Integral points on varieties defined by matrix factorization into elementary matrices

TL;DR

This paper studies varieties parametrizing factorizations of matrices into elementary matrices over rings of S-integers, showing that for sufficiently many factors, the integral points are Zariski dense under certain conditions, revealing infinite factorizations.

Contribution

It introduces matrix-factorization varieties $V_k(A)$, analyzes their geometric structure, and proves Zariski density of integral points for $k \,\geq\, 4$, extending understanding of matrix factorizations over rings of integers.

Findings

$V_k(A)$ are rational $(k-3)$-folds with fibration structure

For $k\geq 4$, integral points are Zariski dense if non-empty and units are infinite

Density holds for $k\geq 9$ under the same conditions

Abstract

Let ${\mathcal O}$ be the ring of $S$-integers in a number field $K$. For $A\in\rm{SL}_{2}(\mathcal{O})$ and $k\geq 1$, we define matrix-factorization varieties $V_k(A)$ over ${\mathcal O}$ which parametrize factoring $A$ into a product of $k$ elementary matrices; the equations defining $V_k(A)$ are written in terms of Euler's continuant polynomials. We show that the $V_k(A)$ are rational $(k-3)$-folds with an inductive fibration structure. We combine this geometric structure with arithmetic results to study the Zariski closure of the ${\mathcal O}$-points of $V_k(A)$. We prove that for $k\geq 4$ the ${\mathcal O}$-points on $V_k(A)$ are Zariski dense if $V_{k}(A)({\mathcal O})\neq\emptyset$ assuming the group of units ${\mathcal O}^{\times}$ is infinite. This shows that if $A$ can be written as a product of $k\geq 4$ elementary matrices, then this can be done in infinitely many ways in the strongest sense possible. This can then be combined with results on factoring into elementary matrices for ${\rm SL}_{2}({\mathcal O})$. One result is that for $k\geq 9$ the ${\mathcal O}$-points on $V_{k}(A)$ are Zariski dense if ${\mathcal O}^{\times}$ is infinite.

Equations

$$\mathcal{O}={\mathcal{O}}_S=\{x\in K^{\times }\mid v(x)\ge 0\text{ for all v∈/S}\}.$$

$${\mathcal{V}}_k={\mathcal{V}}_k(1):1=x_1\cdots x_k.$$

$$U(x):=\left(\begin{array}{cc}1& x\\ 0& 1\end{array}\right),L(x):=\left(\begin{array}{cc}1& 0\\ x& 1\end{array}\right),D(x):=\left(\begin{array}{cc}x& 1\\ 1& 0\end{array}\right).$$

$$D(x)t=U(x)\text{mbox}andtD(x)=L(x).$$

$$V_{k}(A):\quad A=\begin{cases}L(x_{1})U(x_{2})L(x_{3})U(x_{4})\cdots L(x_{k})\quad\mbox{if $k$ is odd}\\ L(x_{1})U(x_{2})L(x_{3})U(x_{4})\cdots U(x_{k})\quad\mbox{if $k$ is even.}\end{cases}$$

$$V^{k}(A):\quad A=\begin{cases}U(x_{1})L(x_{2})U(x_{3})L(x_{4})\cdots U(x_{k})\quad\mbox{if $k$ is odd}\\ U(x_{1})L(x_{2})U(x_{3})L(x_{4})\cdots L(x_{k})\quad\mbox{if $k$ is even.}\end{cases}$$

$$\overline{V}{}_k(A):A=D(x_1)D(x_2)D(x_3)\cdots D(x_k)t^k.$$

$$A:=\left(\begin{array}{cc}a& c\\ b& d\end{array}\right)\in {\mathrm{SL}}_2(\mathcal{O}),A^{\prime }:=\left(\begin{array}{cc}d& b\\ c& a\end{array}\right)\in {\mathrm{SL}}_2(\mathcal{O}),\text{mbox}and{A}^{\ast }:=\left(\begin{array}{cc}d& c\\ b& a\end{array}\right)\in {\mathrm{SL}}_2(\mathcal{O}),$$

$$A^{\prime t}=A^{\ast },A^{t\ast }=A^{\prime },A^{\ast \prime }=A^t,A^{\prime \ast }=A^t,A^{t\prime }=A^{\ast },A^{\ast t}=A^{\prime }.$$

$$A$$

$$=D(x_1)ttD(x_2)D(x_3)ttD(x_4)\cdots D(x_{k-1})ttD(x_k)$$

$$=U(x_1)L(x_2)U(x_3)L(x_4)\cdots U(x_{k-1})L(x_k)\text{mbox}using\text{eqref}identities.$$

$$A=D(x_1)\cdots D(x_k)t=U(x_1)L(x_2)U(x_3)L(x_4)\cdots U(x_k),$$

$$[L(x_1)U(x_2)L(x_3)\cdots L(x_k){]}^{\prime }=U(x_1)L(x_2)U(x_3)\cdots U(x_k)$$

$$[L(x_1)U(x_2)L(x_3)\cdots U(x_k){]}^{\prime }=U(x_1)L(x_2)U(x_3)\cdots L(x_k),$$

$$[L(a_1)U(a_2)\cdots L(a_k){]}^t=U(a_k)L(a_{k-1})\cdots U(a_1)=A^t$$

$$K_{-2}=1,K_{-1}$$

$$K_n(x_1,\dots ,x_n)$$

$$\text{mbox}or,equivalently,K_n(x_1,\dots ,x_n)$$

$$K_2(x_1,x_2)$$

$$K_3(x_1,x_2,x_3)$$

$$K_4(x_1,x_2,x_3,x_4)$$

$$K_5(x_1,x_2,x_3,x_4,x_5)$$

$$L(x_1)U(x_2)L(x_3)\cdots L(x_k)=\left(\begin{array}{cc}{K}_{k-1}(x_2,x_3,\dots ,x_k)& K_{k-2}(x_2,x_3,\dots ,x_{k-1})\\ K_k(x_1,x_2,\dots ,x_k)& K_{k-1}(x_1,x_2,\dots ,x_{k-1})\end{array}\right).$$

$$L(x_1)U(x_2)L(x_3)\cdots U(x_k)=\left(\begin{array}{cc}{K}_{k-2}(x_2,x_3,\dots ,x_{k-1})& K_{k-1}(x_2,x_3,\dots ,x_k)\\ K_{k-1}(x_1,x_2,\dots ,x_{k-1})& K_k(x_1,x_2,\dots ,x_k)\end{array}\right).$$

$$\left(\begin{array}{cc}a& c\\ b& d\end{array}\right)=\left(\begin{array}{cc}{K}_{k-1}(x_2,x_3,\dots ,x_k)& K_{k-2}(x_2,x_3,\dots ,x_{k-1})\\ K_k(x_1,x_2,\dots ,x_k)& K_{k-1}(x_1,x_2,\dots ,x_{k-1})\end{array}\right).$$

$$\left(\begin{array}{cc}a& c\\ b& d\end{array}\right)=\left(\begin{array}{cc}{K}_{k-2}(x_2,x_3,\dots ,x_{k-1})& K_{k-1}(x_2,x_3,\dots ,x_k)\\ K_{k-1}(x_1,x_2,\dots ,x_{k-1})& K_k(x_1,x_2,\dots ,x_k)\end{array}\right).$$

$$(x_1,\dots ,x_k)\mapsto (x_4,\dots ,x_k)$$

$$(x_1,\dots ,x_k)\mapsto (x_3,x_5,\dots ,x_k)$$

$$\left(\begin{array}{cc}a& c\\ b& d\end{array}\right)=\left(\begin{array}{cc}{x}_2{x}_3+1& x_2\\ x_1{x}_2{x}_3+x_1+x_3& x_1{x}_2+1\end{array}\right).$$

$$(x_1,x_2,x_3)=\left(\frac{d-1}{c},c,\frac{a-1}{c}\right).$$

$$x_1(v)$$

$$x_2(v)$$

$$x_3(v)$$

$$x_4(v)$$

Full text

1. Introduction

Let $K$ be a number field and let $S$ be a finite set of primes of $K$ containing the archimedean valuations. Denote by $\mathcal{O}=\mathcal{O}_{S}$ the ring of $S$-integers in $K$:

[TABLE]

Factoring $1$ in $\mathcal{O}^{\times}=\operatorname{\operatorname{GL}_{1}}(\mathcal{O})$ is naturally parametrized by associated factorization varieties. For $k\geq 2$, set

[TABLE]

The (affine) variety $\mathcal{V}_{k}$ is smooth, rational, irreducible, and of dimension $k-1$. When $k=2$, $\mathcal{V}_{2}$ is a curve of genus [math] with $2$ points at infinity. Because there are fewer than $3$ points at $\infty$ on this rational curve, Siegel’s Theorem [siegel], [serre-mw, Chapter 7] does not force $\mathcal{V}_{2}$ to have finitely many $\mathcal{O}$-points. And, indeed, $\mathcal{V}_{2}(\mathcal{O})$ is Zariski dense if and only if $\mathcal{O}^{\times}$ is infinite. It is easy to see that this generalizes for $k>2$:

Theorem 1.1**.**

The $\mathcal{O}$-points on $\mathcal{V}_{k}$ for $k\geq 2$ are Zariski dense if and only if the group of units $\mathcal{O}^{\times}$ is infinite.

We would like to generalize this picture to matrices and consider integral points on matrix-factorization varieties. In this paper we give one such generalization to $\operatorname{\operatorname{SL}_{2}}(\mathcal{O})$, considering the varieties parametrizing factoring into a product of elementary matrices. For $x\in\mathcal{O}$ define the upper triangular matrix $U(x)$, the lower triangular matrix $L(x)$, and the matrix $D(x)$ by

[TABLE]

The elementary matrices over $\mathcal{O}$ are the matrices $U(x)$, $L(x)$ for $x\in\mathcal{O}$; $\det D(x)=-1$. Set $t:=\left(\begin{smallmatrix}0&1\\ 1&0\end{smallmatrix}\right)$ and note the identities

[TABLE]

We consider the (affine) matrix-factorization varieties defined by writing $A\in\operatorname{\operatorname{SL}_{2}}(\mathcal{O})$ as a product of $k$ matrices from (2):

Definition 1.2**.**

Suppose $k\geq 1$.

1. Define the variety $V_{k}(A)$ by writing $A$ a product of $k$ elementary matrices beginning with a lower triangular matrix:

[TABLE]

2. Define the variety $V^{k}(A)$ by writing $A$ a product of $k$ elementary matrices beginning with an upper triangular matrix:

[TABLE]

3. Put $\mathbb{V}_{k}(A)=V_{k}(A)\coprod V^{k}(A)$. Note that $\mathbb{V}_{k}(A)(\mathcal{O})\neq\emptyset$ if and only if $A$ can be written as the product of $k$ elementary matrices.

4. Define the variety $\overline{V}\!_{k}(A)$ by writing $A$ as product of $k$ of the $D(x)$’s and $t^{k}$:

[TABLE]

In this paper our main result is the following analogue for matrix-factorization varieties of Theorem 1.1:

Theorem 1.3**.**

*Suppose $A\in\operatorname{\operatorname{SL}_{2}}(\mathcal{O})$ and let $\mathscr{V}_{k}(A)$ be $V_{k}(A)$, $V^{k}(A)$, or $\overline{V}\!_{k}(A)$. Assume the group of units $\mathcal{O}^{\times}$ is infinite.

1) Suppose $k\geq 9$. Then the $\mathcal{O}$-points on the rational and irreducible $(k-3)$-fold $\mathscr{V}_{k}(A)$ are Zariski dense.

2) Suppose $S$ contains a real archimedean or a finite prime. Then the $\mathcal{O}$-points on $\mathscr{V}_{k}(A)$ are Zariski dense for $k\geq 8$.

3) Assume the Generalized Riemann Hypothesis as in [jz, Section 4]. Then the $\mathcal{O}$-points on $\mathscr{V}_{k}(A)$ are Zariski dense for $k\geq 5$ if $S$ contains a real archimedean prime, for $k\geq 6$ if $S$ contains a finite prime, and for $k\geq 7$ in general.*

Roughly speaking, Theorem 1.3 says that if $\#\mathcal{O}^{\times}=\infty$, then for $k$ sufficiently large, not only can every $A\in\operatorname{\operatorname{SL}_{2}}(\mathcal{O})$ be written as the product of $k$ elementary matrices over $\mathcal{O}$, but this can be done in (infinitely) many ways in the strongest possible sense.

2. First properties of the Matrix-Factorization Varieties

$V_{k}(A)$, $V^{k}(A)$, and $\overline{V}\!_{k}(A)$

In fact we only have to study one of the three matrix-factorization varieties in Definition 1.2 and the three factorization problems are equivalent. Let

[TABLE]

with $A^{t}=\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)$. Then we have $A^{\prime\prime}=A^{tt}=A^{\ast\ast}=A$ and

[TABLE]

In other words, $\{\prime,\,t,\ast=\prime t\}$ are the elements of order $2$ in a Klein Vierergruppe acting on the set $\operatorname{\operatorname{SL}_{2}}(\mathcal{O})$.

Proposition 2.1**.**

1. *Suppose $A\in\operatorname{\operatorname{SL}_{2}}(\mathcal{O})$. The polynomial equations in $x_{1},\,x_{2},\,\ldots,x_{k}$ defining $V_{k}(A)$, $V^{k}(A^{\prime})$, and $\overline{V}\!_{k}(A^{\prime})$ in $\mathbb{A}^{k}_{/\mathcal{O}}$ are identical. The following are equivalent:

*a) * $(a_{1},a_{2},\ldots,a_{k})\in V_{k}(A)(\mathcal{O})\subseteq\mathbb{A}^{k}(\mathcal{O})$.

*b) * \begin{cases}A=L(a_{1})U(a_{2})\cdots L(a_{k})\quad\mbox{if kis odd}\\ A=L(a_{1})U(a_{2})\cdots U(a_{k})\quad\mbox{ ifk is even}.\end{cases}

*c) * $(a_{1},a_{2},\ldots,a_{k})\in V^{k}(A^{\prime})(\mathcal{O})\subseteq\mathbb{A}^{k}(\mathcal{O})$.

*d) * \begin{cases}A^{\prime}=U(a_{1})L(a_{2})\cdots U(a_{k})\quad\mbox{if kis odd}\\ A^{\prime}=U(a_{1})L(a_{2})\cdots L(a_{k})\quad\mbox{ifk is even}.\end{cases}

*e) * $(a_{1},a_{2},\ldots,a_{k})\in\overline{V}\!_{k}(A^{\prime})(\mathcal{O})\subseteq\mathbb{A}^{k}(\mathcal{O})$

*f) * $A^{\prime}=D(a_{1})D(a_{2})\cdots D(a_{k})t^{k}$.

*2. * In particular, the following are equivalent:

*a) * For every matrix $A\in\operatorname{\operatorname{SL}_{2}}(\mathcal{O})$, $V_{k}(A)(\mathcal{O})\neq\emptyset$.

*b) * For every matrix $A\in\operatorname{\operatorname{SL}_{2}}(\mathcal{O})$, $V^{k}(A)(\mathcal{O})\neq\emptyset$.

c) * For every matrix $A\in\operatorname{\operatorname{SL}_{2}}(\mathcal{O})$, $\overline{V}\!_{k}(A)(\mathcal{O})\neq\emptyset$.

Proof.

1. The varieties $\overline{V}\!_{k}(A)$ and $V^{k}(A)$ are defined by the same equations.

If $k$ is even, then

[TABLE]

If $k$ is odd, then similarly

[TABLE]

proving 1).

2. The varieties $V_{k}(A)$ and $V^{k}(A^{\prime})$ are defined by the same equations.

We have $L(x)^{\prime}=U(x)$ and $U(x)^{\prime}=L(x)$. Verify by induction that for $k$ odd

[TABLE]

and for $k$ even

[TABLE]

proving 2). ∎

Proposition 2.2**.**

*Suppose $A\in\operatorname{\operatorname{SL}_{2}}(\mathcal{O})$. The following are equivalent:

*a) * $(a_{1},a_{2},\ldots,a_{k})\in V_{k}(A)(\mathcal{O})\subseteq\mathbb{A}^{k}(\mathcal{O})$.

b) * \begin{cases}(a_{k},a_{k-1},\ldots,a_{1})\in V_{k}(A^{\ast})(\mathcal{O})\subseteq\mathbb{A}^{k}(\mathcal{O})\quad\mbox{if kis odd}\\ (a_{k},a_{k-1},\ldots,a_{1})\in V_{k}(A^{t})(\mathcal{O})\subseteq\mathbb{A}^{k}(\mathcal{O})\quad\mbox{ifk is even}.\end{cases}

Proof.

Proposition 2.2(a) is equivalent to Proposition 2.1(b). We need only note that if $k$ is odd and $A=L(a_{1})U(a_{2})\cdots L(a_{k})$, then

[TABLE]

which is equivalent to $(a_{k},a_{k-1},\ldots,a_{1})\in V_{k}(A^{t\prime})(\mathcal{O})=V_{k}(A^{\ast})(\mathcal{O})$ by Proposition 2.1(d). Likewise if $k$ is even and $A=L(a_{1})U(a_{2})\cdots U(a_{k})$, then $L(a_{k})U(a_{k-1})\cdots U(a_{1})=A^{t}$, equivalently, $(a_{k},a_{k-1},\ldots,a_{1})\in V_{k}(A^{t})$. ∎

Henceforth we restrict our attention to $V_{k}(A)$; in light of Proposition 2.1 and its proof all results are easily transferred to the matrix factorization problems associated to $V^{k}(A)$ and $\overline{V}\!_{k}(A)$. In particular, note that we could equally well put $\mathbb{V}_{k}(A)=V_{k}(A)\coprod V_{k}(A^{\prime})$ since $V^{k}(A)$ and $V_{k}(A^{\prime})$ are defined by identical equations. The explicit equations defining $V_{k}(A)$ over $\mathcal{O}$ can be written down in terms of the continuant polynomials $K_{n}$ of Euler [e]. The $K_{n}$ are defined recursively by

[TABLE]

For example,

[TABLE]

Proposition 2.3**.**

1. • If $k$ is odd,*

[TABLE]

2. If $k$ is even,

[TABLE]

Proof.

Proceed by induction on $k$. ∎

Hence we deduce:

Proposition 2.4**.**

1. If $k$ is odd, the equations defining $V_{k}(A)$ are

[TABLE]

2. If $k$ is even, the equations defining $V_{k}(A)$ are

[TABLE]

3. The geometry of

$V_{k}(A)$

An initial observation on $V_{k}(A)$ is that we know its dimension; a further observation is that $V_{k}(A)$ is rational:

Proposition 3.1**.**

For $k\geq 3$, we have $\dim V_{k}(A)=k-3$ unless $k=3$ and $c=0$. It is irreducible unless $k=4$ and $a=1$ or $k=5$ and $A=L(b)$. In the case $k\geq 3$ with $c\neq 0$ if $k=3$, $V_{k}(A)$ is also rational (or the union of two irreducible rational varieties) with a birational map $V_{k}(A)\to\mathbb{A}^{k-3}$ given by

[TABLE]

(with the other component given by

[TABLE]

in the case when $V_{k}(A)$ is reducible)* and inductively fibered.*

Proof.

We proceed by induction on $k$. For $k=3$ by Proposition 2.4 $V_{3}(A)$ is given by

[TABLE]

Hence unless $c=0$ the unique solution is given by

[TABLE]

Notice that in the case when $c=0$, the variety is empty unless we also have $a=1$ in which case it’s a rational curve with a birational map to $\mathbb{A}^{1}$ given by $x_{3}$.

Now assume the result holds for $k-1$ by induction. Consider the fibration $V_{k}(A)\to\mathbb{A}^{1}$, given by $(x_{1},x_{2},\ldots,x_{k})\mapsto x_{k}$. Notice the fibers of this map are given by $V_{k-1}(AL(x_{k})^{-1})$ for $k$ odd and $V_{k-1}(AU(x_{k})^{-1})$ for $k$ even. Either way the generic fibers will be generically inductively fibered rational varieties of dimension $k-4$ by the induction hypothesis. Hence, $V_{k}(A)$ is an inductively fibered rational variety of dimension $k-3$.

In fact the fibers will be rational and irreducible unless $k\leq 6$. We now describe the cases. If $k=4$, the fibers will be either a point or empty unless $a=1$ in which case one of the fibers is a rational curve. Hence if $a=1$, then $V_{4}(A)$ is the reducible union of two rational curves with maps to $\mathbb{A}^{1}$ given by $x_{4}$ and $x_{3}$ respectively.

If $k=5$, the generic fiber is irreducible with birational map given by (11) unless $A=L(b)$. Hence for $A\neq L(b)$, $V_{5}(A)$ is either irreducible or has a component of dimension $1$ which is impossible since it is given by $3$ equations (the fourth is redundant since the determinant is $1$), and thus can’t have any components of dimension less than $5-3=2$. On the other hand if $A=L(b)$ then all the components are reducible; hence here $V_{5}(A)$ is the union of two rational surfaces with birational maps given by (11) and (12).

If $k\geq 6$, the generic fiber is always irreducible; therefore, $V_{6}(A)$ is always irreducible with birational map given by (11) by the same logic as above. ∎

4. The arithmetic of $V_{k}(A)$

We begin with a rather remarkable identity, which can be verified simply by multiplying matrices.

Lemma 4.1**.**

For $v\in K^{\times}$, set

[TABLE]

Then we have an equality in $\operatorname{Mat}_{2\times 2}(K(x_{1},x_{2},x_{3},x_{4}))$:**

[TABLE]

Suppose $P=(x_{1},x_{2},x_{3},x_{4})\in V_{4}(A)(\mathcal{O})$ with $A=\left(\begin{smallmatrix}a&c\\ b&d\end{smallmatrix}\right)\in\operatorname{\operatorname{SL}_{2}}(\mathcal{O})$ as in (7). Then by Proposition 2.4(2), we have $a=K_{2}(x_{2},x_{3})=1+x_{2}x_{3}$. Hence if $v\equiv 1\bmod a$, then by (14) of Lemma 4.1

[TABLE]

Proposition 4.2**.**

Suppose $A\in\operatorname{\operatorname{SL}_{2}}(\mathcal{O})$ and suppose $\mathcal{O}^{\times}$ is infinite. If $V_{4}(A)(\mathcal{O})\neq\emptyset$, then the $\mathcal{O}$-points are Zariski dense.

Proof.

First assume $a\neq 0$. Set $\mathcal{U}_{1}(a)=\{v\in\mathcal{O}^{\times}\mid v\equiv 1\bmod a\}$. Since $\mathcal{O}^{\times}$ is infinite by assumption, $\mathcal{U}_{1}(a)\subseteq\mathcal{O}^{\times}$ is infinite too. If $P\in V_{4}(A)(\mathcal{O})$, then $v.P$ for $v\in\mathcal{U}_{1}(a)$ as in (15) is an infinite set of points in $V_{4}(A)(\mathcal{O})$.

Now suppose $a=0$, then $x_{3}=-x_{2}^{-1}$ and

[TABLE]

Hence for any $u\in\mathcal{O}$, we get $(x_{1}+ux_{3},x_{2},x_{3},x_{4}-ux_{2})\in V_{4}(A)(\mathcal{O})$. Thus we again get an infinite set of points in $V_{4}(A)(\mathcal{O})$.

In the case when $V_{4}(A)$ is irreducible this implies that the $\mathcal{O}$-points are Zariski dense. By Proposition 3.1 this leaves us with the case when $a=1$. In that case we get $x_{2}x_{3}=0$ so the two components correspond to $x_{2}=0$ and $x_{3}=0$. For $u\in\mathcal{O}$, the formulas $(u,0,b-u,c)$ and $(b,c-u,0,u)$ give infinitely many points on each of the two components. ∎

Theorem 4.3**.**

*Suppose $A\in\operatorname{\operatorname{SL}_{2}}(\mathcal{O})$ and suppose the group of units $\mathcal{O}^{\times}$ is infinite.

1) If the $\mathcal{O}$-points of $V_{k}(A)$ are Zariski dense for some $k\geq 4$, then the $\mathcal{O}$-points of $V_{k^{\prime}}(A)$ are Zariski dense for all $k^{\prime}\geq k$.

2) If $k\geq 4$ and $V_{k}(A)(\mathcal{O})\neq\emptyset$, then the $\mathcal{O}$-points are Zariski dense.*

Proof.

We start by proving (2). We proceed by induction. The $k=4$ case is Lemma 4.1. Now consider the subvariety of $W\subset V_{k}(A)$ given by fixing $(x_{1},\ldots,x_{k-4})$. Note that $W$ is given by $V_{4}(B)$ for some matrix $B$ and is nonempty since $(x_{k-3},\ldots,x_{k})\in W$. Thus by Lemma 4.1 the $\mathcal{O}$-points are Zariski dense in $W$ and in particular the images of the $\mathcal{O}$-points under the map to $\mathbb{A}^{1}$ given by $x_{k}$ from Proposition 3.1 are Zariski dense. Also, by the induction hypothesis the $\mathcal{O}$-points of $V_{k}(A)$ are Zariski dense in each of the fibers above each of these points in $\mathbb{A}^{1}$. Hence, the $\mathcal{O}$-points are Zariski dense in $V_{k}(A)$.

Now (1) follows trivially from (2) and the fact that if $V_{k}(A)(\mathcal{O})\neq\emptyset$, then $V_{k^{\prime}}(A)(\mathcal{O})\neq\emptyset$ for $k^{\prime}\geq k$ since $(x_{1},\ldots,x_{k})\in V_{k}(A)$ implies $(x_{1},\ldots,x_{k},0,\ldots,0)\in V_{k^{\prime}}(A)$. ∎

Remark 4.4*.*

There has been recent work on factoring matrices in $\operatorname{\operatorname{SL}_{2}}(\mathcal{O})$ into products of elementary matrices. Recall that the statement that $A\in\operatorname{\operatorname{SL}_{2}}(\mathcal{O})$ can be written as the product of $k$ elementary matrices is equivalent to $\mathbb{V}_{k}(A)(\mathcal{O})\neq\emptyset$. However, the constructions in [mrs] and [v] all write any $A\in\operatorname{\operatorname{SL}_{2}}(\mathcal{O})$ as a product of elementary matrices beginning with a lower triangular matrix – so in these cases we can deduce that $V_{k}(A)(\mathcal{O})\neq\emptyset$. The construction of [jz, Proof of Theorem 2.6] writes any $A\in\operatorname{\operatorname{SL}_{2}}(\mathcal{O})$ as a product of $k$ elementary matrices beginning with an upper triangular matrix if $k$ is odd and a lower triangular matrix if $k$ is even. Hence in either case here we can also conclude that $V_{k}(A)(\mathcal{O})\neq\emptyset$.

Below we summarize known results:

1. For all $A\in\operatorname{\operatorname{SL}_{2}}(\mathcal{O})$, $V_{k}(A)(\mathcal{O})\neq\emptyset$ if $\mathcal{O}^{\times}$ is infinite and $k\geq 9$ ([mrs, Remark (1), page 1969]).

2. Suppose $\mathcal{O}^{\times}$ is infinite and $S$ contains a real archimedean prime or a finite prime. Then for all $A\in\operatorname{\operatorname{SL}_{2}}(\mathcal{O})$ and for all $k\geq 8$ we have $V_{k}(A)(\mathcal{O})\neq\emptyset$ ([jz, Theorem 1.2 and proof]).

3. Assume the Generalized Riemann Hypothesis and suppose $\mathcal{O}^{\times}$ is infinite. Then for all $A\in\operatorname{\operatorname{SL}_{2}}(\mathcal{O})$ we have $V_{k}(A)(\mathcal{O})\neq\emptyset$ for $k\geq 5$ if $S$ contains a real archimedean prime, for $k\geq 6$ if $S$ contains a finite prime, and for $k\geq 7$ in general ([jz, Theorem 1.3 and proof]).

4. Suppose $p$ is a prime. Then for all $A\in\operatorname{\operatorname{SL}_{2}}(\mathbb{Z}[1/p])$ we have $V_{k}(A)(\mathbb{Z}[1/p])\neq\emptyset$ for $k\geq 5$ ([v, Remark 1.1]).

5. There is a prime $p$ and matrix $A\in\operatorname{\operatorname{SL}_{2}}(\mathbb{Z}[1/p])$ such that $V_{4}(A)(\mathbb{Z}[1/p])=\emptyset$ ([mrs, Proposition 5.1]).

Proof of Theorem 1.3.

Combining these known results with Proposition 2.1 and Theorem 4.3 then proves our main Theorem 1.3. ∎