeq |\mathrm{disc}\mathcal{O}|^{\frac{1}{3}} \mathcal{O}\) under the mapping \(\sigma : K \otimes \mathbb{R} \rightarrow \mathbb{R}^{3}\).
Since one has the following equality of Abelian groups \[\begin{equation} \{ x \in K \otimes \mathbb{R} \mid x \mathcal{O} = \mathcal{O} \} = \mathcal{O}^{\times}, \end{equation}\]
One can then see the action of \((K \otimes \mathbb{R})^{(1)} = \{ x \in K \otimes \mathbb{R} \mid |\mathrm{N}(x)|=1 \}\) on \(K\otimes \mathbb{R}\) is via \[\begin{equation} x \mapsto \left( \begin{matrix} \sigma_{1}(x) & & \\ & \sigma_{2}(x) & \\ & & \sigma_{3}(x) \end{matrix} \right) \end{equation}\] The image of \((K\otimes \mathbb{R})^{(1)}\) under this map is the group \(M \cdot A \subseteq G\) where \(M\) is the sign group and the orbit of \(F = M \cdot A h_{\mathcal{O}} \Gamma\) is periodic whose shape is \(sh(\mathcal{O})\).
One can also show that each \(sh(F)\) for a periodic torus \(F\) is \(sh(\mathcal{O})\) for some totally real cubic order \(\mathcal{O}\).
Here is why. Given a periodic torus \(F = M \cdot A \cdot g \cdot \Gamma\), observe that if \(y \in A\) fixes \(g \Gamma\), then \(g^{-1} y g \in \Gamma\) and therefore the characterstic polynomial of \(y\) is a monic polynomial in \(\mathbb{Z}[X]\) with constant term \(-1\). By compactness of \(F\), we know that \(\exp P(F) = \{ u_{1}^{n_{1}} u_{2}^{n_{2}} \mid (n_{1},n_{2}) \in \mathbb{Z} \}\) for two totally real algebraic units \(u_{1},u_{2}\). We consider \(K = \mathbb{Q}[u_{1},u_{2}]\). This is a field. The questions is what is \(\deg K\)?
Note that \(\mathbb{Q}[u_{1},u_{2}]\) acts on \(g \mathbb{Q}^{3}\) and hence it can be faithfully embedded in \(\mathrm{M}_{3}(\mathbb{Q})\). Hence \(K\) must be a number field of degree \(3\). We can also define \[\begin{equation} \mathcal{O} = \{ k \in K \mid k g \mathbb{Z}^{3} \subseteq g \mathbb{Z}^{3}\}, \end{equation}\] to recover the order. The three diagonal entries of \(g^{-1} K g\) define the three real embeddings of \(K\).
As \(K,\mathcal{O}, (\sigma_{1},\sigma_{2},\sigma_{3})\) vary across totally real cubic fields \(K\), orders \(\mathcal{O} \subseteq K\) and embeddings \(\sigma_{1},\sigma_{2},\sigma_{3} : K \rightarrow \mathbb{R}\), what are the shape points \(sh(\mathcal{O})\) in \(\mathrm{SL}_{2}(\mathbb{R})/\mathrm{SL}_{2}(\mathbb{Z})\)?
In general, it is a difficult question of finding generators of \(\mathcal{O}^{\times}\) for a general \(\mathcal{O}\). But there is a good family of totally real orders whose generators are known.
Take \(0 < a_{1} <a_{2}\) to be integers \[\begin{equation} f(X) = X(X-a_{1})(X-a_{1}) - 1 \end{equation}\] Then there is a choice of \(C>0\) such that for all \(a_{1},a_{2}\) satisfying \(a_{1},a_{2},a_{2}-a_{1} > C\) one has for \(\mathcal{O} = \mathbb{Z}[X]/\langle f(X)\rangle\) the unit group \[\begin{equation} \mathcal{O}^{\times} = \{ \pm X^{n_{1}} (X-a_{1})^{n_{2}} \mid n_{1},n_{2} \in \mathbb{Z} \}. \end{equation}\]
Let us quickly sketch the proof. The polynomial \(f(X)= X(X-a_{1})(X-a_{2}) - 1\) has roots very close to \(0,a_{1},a_{2}\). Hence, it is possible to have a good approximation of \(\mathrm{disc}\mathcal{O} = \mathrm{disc}f\). On the other hand, one knows that \(X,X-a_{1},X-a_{2}\) generate a finite index subgroup \(H_{\mathcal{O}} \subseteq \mathcal{O}^{\times}\) so we have an upper bound on the regulator \(\mathrm{reg}\mathcal{O}^{\times}\). One then has Cusick’s inequality [3] \[\begin{equation} \mathrm{reg}\mathcal{O} \geq \tfrac{1}{16} \log ^{2} \left( \tfrac{1}{4} |\mathrm{disc}\mathcal{O}|\right) \end{equation}\] Then, on computation one sees that for the given conditions on \(a_{1},a_{2},a_{2}-a_{1}\), the upper bound and lower bound are at a ratio of \(\sim 4/3\). Hence if the index of \(H_{\mathcal{O}} \subseteq \mathcal{O}^{\times}\) were more than \(1\), we would have a contradiction.
One however fails to solve the problem with these polynomials. SinceThe polynomials \(f(X)\) of Cusick generate a compact set of shape points \(sh(\mathcal{O})\) for \(\mathcal{O} = \mathbb{Z}[X]/\langle f(X)\rangle\).
However, this is a feature not a bug of our proof.
One can then try to take \(\mathcal{O}_{(n)} = \mathbb{Z}+ n \mathcal{O}\). We work with \(n= p_{1}^{c} p_{2}^{d} p_{3}^{e}\) for \(1 \leq c,d,e \leq N\) for \(N \rightarrow \infty\). We want to know what is \(\mathcal{O}^{\times}_{(n)}\). Take \(b_{1} = X, b_{2} = (X-a_{1})\), which are the generators of units in \(\mathcal{O}^{\times}\). Then \[\begin{equation} \mathcal{O}_{(n)}^{\times} = \{ \pm b_{1}^{n_{1}} b_{2}^{n_{2}} \mid (n_{1},n_{2}) \in \Lambda_{(c,d,e)} \subseteq \mathbb{Z}^{2} \}. \end{equation}\] The goal is then to understand these lattices \(\Lambda_{(c,d,e)}\).
This is done by using the following proposition.
Let \[\begin{align} a_{1} \equiv 0 & \equiv a_{2} - 1\pmod{p_{1}^{c}} \\ a_{1}-1 \equiv 0 & \equiv a_{2} \pmod{p_{2}^{c}} \\ a_{1} -1 \equiv 0 & \equiv a_{2} - 1\pmod{p_{3}^{c}} \ \ \ \ \ \ \ \ (**) \end{align}\] Then the lattice \(\Lambda_{(c,d,e)}\) defined above are \[\begin{align} m+n \equiv & 0\pmod{l_{1} p_{1}^{ \max(c-l_{1}',0)}} \\ m-2n \equiv & 0\pmod{l_{2} p_{2}^{ \max(c-l_{2}',0)}} \\ n -2m\equiv & 0\pmod{l_{3} p_{3}^{ \max(c-l_{3}',0)}}, \end{align}\] for some fixed constants \(l_{1},l_{2},l_{3},l_{1}',l_{2}',l_{3}'\).
Let \(b \in \mathcal{O}^{\times}\) be a non-torsional unit and let \(p \in \mathbb{P}\). Then \(\exists l,l' \in \mathbb{Z}_{\geq 1}\) some constants such that \[\begin{equation} b^{n} \in \mathbb{Z} + p^{c} \mathcal{O} \Leftrightarrow n \in N \mathbb{Z} \text{ where } N = l p^{\max{c-l',0}}. \end{equation}\]
The proof is via induction on \(c\).
This is then used to finish the proof.
Here is how to show the main theorem.
Step 1: We first take \(N \gg 1\). Take \(0<a_{1}<a_{2}\) smallest satisfying the congruence conditions \((**)\) for all \(1 \leq c,d,e \leq N\). Construct \(\mathcal{O}\) (depending on \(N\)) from the polynomial \(f(X)\) using \(a_{1},a_{2}\).
Step 2: Take \(\mathcal{O}_{(n)} = \mathbb{Z} + n \mathcal{O}\) where \(n = p_{1}^{c} p_{2}^{d} p_{3}^{e}\). We vary \(c,d,e\) in \(1,\dots,N\). This creates \(\sim \mathbb{Z}^{3}\). Then \(sh(\mathcal{O}_{(n)})\) is in Hecke correspondence with \(sh(\mathcal{O})\).
Step 3: Because \(sh(\mathcal{O})\) live in a compact set as \(a_{1},a_{2}\) vary, we can assume that upto taking a subsequence of \(N\), \(sh(\mathcal{O})\) is a fixed point in \(\mathrm{SL}_{2}(\mathbb{R})/ \mathrm{SL}_{2}(\mathbb{Z})\).
Step 4: Enough to now that the lattices \(\{ \Lambda_{(c,d,e)}\}_{1 \leq c,d,e \leq N}\) are dense for \(N \gg 1\). All of them are Hecke neighbours of \(\mathbb{Z}^{2}\).
We finish the fourth step by showing that the closure of the union of points \(\{ \Lambda_{(c,d,e)}\}_{1 \leq c,d,e \leq N} \subseteq \mathrm{SL}_{2}(\mathbb{R})/\mathrm{SL}_{2}(\mathbb{Z})\) as \(N \rightarrow \infty\) contains a horopsherical piece and is invariant under diagonal action.
See [1] for the details!
This page was updated on January 22, 2026.
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