# Arithmetic of arithmetic Coxeter groups

Edited by Kenneth A. Ribet, University of California, Berkeley, CA, and approved November 16, 2018 (received for review June 3, 2018)
December 26, 2018
116 (2) 442-449

## Significance

Conway’s topograph provided a combinatorial-geometric perspective on integer binary quadratic forms—quadratic functions of two variables with integer coefficients. This perspective is practical for solving equations and easily bounds the minima of binary quadratic forms. It appears that Conway’s topograph is just the first in a series of applications of arithmetic Coxeter groups to arithmetic. Four other applications are described in this article, and dozens more may be possible.

## Abstract

In the 1990s, J. H. Conway published a combinatorial-geometric method for analyzing integer-valued binary quadratic forms (BQFs). Using a visualization he named the “topograph,” Conway revisited the reduction of BQFs and the solution of quadratic Diophantine equations such as Pell’s equation. It appears that the crux of his method is the coincidence between the arithmetic group $No alternative text available$ and the Coxeter group of type $No alternative text available$. There are many arithmetic Coxeter groups, and each may have unforeseen applications to arithmetic. We introduce Conway’s topograph and generalizations to other arithmetic Coxeter groups. This includes a study of “arithmetic flags” and variants of binary quadratic forms.
Binary quadratic forms (BQFs) are functions $No alternative text available$ of the form $No alternative text available$, for some integers $No alternative text available$. The discriminant of such a form is the integer $No alternative text available$. In ref. 1, J. H. Conway visualized the values of a BQF through an invention he called the topograph.

## 1. Conway’s Topograph

### The Geometry of the Topograph.

The topograph is an arrangement of points, edges, and faces, as described below.
Faces correspond to primitive lax vectors: coprime ordered pairs $No alternative text available$, modulo the relation $No alternative text available$. Such a vector is written $No alternative text available$.
Edges correspond to lax bases: unordered pairs $No alternative text available$ of primitive lax vectors which form a $No alternative text available$ basis of $No alternative text available$. (Clearly this is independent of sign choices.)
Points correspond to lax superbases: unordered triples $No alternative text available$, any two of which form a lax basis.
Incidence among points, edges, and faces is defined by containment. A maximal arithmetic flag in this context refers to a point contained in an edge contained in a face. The geometry is displayed in Fig. 1; the points and edges form a ternary regular tree, and the faces are $No alternative text available$-gons. The group $No alternative text available$ acts simply–transitively on maximal arithmetic flags.
Fig. 1.
On the other hand, the geometry of Fig. 1 also arises as the geometry of flags in the Coxeter group of type $No alternative text available$. This is the Coxeter group with a diagram . The group $No alternative text available$ encoded by such a diagram is generated by elements $No alternative text available$ corresponding to the nodes, modulo the relations $No alternative text available$ (for i = 0, 1, 2), $No alternative text available$, $No alternative text available$. If $No alternative text available$ is a subset of nodes, write $No alternative text available$ for the subgroup generated by $No alternative text available$; it is called a parabolic subgroup. The flags of type $No alternative text available$ are the cosets$No alternative text available$ The Coxeter group $No alternative text available$ acts simply–transitively on the maximal flags; i.e., the cosets$No alternative text available$.
The geometric coincidence reflects the fact that $No alternative text available$ is isomorphic to the Coxeter group $No alternative text available$ of type $No alternative text available$, a classical result known to Poincaré and Klein. But Conway’s study of lax vectors, bases, and superbases goes further, giving an arithmetic interpretation of the flags for the Coxeter group. This raises the natural question: Given a coincidence between an arithmetic group and a Coxeter group, is there an arithmetic interpretation of the flags in the Coxeter group?

If one draws the values $No alternative text available$ on the faces labeled by the primitive lax vectors $No alternative text available$, one obtains Conway’s topograph of $No alternative text available$. Figs. 2 and 3 display examples. If $No alternative text available$ appear on the topograph of $No alternative text available$, in a local arrangement we call a cell, then Conway observes that the integers $No alternative text available$ form an arithmetic progression.
Fig. 2.
Fig. 3.
The discriminant of $No alternative text available$ can be seen locally in the topograph, at every cell, by the formula $No alternative text available$.
A consequence of the arithmetic progression property is Conway’s climbing principle; if all values in a cell are positive, place arrows along the edges in the directions of increasing arithmetic progressions. Then every arrow propagates into two arrows; the resulting flow along the edges can have a source, but never a sink. This implies the existence and uniqueness of a well for positive-definite forms: a triad or cell which is the source for the flow. The well gives the unique Gauss-reduced form $No alternative text available$ in the $No alternative text available$-equivalence class of $No alternative text available$. More precisely, every well contains a triple $No alternative text available$ of positive integers satisfying $No alternative text available$, with strict inequality at triad wells and equality at cell wells (Fig. 2). Depending on the orientation of $No alternative text available$ at the well, the Gauss-reduced form is given below; in the ambiguously oriented case with $No alternative text available$, $No alternative text available$. If $No alternative text available$, both orientations occur in a cell well, and $No alternative text available$.
When $No alternative text available$ is a nondegenerate indefinite form, Conway defines the river of $No alternative text available$ to be the set of edges which separate a positive value from a negative value in the topograph of $No alternative text available$. Since all values on the topograph of $No alternative text available$ must be positive or negative, the river cannot branch or terminate. The climbing principle implies uniqueness of the river. Thus, the river is a set of edges composing a single endless line. Bounding the values adjacent to the river implies periodicity of values adjacent to the river and thus the infinitude of solutions to Pell’s equation. This is described in detail in ref. 1.
Riverbends—cells with a river as drawn below—correspond to Gauss’s reduced forms in the equivalence class of Q.

### Proposition 1.

Let $No alternative text available$ be the cell values at a riverbend, with $No alternative text available$, $No alternative text available$. Then a Gauss-reduced form in the $No alternative text available$-equivalence class of $No alternative text available$ is given by
$No alternative text available$

#### Proof:

Let $No alternative text available$ be the common difference at the cell. Then $No alternative text available$ is $No alternative text available$ equivalent to $No alternative text available$ and we must prove Gauss’s reduction conditions (ref. 2, article 183):
$No alternative text available$
Note that $No alternative text available$ since $No alternative text available$. Since $No alternative text available$, and $No alternative text available$ and $No alternative text available$ have opposite sign, we find that $No alternative text available$. Gauss’s first reduction condition $No alternative text available$ follows.
Since $No alternative text available$, and $No alternative text available$ and $No alternative text available$ have opposite sign, we find that $No alternative text available$. Since $No alternative text available$ and $No alternative text available$ have opposite sign, this implies $No alternative text available$. Multiplying by $No alternative text available$ yields $No alternative text available$. Replace $No alternative text available$ by $No alternative text available$ to obtain $No alternative text available$. Rearranging yields
$No alternative text available$
Hence $No alternative text available$, and so $No alternative text available$. This verifies Gauss’s second reduction condition. □
The existence of riverbends gives a classical bound, by an argument we learned from Gordan Savin.

### Theorem 2.

The minimum nonzero absolute value $No alternative text available$ of a nondegenerate indefinite BQF $No alternative text available$ satisfies $No alternative text available$.

#### Proof:

At a riverbend, one finds $No alternative text available$, the sum of five positive integers. It follows that one of $No alternative text available$ must be bounded by $No alternative text available$. Among $No alternative text available$, one must be bounded by $No alternative text available$.□
These are some highlights and applications of Conway’s topograph. In the next sections, we describe generalizations.

## 2. Gaussian and Eisenstein Analogues

Let $No alternative text available$ denote the Gaussian integers: $No alternative text available$. Let $No alternative text available$ denote the Eisenstein integers: $No alternative text available$.

### Arithmetic Flags and Honeycombs.

One may generalize Conway’s vectors, bases, and superbases to arithmetic structures in $No alternative text available$ and $No alternative text available$. Guiding this are embeddings of $No alternative text available$ and $No alternative text available$ into hyperbolic Coxeter groups. In ref. 3, sections 1.I, 1.II, 3, and 6, Bianchi describes generators for $No alternative text available$ and $No alternative text available$ and fundamental polyhedra for their action on hyperbolic 3-space. Using reflections in the faces of these polyhedra, one may write explicit presentations of these groups; Fricke and Klein carry this out for $No alternative text available$ in ref. 4, section I.8, where one finds a connection to the (later-named) Coxeter group of type $No alternative text available$. Schulte and Weiss give a detailed treatment, proving the following in ref. 5, theorems 7.1 and 9.1.

### Theorem 3.

$No alternative text available$ is isomorphic to an index-two subgroup of $No alternative text available$. $No alternative text available$ is isomorphic to an index-two subgroup of $No alternative text available$.
Here $No alternative text available$ denotes the even subgroup of the Coxeter group of type $No alternative text available$. As the Coxeter groups of types $No alternative text available$ and $No alternative text available$ are commensurable to $No alternative text available$ and $No alternative text available$, respectively, we expect an arithmetic interpretation of the Coxeter geometries. Such an arithmetic incidence geometry is described below.
Cells correspond to primitive lax vectors: coprime ordered pairs $No alternative text available$ (respectively $No alternative text available$), modulo the relation $No alternative text available$ for all $No alternative text available$ (resp., $No alternative text available$).
Faces correspond to lax bases: unordered pairs $No alternative text available$ of primitive lax vectors which form a $No alternative text available$ basis of $No alternative text available$ (respectively $No alternative text available$ basis of $No alternative text available$).
Edges correspond to lax superbases: unordered triples $No alternative text available$, any two of which form a lax basis.
Points of the Eisenstein topograph* correspond to lax tetrabases: unordered quadruples $No alternative text available$, any three of which form a lax superbasis.
Points of the Gaussian topograph correspond to lax cubases: sets of three two-element sets $No alternative text available$, such that all eight choices of $No alternative text available$ give a lax superbasis $No alternative text available$.
Incidence is given by the obvious containments described above. We call this incidence geometry the topograph for $No alternative text available$ or $No alternative text available$, and it is equipped with an action of $No alternative text available$ and $No alternative text available$, respectively. The terms tetrabasis and cubasis reflect the residual geometry around a point (Fig. 4). Both geometries produce regular hyperbolic honeycombs (ref. 6, chap. IV); the points, edges, and faces around each cell form square or hexagonal planar tilings in the Gaussian or the Eisenstein case, respectively.
Fig. 4.
The Gaussian and Eisenstein topographs are described by Bestvina and Savin in ref. 7, sections 7 and 8. Both topographs, and the following link to Coxeter geometries, are given in the PhD thesis of the second author (8).

### Theorem 4.

The topographs for $No alternative text available$ and $No alternative text available$ are equivariantly isomorphic to the Coxeter geometries of types (3,3,6) and (3,4,4), respectively.
By equivariance, we mean that the isomorphism intertwines the natural actions of $No alternative text available$ and $No alternative text available$ on one hand with the actions of the Coxeter groups on the other, via the inclusion described in Theorem 3.

### Binary Hermitian Forms.

An integer-valued binary Hermitian form (BHF), over $No alternative text available$ or $No alternative text available$, is a function $No alternative text available$ or $No alternative text available$, of the form
$No alternative text available$
Here we assume $No alternative text available$, and $No alternative text available$ or $No alternative text available$ (the inverse different of $No alternative text available$ or $No alternative text available$, respectively). The discriminant of $No alternative text available$ is the integer defined by
$No alternative text available$
Fricke and Klein discuss reduction theory of Hermitian forms over $No alternative text available$, using the geometry of $No alternative text available$, in ref. 4, section III.1,1–8. The topographs give a new approach, pursued by Bestvina and Savin (7).
Let $No alternative text available$ be a BHF over $No alternative text available$ or $No alternative text available$. Recalling that cells of the topographs correspond to primitive lax vectors, we define the topograph of $No alternative text available$ to be the result of placing the value $No alternative text available$ at the topograph cell marked by the primitive lax vector $No alternative text available$. The most interesting case occurs when $No alternative text available$ is nondegenerate indefinite (taking positive and negative values, but never zero on a nonzero vector input). In this case, Conway’s river is replaced by the ocean—the set of faces separating a cell with positive value from one with negative value. Bestvina and Savin prove (ref. 7, theorems 5.3 and 6.1) that this ocean is topologically an open disk, locally CAT(0) as a metric space, and the unitary group $No alternative text available$ acts cocompactly on the ocean.
Reduced indefinite BQFs correspond to riverbends in Conway’s topograph. In a similar way, one finds reduced indefinite BHFs at the points of the ocean of negative curvature, i.e., where more than four ocean squares (for $No alternative text available$) or more than three ocean hexagons (for $No alternative text available$) meet at a point. From this, Bestvina and Savin (ref. 7, theorem 8.7) recover the optimal bound on the minima of nondegenerate indefinite BHFs over $No alternative text available$. The bound for $No alternative text available$ can be obtained by the same method.

### Theorem 5.

Let $No alternative text available$ be a nondegenerate indefinite BHF. Then the minimum nonzero absolute value $No alternative text available$ satisfies $No alternative text available$.

#### Proof:

The Eisenstein case is proved in ref. 7, so we prove the Gaussian case. Consider a vertex at which the ocean of $No alternative text available$ has negative curvature; such a point exists by ref. 7, corollary 6.2. The residue of the topograph at this vertex is a cube, whose faces are labeled by the values of $No alternative text available$. The intersection of the ocean with this cube forms a simple closed path on the edges, separating positive-valued faces from negative ones.
Form I corresponds to a Euclidean vertex; forms II, III, and IV correspond to ocean vertices of negative curvature. Label the values of $No alternative text available$ on the cube by $No alternative text available$, with $No alternative text available$ opposite $No alternative text available$, $No alternative text available$ opposite $No alternative text available$, and $No alternative text available$ opposite $No alternative text available$. In ref. 7, proposition 7.1, Bestvina and Savin demonstrate that $No alternative text available$. This excludes form IV, since the sum of two positive numbers cannot equal the sum of two negative numbers. Ref. 7, section 7 also gives a formula for the discriminant,
$No alternative text available$
In forms II and III, we may place $No alternative text available$ so that $No alternative text available$ and $No alternative text available$ have opposite signs, and $No alternative text available$ and $No alternative text available$ have opposite signs. Expressing $No alternative text available$ as $No alternative text available$ yields
$No alternative text available$
As the right side is a sum of positive terms, we find
$No alternative text available$

### Dilinear Algebra.

Here we introduce a family of “dilinear groups” associated to certain real quadratic rings. Let $No alternative text available$ be a square-free positive integer, and let $No alternative text available$ be the quadratic ring of discriminant $No alternative text available$. We define the dilinear group $No alternative text available$ to be the group of all matrices $No alternative text available$ such that
$No alternative text available$
Let $No alternative text available$ denote its subgroup consisting of matrices with $No alternative text available$ and $No alternative text available$. While the dilinear groups seem a bit mysterious at first, $No alternative text available$ is $No alternative text available$ conjugate to a congruence subgroup of $No alternative text available$: If $No alternative text available$, then
$No alternative text available$
We thank the referee for this insight.
A divector over $No alternative text available$ will mean a vector in $No alternative text available$ of red or blue type. Red divectors are those of the form $No alternative text available$ for some $No alternative text available$. Blue divectors are those of the form $No alternative text available$ for some $No alternative text available$. A red divector $No alternative text available$ is called primitive if $No alternative text available$. A blue divector $No alternative text available$ is called primitive if $No alternative text available$.

### Theorem 6.

The dilinear group $No alternative text available$ acts transitively on the set of primitive divectors, and $No alternative text available$ acts transitively on the set of primitive red (or blue) divectors.

#### Proof:

A matrix in $No alternative text available$ has a determinant in $No alternative text available$. It follows that a matrix in $No alternative text available$ sends primitive divectors to primitive divectors. For transitivity, consider a primitive red divector $No alternative text available$. Since $No alternative text available$, there exist $No alternative text available$ such that $No alternative text available$. Observe that
$No alternative text available$
Hence $No alternative text available$ acts transitively on the set of primitive red vectors. Since the matrix $No alternative text available$ swaps primitive red and blue divectors, the result follows. □
Define $No alternative text available$. When $No alternative text available$ or $No alternative text available$, Johnson and Weiss (ref. 9, section 4) present $No alternative text available$ by generators and relations, proving the following.

### Theorem 7.

If $No alternative text available$ or $No alternative text available$, then $No alternative text available$ is isomorphic to the Coxeter group of type $No alternative text available$.
Explicitly, Johnson and Weiss (8) note that $No alternative text available$ is generated by the triple of matrices (modulo $No alternative text available$),
$No alternative text available$
which satisfy the Coxeter relations $No alternative text available$, $No alternative text available$, $No alternative text available$. From their result, we were led to “dilinear” arithmetic interpretations of the geometries of types $No alternative text available$ and $No alternative text available$.

### Arithmetic Flags.

The “dilinear” variant of Conways’s topograph is as follows. Assume $No alternative text available$ or $No alternative text available$.
Faces correspond to primitive lax divectors over $No alternative text available$, i.e., primitive divectors modulo $No alternative text available$.
Edges correspond to lax dibases: unordered pairs of lax divectors generating $No alternative text available$ as an $No alternative text available$ module. This implies that the divectors are primitive, have opposite color, and form the rows of a matrix in $No alternative text available$.
Points correspond to lax pinwheels: cyclically ordered $No alternative text available$-tuples of lax divectors such that any adjacent pair forms a lax dibasis (and hence has opposite color).

### Theorem 8.

The geometry of primitive lax divectors, lax dibases, and pinwheels for $No alternative text available$ is equivariantly isomorphic to the Coxeter geometry of type $No alternative text available$ (Fig. 5).
Fig. 5.

Let us return to the general case of a square-free positive integer $No alternative text available$ again. A binary quadratic diform (BQD) is a function of the form
$No alternative text available$
We restrict $No alternative text available$ to be a divector in $No alternative text available$, so the values of $No alternative text available$ are integers. We define the discriminant of $No alternative text available$ by $No alternative text available$.
Restricting $No alternative text available$ to red and blue divectors yields a pair $No alternative text available$ of BQFs over $No alternative text available$ of discriminant $No alternative text available$; explicitly,
$No alternative text available$
$No alternative text available$
This pair of BQFs can be related through an accessory form we call $No alternative text available$. Namely, whenever $No alternative text available$, define
$No alternative text available$
Write $No alternative text available$ for the group of $No alternative text available$-equivalence classes of primitive BQFs of discriminant $No alternative text available$, following Bhargava (ref. 10, theorem 1). If $No alternative text available$ is a BQF of discriminant $No alternative text available$, write $No alternative text available$ for its $No alternative text available$-equivalence class. Since $No alternative text available$ is an ambiguous form (its first coefficient divides its middle coefficient), its class in $No alternative text available$ satisfies $No alternative text available$. The class of $No alternative text available$ has another characterization below.

### Lemma 9.

If $No alternative text available$ is a BQF of discriminant $No alternative text available$ that represents $No alternative text available$, and $No alternative text available$, then $No alternative text available$.

#### Proof:

The square-freeness of $No alternative text available$ is used repeatedly in what follows. If $No alternative text available$ represents $No alternative text available$, then $No alternative text available$ is $No alternative text available$ equivalent to a BQF of the form $No alternative text available$. Since $No alternative text available$, we find that $No alternative text available$. It follows that $No alternative text available$ is ambiguous and equivalent to $No alternative text available$ for some $No alternative text available$ and $No alternative text available$.
If $No alternative text available$, then $No alternative text available$ and $No alternative text available$ mod 4. In this case $No alternative text available$. If $No alternative text available$, then $No alternative text available$ and $No alternative text available$ mod 4. In this case $No alternative text available$.□
Thus, with $No alternative text available$ fixed and $No alternative text available$, we find that $No alternative text available$ is the unique class in $No alternative text available$ which represents $No alternative text available$; it happens to be a 2-torsion element in the class group. The following relates $No alternative text available$ and $No alternative text available$ via $No alternative text available$.

### Theorem 10.

Suppose that $No alternative text available$, $No alternative text available$, and $No alternative text available$ are pairwise coprime. In $No alternative text available$, one has $No alternative text available$. Conversely, if $No alternative text available$ are primitive BQFs of discriminant $No alternative text available$, and $No alternative text available$, and $No alternative text available$, there exists a BQD $No alternative text available$ such that $No alternative text available$ and $No alternative text available$.

#### Proof:

Consider the cube of integers below.
Let $No alternative text available$ be the partition of this cube into a pair of two-by-two matrices, in a front–back, left–right, and top–bottom fashion according to whether $No alternative text available$, respectively, as in ref. 10, section 2.1. From these matrices, Bhargava constructs a triple of binary quadratic forms $No alternative text available$:
$No alternative text available$
By ref. 9, theorem 1, we have $No alternative text available$ in $No alternative text available$. Observe that $No alternative text available$ is precisely $No alternative text available$. Next, observe that $No alternative text available$ is related to $No alternative text available$ by switching $No alternative text available$ and $No alternative text available$; it follows that $No alternative text available$. By Lemma 9, $No alternative text available$. Since $No alternative text available$, we have
$No alternative text available$
For the converse, suppose that $No alternative text available$ and $No alternative text available$ are primitive BQFs of discriminant $No alternative text available$, $No alternative text available$, and $No alternative text available$. Write $No alternative text available$, so $No alternative text available$. If $No alternative text available$, then $No alternative text available$, and $No alternative text available$ for the diform
$No alternative text available$
If $No alternative text available$ does not divide $No alternative text available$, then there exists an integer $No alternative text available$ satisfying the congruence $No alternative text available$. One may check this by working one prime divisor of $No alternative text available$ at a time; the quadratic formula applies for odd prime divisors. Modulo two, $No alternative text available$ implies that $No alternative text available$ is even and the congruence has a solution.
Hence $No alternative text available$ mod $No alternative text available$. Since $No alternative text available$ represents a multiple of $No alternative text available$, $No alternative text available$ is equivalent to a form $No alternative text available$. The fact that $No alternative text available$ divides the discriminant implies $No alternative text available$ for some $No alternative text available$.
Thus, whether $No alternative text available$ divides $No alternative text available$ or not, $No alternative text available$ for some diform $No alternative text available$. Since $No alternative text available$, and $No alternative text available$, we find that $No alternative text available$. □
Let $No alternative text available$ be the subgroup of $No alternative text available$ consisting of matrices of determinant one. We say that two diforms $No alternative text available$ are $No alternative text available$ equivalent if there exists $No alternative text available$ satisfying $No alternative text available$ for all divectors $No alternative text available$. We write $No alternative text available$ when the diforms $No alternative text available$ and $No alternative text available$ are $No alternative text available$ equivalent. One may check directly that $No alternative text available$ implies $No alternative text available$. From this, we may reframe Theorem 10 in terms of equivalence classes.

### Corollary 11.

Assume $No alternative text available$. The map $No alternative text available$ yields a surjective function from
the set of $No alternative text available$-equivalence classes of binary quadratic diforms of discriminant $No alternative text available$ to…
the set of ordered pairs $No alternative text available$ in $No alternative text available$ satisfying $No alternative text available$.
It seems interesting to determine the fibers of this map. As we will see, for $No alternative text available$ and $No alternative text available$, the question is how two of Conway’s topographs can be interlaced in a single topograph of a diform.

### Dilinear Topographs.

Here we return to the assumption that $No alternative text available$ or $No alternative text available$. The topograph of a binary quadratic diform $No alternative text available$ is obtained by replacing each primitive lax divector by the corresponding value of $No alternative text available$. Every value on the topograph of $No alternative text available$ thus appears on the topograph of $No alternative text available$ or of $No alternative text available$. In this way, values from two of Conway’s topographs interlace in the topograph of a binary quadratic diform. More precisely, we have the following.

### Proposition 12.

If $No alternative text available$ appears on the topograph of $No alternative text available$, then (i) $No alternative text available$ appears on the topograph of $No alternative text available$ or (ii) $No alternative text available$ and $No alternative text available$ appears on the topograph of both $No alternative text available$ and $No alternative text available$. Similarly, if $No alternative text available$ appears on the topograph of $No alternative text available$, then (i) $No alternative text available$ appears on the topograph of $No alternative text available$ or (ii) $No alternative text available$ and $No alternative text available$ appears on the topographs of both $No alternative text available$ and $No alternative text available$.

#### Proof:

Suppose $No alternative text available$ occurs on the topograph of $No alternative text available$. Thus, $No alternative text available$ for some coprime $No alternative text available$. If $No alternative text available$, then $No alternative text available$ is a primitive divector, and $No alternative text available$ appears on the topograph of $No alternative text available$.
If $No alternative text available$, then $No alternative text available$ and $No alternative text available$. We compute $No alternative text available$. Hence $No alternative text available$ appears on the topograph of both $No alternative text available$ and $No alternative text available$. □

### Corollary 13.

Let $No alternative text available$ and $No alternative text available$ be the minimum nonzero absolute values of $No alternative text available$ and $No alternative text available$. Then $No alternative text available$ is the minimum nonzero absolute value of $No alternative text available$.
The discriminant of a binary quadratic diform is locally visible in its topograph, according to the formulas below:
$No alternative text available$
[1]
Polarization for the quadratic form $No alternative text available$ implies the following.

### Theorem 14.

At every cell in the topograph of $No alternative text available$, as in Fig. 6, one finds arithmetic progressions as below.
i)
$No alternative text available$: The triples $No alternative text available$ and $No alternative text available$ are arithmetic progressions of the same step size.
ii)
$No alternative text available$: The triples $No alternative text available$ and $No alternative text available$ are arithmetic progressions of the same step size $No alternative text available$ and the triples $No alternative text available$ and $No alternative text available$ are arithmetic progressions of the same step size $No alternative text available$.
Fig. 6.

#### Proof:

In both cases $No alternative text available$, the integers $No alternative text available$ of a cell in Fig. 6 arise as values of $No alternative text available$ of the form
$No alternative text available$
By the polarization identity for quadratic forms, the sequence
$No alternative text available$
is an arithmetic progression with step size $No alternative text available$. Here $No alternative text available$ is the bilinear form associated to the quadratic form $No alternative text available$. Similarly, the integers $No alternative text available$ arise as values of $No alternative text available$:
$No alternative text available$
The sequence
$No alternative text available$
is an arithmetic progression with step size $No alternative text available$. Note that $No alternative text available$, $No alternative text available$, and $No alternative text available$. Hence $No alternative text available$ and $No alternative text available$ are arithmetic progressions of the same step size.
When $No alternative text available$, the values $No alternative text available$ arise as values of $No alternative text available$: $No alternative text available$ and $No alternative text available$. The polarization identity, applied again, shows that $No alternative text available$ and $No alternative text available$ are arithmetic progressions of the same step size $No alternative text available$. □
Similar computations give linear relations among the values around any vertex in the topograph.

### Proposition 15.

(Refer to the vertex diagrams above.) When $No alternative text available$, $No alternative text available$. When $No alternative text available$, $No alternative text available$ and also $No alternative text available$.
We draw an arrow on each edge to represent the direction of increasing progressions or a circle if all progressions are constant. Fig. 7 displays some examples. The climbing principle is the same as Conway’s: Arrows always propagate when one looks at a cell of positive values. By the same argument as that of Conway, one obtains unique wells—a reduction theory for positive-definite diforms.
Fig. 7.

### Proposition 16.

Let $No alternative text available$ be a positive-definite BQD over $No alternative text available$, with $No alternative text available$ or $No alternative text available$. Then the topograph of $No alternative text available$ exhibits a unique well—either a single vertex or an edge (double well) from which all arrows emanate.
The river of a BQD is the set of segments separating positive values from negative ones in its topograph. The most interesting forms, just as for BQFs, are the nondegenerate indefinite forms.

### Proposition 17.

If $No alternative text available$ is a nondegenerate indefinite diform, then its topograph contains a single endless nonbranching river.

#### Proof:

The existence and uniqueness of a river follows from the same argument as for Conway’s case. Namely, as one travels from a positive face to a negative face, one must at some point cross a river from positive to negative. This gives existence. The climbing principle (propagation of growth arrows) demonstrates that as one travels away from a river, one cannot hit another river, giving uniqueness. The crux of Proposition 17 is that rivers cannot branch.
For if a river branched, the faces around the branch point would alternate signs as they cross each river segment. Hence the rivers may branch only with even degree at a vertex. The possibilities, up to symmetry, are displayed below.
We apply Proposition 15 repeatedly. When $No alternative text available$, the identity $No alternative text available$ yields a contradiction if the signs of $No alternative text available$ and $No alternative text available$ are equal and opposite to the signs of $No alternative text available$ and $No alternative text available$. Similarly, when $No alternative text available$, the identity $No alternative text available$ yields a contradiction if the signs of $No alternative text available$ are equal and opposite to the signs of $No alternative text available$. Branch-forms I and II are excluded.
It also happens that, when $No alternative text available$, then $No alternative text available$. Thus, we find a contradiction if the signs of $No alternative text available$ are equal and opposite to the signs of $No alternative text available$. This excludes form III. We also find a contradiction if the signs of $No alternative text available$ are equal and opposite to the signs of $No alternative text available$. This excludes form IV. Hence the river cannot branch. □
As we have an endless nonbranching river, analysis of riverbends gives a minimum-value bound for diforms.

### Theorem 18.

Let $No alternative text available$ be a nondegenerate indefinite BQD, and let $No alternative text available$ denote its minimum nonzero absolute value.
i)
$No alternative text available$: If Q is not $No alternative text available$ equivalent to a multiple of $No alternative text available$, then $No alternative text available$.
ii)
$No alternative text available$: If Q is not $No alternative text available$ equivalent to a multiple of $No alternative text available$, then $No alternative text available$.

#### Proof:

The entire river cannot be adjacent to a single region, because its values opposite such a region would form a biinfinite quadratic sequence with positive sign and negative acceleration or negative sign and positive acceleration. Hence the river must “bend.” If one finds riverbends as in Figs. 8 and 9, Eq. 1 gives the stated minimum value bound or better (as derived in Figs. 8 and 9). If no such riverbends of those shapes occur, then the river must maintain one of the three shapes of Fig. 10 throughout its entire length.
Fig. 8.
Fig. 9.
Fig. 10.
The isometry group of such a homogeneous river includes a translation along the river. Replacing $No alternative text available$ by a $No alternative text available$-equivalent form if necessary, we may place this river through the segment separating $No alternative text available$ and $No alternative text available$. Translation along the homogeneous rivers is then given by the matrices
$No alternative text available$
in the three cases shown in Fig. 10. Periodicity of the river implies that $No alternative text available$, $No alternative text available$, or $No alternative text available$ is an isometry of $No alternative text available$ for some $No alternative text available$.
The eigenvectors of $No alternative text available$ and $No alternative text available$ are $No alternative text available$ and (1, −1). If $No alternative text available$ and $No alternative text available$ denote their eigenvalues, then
$No alternative text available$
But a quick computation demonstrates that $No alternative text available$ and $No alternative text available$. Hence $No alternative text available$ in the two straight-river cases. Writing the diform as $No alternative text available$, this implies $No alternative text available$. Hence $No alternative text available$ and $No alternative text available$. We have proved that an endless straight river occurs only if $No alternative text available$ is equivalent to a multiple of $No alternative text available$ (when $No alternative text available$ or $No alternative text available$).
It remains to study the third shape of the homogeneous river, on which $No alternative text available$ acts by translation. The eigenvectors of $No alternative text available$ are $No alternative text available$, with eigenvalues $No alternative text available$, respectively. Hence, if $No alternative text available$ is an isometry of $No alternative text available$ for some $No alternative text available$, then $No alternative text available$. In this case, $No alternative text available$. Hence $No alternative text available$ and $No alternative text available$. We have proved that an endless homogeneous river of the third form occurs if and only if $No alternative text available$ is equivalent to a multiple of $No alternative text available$ (displayed in Fig. 7). The discriminant of the diform $No alternative text available$ is 24, while its minimum absolute value is $No alternative text available$. The estimate $No alternative text available$ can be directly checked in this case, finishing the proof. □
The discriminant of the diform $No alternative text available$ is $No alternative text available$ and its minimal value is $No alternative text available$. Thus, when $No alternative text available$, the estimate $No alternative text available$ is violated; when $No alternative text available$, the estimate $No alternative text available$ is violated. Hence the exceptional diforms $No alternative text available$ cannot be removed from Theorem 18.

### Corollary 19.

Suppose that $No alternative text available$ and $No alternative text available$ are nondegenerate indefinite BQFs of discriminant $No alternative text available$, with $No alternative text available$ and $No alternative text available$. Then
i)
$No alternative text available$: If $No alternative text available$ and $No alternative text available$ are not equivalent to a multiple of $No alternative text available$, then $No alternative text available$
ii)
$No alternative text available$: If $No alternative text available$ and $No alternative text available$ are not equivalent to a multiple of $No alternative text available$, then $No alternative text available$

#### Proof:

This follows directly from Theorems 18 and 10, except that $No alternative text available$ has been replaced by $No alternative text available$. This replacement is possible, due to a gap in the Markoff spectrum between $No alternative text available$ and $No alternative text available$; see ref. 11, section 1, proof of theorem 3.3.□

## 4. Conclusion

In each of the discussed examples, there is a coincidence between a Coxeter group and an arithmetic group. For Conway’s topograph, it is the coincidence between the Coxeter group of type $No alternative text available$ and the arithmetic group $No alternative text available$. The dilinear groups, of Coxeter types $No alternative text available$ for $No alternative text available$, are arithmetic subgroups of $No alternative text available$, the projective unitary similitude group for a Hermitian form relative to $No alternative text available$.
When such a coincidence occurs, the Coxeter group is arithmetic, and the following two questions are natural.
i)
Is there an arithmetic interpretation for the flags in the Coxeter group?
ii)
Does the Coxeter geometry give a new reduction theory for a class of quadratic (or Hermitian) forms?
The first question is reminiscent of the classical theory of flag varieties. When $No alternative text available$ is a simple simply connected linear algebraic group over a field $No alternative text available$, one can often identify a “standard” representation of $No alternative text available$ on a $No alternative text available$-vector space $No alternative text available$. Every $No alternative text available$-parabolic subgroup of $No alternative text available$ is the stabilizer of some sort of $No alternative text available$ flag in $No alternative text available$. If $No alternative text available$ is a symplectic or spin or unitary group, these are the isotropic flags in the standard representation. In type $No alternative text available$, these are the nil flags in the split octonions. In an 11-part series of papers (Beziehungen der $No alternative text available$ und $No alternative text available$ zur Oktavenebene I–XI, published 1954–1963, refs. 1216), Freudenthal studied the “metasymplectic” geometry which describes flags in representations of exceptional groups.
Now it appears that arithmetic Coxeter groups provide a parallel industry, examining their representations on various modules over Euclidean domains. Arithmetic flags are generalized bases of these modules. The geometry of arithmetic flag varieties seems (so far) to be the combinatorial geometry of Coxeter groups. We do not yet see algebraic geometry in the picture, as one finds in flag varieties $No alternative text available$.
The applications to arithmetic (the arithmetic of arithmetic Coxeter groups) include Conway’s approach to binary quadratic forms and new generalizations. The reduction theory for quadratic and Hermitian forms is a classical subject, sometimes tedious in its algebra—the Coxeter geometry and Conway’s theory of wells and rivers give an intuitive approach. Beyond reframing old results, it seems unlikely that one would find the reduction theory of our “diforms” (or suitable pairs of binary quadratic forms) without considering the Coxeter group. In this way, arithmetic Coxeter groups offer applications to number theory.
This paper has discussed five arithmetic Coxeter groups, of types $No alternative text available$, $No alternative text available$, $No alternative text available$, $No alternative text available$, and $No alternative text available$. If this is a game of coincidences, when might it end? In ref. 17, Belolipetsky surveys the arithmetic hyperbolic Coxeter groups; following his treatment, we review the classification of such Coxeter groups.
The groups we have studied are simplicial hyperbolic arithmetic Coxeter groups. In ref. 18, Vinberg proves there are 64 such groups in dimension at least 3. These fall into 14 commensurability classes by ref. 19, as shown in Table 1. It would not be surprising if each one offered a notion of arithmetic flags (e.g., superbases, etc.) and quadratic/Hermitian forms. For example, the Coxeter group of type $No alternative text available$ is arithmetic, commensurable with $No alternative text available$ where $No alternative text available$ is the Hurwitz order in the quaternion algebra $No alternative text available$. Arithmetic flags in this case can be interpreted as lax vectors, bases, superbases, 3-simplex bases, 4-simplex bases, and 5-orthoplex bases, in the $No alternative text available$-module $No alternative text available$.
Table 1.
Table 1 displays only groups of dimension at least 3. In dimension 2, we find Conway’s topograph and its dilinear variants. One might also consider arithmetic hyperbolic triangle groups, classified by Takeuchi in refs. 20 and 21. Up to commensurability, there are 19 of these, each associated to a quaternion algebra over a totally real field. Vertices, edges, and triangles in the resulting hyperbolic tilings surely correspond to arithmetic objects—What are they?
If one wishes to depart from the simplicial groups, there are nonsimplicial arithmetic hyperbolic Coxeter groups. In the results of Vinberg (22), all examples occur in dimension at most 30; there are finitely many up to commensurability. One may be able to explore the arithmetic of arithmetic Coxeter groups for a long time—what is currently missing is a general theory of arithmetic flags and forms to make predictions in a less ad hoc manner.
Departing from the setting of Coxeter groups may also be appealing, especially in low dimension. For example, the Coxeter geometry makes the reduction theory of BHFs particularly appealing over $No alternative text available$ and $No alternative text available$. But Bestvina and Savin (7) are able to work over other quadratic imaginary rings although the geometry lacks homogeneity. One might study diforms over other real quadratic rings, in the same way. More arithmetic may be found in “thin” rather than arithmetic groups, e.g., in the work of Stange (23) on Apollonian circle packings. Still, Coxeter groups seem an appropriate starting place, where arithmetic applications are low-hanging fruit.

## Acknowledgments

The authors thank the anonymous referee for comments and insights. M.H.W. is supported by the Simons Foundation Collaboration Grant 426453.

## References

1
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3
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4
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HSM Coxeter Regular Polytopes (Dover Publications, Inc., 3rd Ed, New York, 1973).
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M Bestvina, G Savin, Geometry of integral binary Hermitian forms. J Algebra 360, 1–20 (2012).
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CD Shelley, An arithmetic construction of the Gaussian and Eisenstein topographs. PhD thesis (University of California, Santa Cruz, CA). (2013).
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H Freudenthal, Beziehungen der $No alternative text available$ und $No alternative text available$ zur Oktavenebene [Relationship of $No alternative text available$ and $No alternative text available$ with the Cayley plane], I–II. Nederl Akad Wetensch Proc Ser A 57, 363–368 (1954).
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H Freudenthal, Beziehungen der $No alternative text available$ und $No alternative text available$ zur Oktavenebene [Relationship of $No alternative text available$ and $No alternative text available$ with the Cayley plane], III. Nederl Akad Wetensch Proc Ser A 58, 151–157 (1955).
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H Freudenthal, Beziehungen der $No alternative text available$ und $No alternative text available$ zur Oktavenebene [Relationship of $No alternative text available$ and $No alternative text available$ with the Cayley plane], IV. Nederl Akad Wetensch Proc Ser A 58, 277–285 (1955).
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H Freudenthal, Beziehungen der $No alternative text available$ und $No alternative text available$ zur Oktavenebene [Relationship of $No alternative text available$ and $No alternative text available$ with the Cayley plane], V–IX. Nederl Akad Wetensch Proc Ser A 62, 447–474 (1959).
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H Freudenthal, Beziehungen der $No alternative text available$ und $No alternative text available$ zur Oktavenebene [Relationship of $No alternative text available$ and $No alternative text available$ with the Cayley plane], X–XI. Nederl Akad Wetensch Proc Ser A 66, 455–487 (1963).
17
M Belolipetsky, Arithmetic hyperbolic reflection groups. Bull Am Math Soc 53, 437–475 (2016).
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NW Johnson, R Kellerhals, JG Ratcliffe, ST Tschantz, Commensurability classes of hyperbolic Coxeter groups. Linear Algebra Appl 345, 119–147 (2002).
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K Takeuchi, Commensurability classes of arithmetic triangle groups. J Fac Sci Univ Tokyo Sect IA Math 24, 201–212 (1977).
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K Takeuchi, Arithmetic triangle groups. J Math Soc Jpn 29, 91–106 (1977).
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EB Vinberg, The nonexistence of crystallographic reflection groups in Lobachevskiĭ spaces of large dimension. Funktsional Anal i Prilozhen 15, 67–68 (1981).
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KE Stange, The sensual Apollonian circle packing. Expo Math 34, 364–395 (2016).

## Information & Authors

### Information

#### Published in

Proceedings of the National Academy of Sciences
Vol. 116 | No. 2
January 8, 2019
PubMed: 30587588

#### Submission history

Published online: December 26, 2018
Published in issue: January 8, 2019

#### Acknowledgments

The authors thank the anonymous referee for comments and insights. M.H.W. is supported by the Simons Foundation Collaboration Grant 426453.

#### Notes

*The Eisenstein topograph was first described in 2007, in the unpublished master’s thesis of Andreas Weinert.

### Authors

#### Affiliations

Suzana Milea
Department of Mathematics, University of California, Santa Cruz, CA 95064;
Christopher D. Shelley
Department of Mathematics, University of California, Santa Cruz, CA 95064;

#### Notes

1
To whom correspondence should be addressed. Email: [email protected].
Author contributions: S.M., C.D.S., and M.H.W. designed research, performed research, analyzed data, and wrote the paper.

#### Competing Interests

The authors declare no conflict of interest.

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