Rational configuration problems and a family of curves

Fix $\eta:=\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in\operatorname{GL}_{2}(\mathbb{Q})$, and let $H_{\eta}$ be the curve defined by

in the weighted projective plane where $x,y,z$ have degree $1,2,1$, respectively. Rational points on this curve correspond to vectors $\begin{pmatrix}u\\ v\end{pmatrix}\in\mathbb{Q}^{2}\setminus\{0\}$ such that both $\begin{pmatrix}u\\ v\end{pmatrix}$ and $\eta\begin{pmatrix}u\\ v\end{pmatrix}$ have rational length, and as a result, curves of this form can be used to describe solutions to a collection of rational configuration problems; see Section 1.2 for more details. In this paper we study the loci of points $\eta$ for which $H_{\eta}$ has zero, finitely many, or infinitely many rational points.

First, we show that for most values of $\eta$, the curve $H_{\eta}$ has no rational points.

The proof is given in Section 4. Note that $H_{\eta}\simeq H_{m\eta}$ for any positive integer $m$, so by clearing denominators, every $H_{\eta}$ is isomorphic to one of the curves counted in Theorem 1.1. For the sake of comparison, consider the following result by Bhargava, Cremona, and Fisher.

We see that the subfamily $H_{\eta}$ differs from the larger family, in that far fewer specializations are everywhere locally soluble.

Now suppose we restrict our attention to the collection of points $\eta$ for which $H_{\eta}$ does contain a rational point. In this case we have a stronger classification. Let $\eta^{t}$ denote the transpose of $\eta$.

A proof is given in Section 3.4 using the fact that the isomorphism type of $H_{\eta}$ is invariant under acting on the left and right of $\eta$ by elements of the orthogonal group $\operatorname{O}_{2}(\mathbb{Q})$. An explicit change of variables expressing $r,s$ in terms of $a,b,c,d$ and the rational point $(x_{0}:y_{0}:z_{0})\in H_{\eta}(\mathbb{Q})$ is given by Lemma 3.2.

In particular, for most of the values $\eta$ such that $H_{\eta}(\mathbb{Q})$ is nonempty, $H_{\eta}(\mathbb{Q})$ is actually infinite. The proof of this result is given in Section 3.5. We discuss several applications of this result to rational distance problems in Section 1.2, but mention one here as a representative example.

In fact we prove a stronger result: there is an infinite collection of curves $C_{n}$ in the plane such that the intersection points of the curves $C_{n}$ with any fixed line $y=\frac{2t}{1-t^{2}}x$ (for $t\in\mathbb{Q}\setminus\{-1,0,1\}$) gives a dense set of solutions to the three-distance problem within the given line (Corollary 5.4).

Even in the cases where $(-1,r)\in E_{r,s}(\mathbb{Q})$ is torsion, there are still several cases in which we can prove $E_{r,s}(\mathbb{Q})$ has positive rank. We discuss these in more depth in Section 5.4, but note one special case here. Let

denote the set of slopes of rational right triangles (including negatives and zero).

See Section 5.4 for a proof.

Given a finite simple graph $G=(V,E)$, an embedding $\phi:V\hookrightarrow\mathbb{R}^{n}$ is a rational configuration if the distance $d(\phi(v),\phi(w))$ is rational for all $(v,w)\in E$. We may add some additional constraints to the set of allowable embeddings (for instance, we may require some pairs of edges to be the same length, or to meet at right angles), and in doing so we obtain a corresponding rational configuration problem: to determine whether there exists a rational configuration satisfying the desired constraints, and if so, to classify or count the number of rational configurations. We describe a list of sample rational configurations below; the corresponding graphs can be found in Table 1.

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  “Adjacent rectangles:” Find two rectangles sharing an edge such that the distance between any two vertices is rational.

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  “Detour:” Fix parameters $r,s,t\in\mathbb{Q}^{\times}$. Find a point $x$ such that $(x,0)$ has rational distance to $(0,0)$, $(r,0)$, $(0,s)$, and $(r,t)$. (A traveller is going from $(0,s)$ to $(r,t)$, but has to take a detour to stop at the $x$-axis along the way; can they do so using only two straight paths of rational length?)

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  “Perfect cuboid:” Find a rectangular prism such that the distance between any two vertices is rational.

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  “Body cuboid:” Find a rectangular prism such that the distance between any two vertices that share a face are rational.

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  “Square four-distance:” Find a point $(x,y)\in\mathbb{R}^{2}$ such that the distance to each of $(0,0)$, $(1,0)$, $(0,1)$, and $(1,1)$ is rational.

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  “Square three-distance:” Find a point $(x,y)\in\mathbb{R}^{2}$ such that the distance to each of $(0,0)$, $(0,1)$, and $(1,1)$ is rational.

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  “Rectangle four-distance:” Find $r\in\mathbb{Q}^{\times}$ and a point $(x,y)\in\mathbb{R}^{2}$ such that the distance to each of $(0,0)$, $(0,1)$, $(r,0)$, and $(r,1)$ is rational.

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  “Rational distances under Möbius transformation:” Fix $a,b,c,d\in\mathbb{Q}$ with $ad-bc\neq 0$. Find $z\in\mathbb{C}$ such that $z$ and $\frac{az+b}{cz+d}$ both have rational distance from $0$.

The perfect cuboid problem and square four-distance problem are classic unsolved problems (see Section 2); this paper does not present a solution to either of them. However, we can put all the remaining problems in this list into a common framework. Define

so that whenever $u,v$, not both zero, are the legs of a (possibly degenerate) rational right triangle, the slope of the triangle is in $\mathcal{S}$. Then for distinct $P_{1},P_{2}\in\mathbb{Q}^{2}$, the distance between $P_{1}$ and $P_{2}$ is rational if and only if the line between $P_{1}$ and $P_{2}$ has slope in $\mathcal{S}$. Using this observation, we can parametrize solutions to many rational configuration problems by finding elements of $\mathcal{S}$ satisfying simple polynomial relations.

Similar polynomial constraints for each of the problems above are listed in Table 1. Note that for every problem in Table 1 besides the perfect cuboid problem and the square four-distance problem, rational configurations correspond to solutions in $\mathcal{S}$ to a single polynomial in multiple variables that is linear in each variable.

This follows from Proposition 5.1. Proposition 1.8 shows that for a wide collection of problems, rational configurations can be classified using rational points on curves of the form $H_{\eta}$. We can use this observation to show that some rational configuration problems have infinitely many rational configurations. In some cases, such as the detour problem and the square three-distance problem, the infinitude of solutions will be a consequence of Theorem 1.4. For others, including the adjacent rectangles, body cuboid, and rectangle four-distance problems, the corresponding curve $E_{r,s}$ lands in one of the exceptional cases of Theorem 1.4, and so we cannot immediately conclude that there are infinitely many solutions.

We begin with a discussion of some related problems and their histories in Section 2. In Section 3 we analyze the algebraic structure of the family $H_{\eta}$, in particular showing that the isomorphism type of $H_{\eta}$ is invariant under a left- and right-action of the orthogonal group (Section 3.3). We then analyze the singular fibers (Section 3.4), followed by the non-singular fibers that contain a rational point (Section 3.5), proving that most fibers of this type have infinitely many rational points (Theorem 1.4). Completing our study of rational points on the fibers, Section 4 contains a proof the set of fibers containing a rational point has low density (Theorem 1.1). Note that Section 4 only requires Section 3.1 and Section 3.2 from Section 3.

We conclude with some applications of these results in Section 5, focusing primarily on the square three-distance problem.

The author was supported by a CRM-ISM Postdoctoral Fellowship during the writing of this article, and would like to thank Andrew Granville, Sun-Kai Leung, Michael Lipnowski, Henri Darmon, Eyal Goren, Allysa Lumley, Olivier Mila, Wanlin Li, and Valeriya Kovaleva for helpful discussions.

While the perfect cuboid problem is open, significant progress has been made towards studying the “body cuboid” problem, which is to give a cuboid in which all edges and all face diagonals (but not necessarily the body diagonal) have rational lengths. If $1,\alpha_{1},\alpha_{2}$ are the edge lengths of a body cuboid, then $\alpha_{1}^{2}+1$, $\alpha_{2}^{2}+1$, and $\alpha_{1}^{2}+\alpha_{2}^{2}$ are all perfect squares; the first two conditions say that $\alpha_{1},\alpha_{2}\in\mathcal{S}$ and the third is equivalent to requiring $\frac{\alpha_{2}}{\alpha_{1}}\in\mathcal{S}$.

For each fixed $\alpha_{3}\in\mathcal{S}$, the values $\alpha_{1},\alpha_{2}\in\mathcal{S}$ satisfying $\alpha_{1}\alpha_{3}=\alpha_{2}$ are parametrized by an elliptic curve (Proposition 1.8 and Proposition 1.3). This association between body cuboids and a family of elliptic curves is well-studied; Luijk has an in-depth survey [13] that mentions this association as well as many other known results about perfect cuboids. Halbeisen and Hungerbüler [10] investigate this problem as well. Given a fixed $\alpha_{3}=\frac{b}{a}$, they associate solutions $\alpha_{1},\alpha_{2}\in\mathcal{S}\setminus\{0\}$ satisfying $\alpha_{1}\alpha_{3}=\alpha_{2}$ to rational points on the elliptic curve

Proposition 1.8 and Proposition 1.3 recovers this classification. They show that there is a subgroup of $E(\mathbb{Q})$ isomorphic to $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/4\mathbb{Z}$ which give degenerate solutions to the corresponding rational distance problem. Ruling out other possible torsion points, they conclude [10, Theorem 8] that nondegnerate solutions exist if and only if $E(\mathbb{Q})$ has positive rank. In this case they call $(a,b)$ a double-pythapotent pair.

As with the perfect cuboid problem, the four-distance problem is currently out of reach, but a slightly weaker variant has many known solutions. The three-distance problem is to find points $P=(x,y)\in\mathbb{R}^{2}$ with rational distance to $(0,0)$, $(0,1)$, and $(1,1)$. The coordinates $x,y$ are not a priori assumed to be rational, but since $x^{2}+y^{2}$, $x^{2}+(1-y)^{2}$, and $(1-x)^{2}+(1-y)^{2}$ must all be rational, the differences $2y-1$ and $2x-1$ must also be rational, so in fact $P\in\mathbb{Q}^{2}$. We can then scale by an element of $\mathbb{Q}^{\times}$ so that $x=1$, and a solution to the square three-distance problem is equivalent to the existence of $\alpha_{1},\alpha_{2},\alpha_{3}\in\mathcal{S}$ satsifying $1+\alpha_{1}\alpha_{2}=\alpha_{1}+\alpha_{3}$.

For many years it was believed that there were no solutions to the three-distance problem aside from points on the coordinate axes. The first one-parameter family of nontrivial solutions was found in 1967 by J.H. Hunter, and then many more infinite families were found in rapid succession; a historical overview is given by Berry, who also presents an “extraordinary abundance” of solutions lying in infinitely many one-parameter families [3]. We observe that the families of solutions obtained in Corollary 1.5 are distinct from those that appear in [3, Table 4], though it is unclear whether any (or all) of the one-parameter families we consider are eventually accounted for by Berry’s construction.

A rational number $n\in\mathbb{Q}$ is a congruent number if it is the area of a right triangle with rational edge lengths; that is, if there is a solution to

The “congruent number problem” is to determine whether a given $n\in\mathbb{Q}$ is a congruent number. This problem is not a rational configuration problem, but the underlying methods used to study these two problems are similar enough that a comparison is worthwhile.

There is a well-known approach to studying the congruent number problem; see for example the expositions [6] and [5]. For fixed $n$, any solution to Eq. 3 corresponds to a rational point on an elliptic curve over $\mathbb{Q}$ defined by

There are “degenerate points” in $E^{(n)}(\mathbb{Q})$ that do not correspond to solutions; it can be shown that the set of degenerate points equals the torsion subgroup of $E^{(n)}(\mathbb{Q})$. Thus $n$ is a congruent number if and only if $E^{(n)}(\mathbb{Q})$ has positive rank. A formula due to Tunnell can be used to determine whether the analytic rank of $E^{(n)}$ is zero or positive [16], so by assuming the Birch and Swinnerton-Dyer conjecture, this gives a criterion that determines whether a given number is congruent.

Many aspects of this paper are modeled off of the approach described for studying the congruent number problem. To put the two problems on a common footing, note that $n$ is a congruent number if and only if $x_{1}=a^{2}$ and $x_{2}=\frac{b}{a}$ give a solution to

Both $\mathcal{S}$ and $(\mathbb{Q}^{\times})^{2}$ can be represented as the image of $\mathbb{A}^{1}(\mathbb{Q})$ under the image of a degree $2$ rational map $\mathbb{A}^{1}\to\mathbb{A}^{1}$. The curve $E^{(n)}$ comes equipped with a degree $4$ rational map to the variety defined by $x_{1}x_{2}-2n=0$, and non-degenerate points in $E^{(n)}(\mathbb{Q})$ map to solutions to Eq. 5. This is directly analogous to the relation between $H_{\eta}(\mathbb{Q})$ and solutions to rational distance problems (Proposition 1.8).

However, it is worth highlighting a few key differences between the congruent number problem and the family of rational distance problems we consider.

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  Size of parameter space. The isomorphism class of $E^{(n)}$ is determined by the class of $n$ in $\mathbb{Q}^{\times}/(\mathbb{Q}^{\times})^{2}$, while the isomorphism class of $H_{\eta}$ is determined by the class of a corresponding matrix in a double quotient of $\operatorname{GL}_{2}(\mathbb{Q})$.

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  Existence of rational points. Every $n$ determines an elliptic curve $E^{(n)}$, which has a rational point. By contrast, the genus one curves $H_{\eta}$ typically have no rational points (Theorem 1.1).

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  Closure under addition of degenerate points. In both problems, the corresponding genus one curve has a set of “degenerate” rational points, which do not yield valid solutions to the original problem. For the congruent number problem, the set of degenerate points equals the torsion subgroup of $E^{(n)}(\mathbb{Q})$. For rational configuration problems, however, even if $\mathcal{H}_{\eta}$ is isomorphic to an elliptic curve (Proposition 1.3), the degenerate points in ${H}_{\eta}(\mathbb{Q})$ may not form a subgroup. This is to our advantage: we can often add together degenerate points to produce non-degenerate points, something that is not possible in the congruent number problem. This is the key idea behind Theorem 1.4.

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  Geometric variation in the family. The curves $E^{(n)}$ are quadratic twists of the curve $y^{2}=x^{3}-x$, and are therefore all isomorphic over $\overline{\mathbb{Q}}$. This fact is used in a key way in the proof of Tunnell’s theorem, as he applies a result due to Waldspurger [17] relating the central value of the $L$-function of an elliptic curve with that of each of its quadratic twists. By contrast, the curves $H_{\eta}$ do not have constant $j$-invariant. This means that Tunnell’s approach to computing the analytic rank does not apply to this family.

Let $K$ be a field of characteristic not equal to $2$, in which $-1$ is not a square; later we will restrict to $K=\mathbb{Q}$, but many of our results hold in more generality. Throughout this paper, all schemes will be defined over $K$ unless otherwise indicated, and if $X$ and $Y$ are schemes then $X\times Y:=X\times_{K}Y$.

Throughout, $\mathbb{P}^{1}$ will denote the projective line over $K$, while $\mathbb{P}^{2}$ will denote a weighted projective space over $K$, where the variables $x,y,z$ have weights $1,2,1$, respectively. We use the notation $(x:z)$ and $(x:y:z)$ to denote elements of $\mathbb{P}^{1}(K)$ and $\mathbb{P}^{2}(K)$, respectively. That is, for $(x,z)\in K^{2}\setminus\{(0,0)\}$ we have

and for $(x,y,z)\in K^{3}\setminus\{(0,0,0)\}$ we have

Let $\operatorname{GL}_{2}=\operatorname{Spec}K[a,b,c,d,(ad-bc)^{-1}]$ denote the algebraic group of $2\times 2$ invertible matrices, with identity element $I$. Given a matrix $\eta\in\operatorname{GL}_{2}(K)$, its transpose will be denoted $\eta^{t}$. Let $\operatorname{O}_{2}$ denote the orthogonal group of $2\times 2$ matrices, that is, the algebraic subgroup of ${\operatorname{GL}_{2}}$ defined by the condition that $M\in{\operatorname{GL}_{2}}(\overline{K})$ is in $\operatorname{O}_{2}(\overline{K})$ if and only if $MM^{t}=M^{t}M=I$.

Using the coordinates $\left((x:y:z),\begin{pmatrix}a&b\\ c&d\end{pmatrix}\right)$ on $\mathbb{P}^{2}\times\operatorname{GL}_{2}$, define the variety $\mathcal{H}$ by the equation

If we let $N:K^{2}\to K$ be defined by $N(u,v)=u^{2}+v^{2}$, then this can equivalently be written

This variety comes equipped with a morphism $\pi:\mathcal{H}\to\operatorname{GL}_{2}$, which equips $\mathcal{H}$ with the structure of a flat family of curves. Given $\eta\in\operatorname{GL}_{2}(K)$, ${H}_{\eta}$ is the fiber of $\pi$ over $\eta$.

The generic fiber of $\pi$ is a genus one hyperelliptic curve over the function field $K(a,b,c,d)$, with discriminant

The Jacobian variety of this curve is an elliptic curve over $K(a,b,c,d)$, which by classical invariant theory (see for example [18, 2]) has a model

We have two commuting involutions on $\mathcal{H}$ as a scheme over $\operatorname{GL}_{2}$, given by

generating a Klein four-group

acting on $\mathcal{H}$. (Note that $\sigma_{2}$ is an involution because $y$ has weight $2$, and so $(-x:y:-z)=(x:y:z)$.)

We show that the isomorphism class of $\eta\in\operatorname{GL}_{2}(K)$ is invariant on double cosets in

and use this to show that $\mathcal{H}_{\eta}$ has a $K$-point if and only if $\eta$ is in the same double coset as $\begin{pmatrix}1&r\\ 0&s\end{pmatrix}$ for some $r,s\in K$.

Given $\eta\in{\operatorname{GL}_{2}}(K)$, suppose $\eta$ is in the same double coset as an element of the form $\eta^{\prime}=\begin{pmatrix}1&r\\ 0&s\end{pmatrix}\in\operatorname{GL}_{2}(K)$. We have $(0:1:1)\in\mathcal{H}_{\eta^{\prime}}(K)$, so by Lemma 3.1, we can conclude that $\mathcal{H}_{\eta}(K)$ is nonempty. The following lemma gives us the converse result: if $\mathcal{H}_{\eta}(K)$ is nonempty then $\eta$ is in the same double-coset as a matrix of the form $\eta^{\prime}=\begin{pmatrix}1&r\\ 0&s\end{pmatrix}$.

The curve $H_{\eta}$ is singular when the discriminant (Eq. 8) vanishes. Since $ad-bc\neq 0$ for all $\eta\in\operatorname{GL}_{2}(K)$ and $K$ does not contain a square root of $-1$, this can only occur if $a=-d$ and $b=c$, or if $a=d$ and $b=-c$. One of these two conditions holds if and only if $a^{2}+b^{2}=c^{2}+d^{2}$ and $ac+bd=0$; thus the singular fibers $H_{\eta}$ are exactly those with

In this case $H_{\eta}$ reduces to the form

If $a^{2}+b^{2}=\lambda^{2}$ then $H_{\eta}$ splits into two conics, $y=\pm\lambda(z^{2}+x^{2})$. If $a^{2}+b^{2}$ is not a square in $K$, then there are no solutions in $\mathbb{P}^{2}(K)$.

If $H_{\eta}$ has a rational point and the discriminant (Eq. 8) does not vanish at $\eta$, then $H_{\eta}$ is isomorphic to its Jacobian. Using Lemma 3.2 and Eq. 9, we can conclude that $H_{\eta}$ is isomorphic to

for some $r,s\in K$; the non-vanishing of the discriminant says that $s\neq 0$ and $(r,s)\neq(0,\pm 1)$. This completes the proof of Proposition 1.3.

We now restrict our attention to $K=\mathbb{Q}$ in order to prove Theorem 1.4, which we recall for convenience.

See 1.4