Topology of Numbers

Topology of Numbers

By Allen Hatcher. American Mathematical Society, 2022, 341 pp. https://bookstore.ams.org/mbk-145

The fundamental object of study in this book is a quadratic form, i.e., any expression $Q(x,y) = a x^2 + b x y + c y^2$ where $a, b, c$ are rational integers.⁠¹ This is an area studied by Euler, Lagrange, and Gauss in the eighteenth century and Hatcher juxtaposes this rich theory with number theory in order to get, in his words, “a deeper sense of the beauty and subtlety of number theory.” With a minimal collection of prerequisites the book develops a list of topics including Pythagorean triples, the Euclidean algorithm, Pell’s equation, continued fractions, and Farey sequences, leading to the high-water mark, the Class Group, a notion first introduced by Gauss.

¹

These are the integers you learnt about at your mother’s knee $\{ \dots , -1, 0, 1, 2, \dots \}$ and so on. Number theorists use the term integer for any root of a monic equation with rational integer coefficients.

✖

Quadratic forms arise in many contexts in number theory, for example triples of integers which make the form $x^2 + y^2 - z^2$ equal to zero are usually called Pythagorean triples, with everyone’s favourite such triple being $(3,4,5)$. It’s not difficult to give a formula for all such triples, and this was known to the Ancient Greeks. Of course, one can, and should, make certain simplifying nondegeneracy assumptions. For example, if $(x,y,z)$ is a Pythagorean triple, and if two of the three numbers have a common factor $p$, then in fact $p$ divides the remaining number. This means one can restrict attention to primitive triples, which is to say triples with no common factor. In this way one is led to considering the equation $(x/z)^2 + (y/z)^2 = 1$ since for primitive triples, the fractions $x/z$ and $y/z$ are now in their lowest terms and this recasts the question of Pythagorean triples into the question of finding rational points on the circle $X^2 + Y^2 = 1$. Taking this as a starting point makes the analysis of the equation quite easy: There is a trivial solution, $X=0$ and $Y=1$ and if one considers a line of slope $m$ through this point, it has equation $Y= mX + 1$. Now, draw the picture and you’ll see that with the exception of the cases that the slope $m$ is one of $0$ or $\infty$, this line meets $X$-axis at one nonzero point. Denoting this point by $(r ,0)$, some gentle algebra reveals that the line meets the circle at the point $(\frac{2r}{r^2+1} , \frac{r^2-1}{r^2+1})$. Clearly, any rational choice of $r$ yields a rational point, hence an integral solution to Pythagorus’s equation and one sees easily that the converse is also true. However, the method also shows more, for example, any arc on the circle contains infinitely many rational points hence infinitely many such solutions.

This is the first such instance of a quadratic form which yields interesting and nontrivial mathematics, but number theory is replete with such examples. Another such example is the equation $x^2 - D y^2 = 1$ for some fixed nonsquare rational integer $D$; this is known as Pell’s equation. It arises in multiple contexts in number theory, and its study also goes back hundreds of years. Its subtlety and interest is highlighted by the fact that quite small values of $D$ only have very large integer solutions. For example when $D=61$, the smallest positive integer solution is $(1766319049, 226153980)$. Nonetheless, one can find all the solutions using variations on the ideas presented in the previous paragraph and it is a theme of Hatcher’s book that one can use this kind of geometric thinking to study the solutions to certain equations.

To this end, he first introduces the Farey diagram, an idea which goes back to Hurwitz.

Figure 1.

Farey graph.

The Farey diagram is a deceptively simple yet rich way of organizing the rational numbers together with infinity (which is regarded as both $1/0$ and $-1/0$). This diagram may be constructed as follows: If there is an edge connecting $p/q$ and $r/s$, then there is a triangle in the tessellation whose vertices are $( p/q, (p+r)/(q+s) , r/s)$. This provides an inductive scheme for generating the whole picture, by beginning with the edge joining $1/0$ and $0/1$ for positive rationals and taking the same edge regarded as joining $-1/0$ and $0/1$ for the negative rationals. It’s easily checked that if $p/q < r/s$, then $p/q < (p+r)/(q+s) < r/s$ but this kind of inductive description makes it hard to determine whether a given pair of fractions is connected by an edge in the Farey graph, however there is a clean criterion: $p/q$ and $r/s$ are connected by an edge if and only if the determinant of the matrix $\left( \begin{array}{cc}p&r\\q& s\end{array}\right)$ is equal to $\pm 1$. This is the first hint that something geometric in nature is connected to something algebraic. Moreover, while the proof is elementary it has nontrivial implications, for example, it shows the inductive rule always produces fractions in their lowest terms. It can also be used to show (it’s a nice exercise using the Euclidean algorithm) that every such fraction occurs in the Farey diagram.

Another important feature of the Farey diagram is its symmetry group, by which we mean any invertible transformation which carries vertices to vertices, edges to edges, and triangles to triangles. It is not hard to show that given a matrix $M = \left( \begin{array}{cc}p&r\\q& s\end{array}\right)$ with integer entries and of determinant $\pm 1$, that the transformation given by $T(x/y) = \frac{px + ry}{qx+sy}$ is a transformation in this sense. All such transformations have this form (notice that $M$ and $-M$ have the same action) and the group of such is usually called linear fractional transformations. It plays an important role in other areas of number theory as well as in hyperbolic geometry.

The next ingredient is the topograph. One can superimpose a dual tree on the Farey diagram, i.e., take a vertex in the center of each triangle of the Farey diagram and connect any two such vertices if their Farey triangles are adjacent.

Figure 2.

Farey graph plus dual tree.

This dual graph divides the interior of the circle into regions, each of which is adjacent to one rational number in the original diagram. Given a quadratic form $Q(x,y)$ one forms the topograph of $Q$ by labelling the region adjacent to $p/q$ with the value $Q(p,q)$. Figure 3 shows the topograph for $Q(x,y) = x^2+y^2$, so that, for example, the region labelled $13 = 2^2 + 3^2$ is adjacent to $2/3$.

Figure 3.

Topograph picture.

The power of this idea derives in part from the fact that there is a simple inductive rule which generates the labelled picture from a small amount of initial data: If the values of $Q(x,y)$ are as shown in the regions adjacent to an edge, then the integers $p, q+r, s$ form an arithmetic progression (see Figure 4). It’s not difficult to convince oneself that this rule, coupled with the three values in the regions adjacent to a single vertex, determines the value in every region.

Figure 4.

Edge picture.

The topograph offers a geometric avenue into problems associated with quadratic forms, and the book explores some of these. Let us illustrate with the example of Pell’s equation; here $Q(x,y) = x^2 - D y^2$ and we are interested in regions in the topograph which carry the label $1$.

One finds that there is a separator line in the graph separating the positive regions from the negative and the line organises the picture into a pattern which is infinitely repeating with respect to translation along the line. The separator line can be constructed easily from the three values $Q(1,0)=1$, $Q(0,1)= - D$ and $Q(1,1) = 1-D$; we remark that in fact it is closely connected to the continued fraction for the nonsquare integer $\sqrt {D}$. One can show that every periodic line in the dual tree of the Farey diagram occurs as the separator line for some form.

Not unexpectedly, one can show that in a topograph which has a separator line, as one moves away from this line the values increase monotonically towards $+ \infty$ on the positive side and $- \infty$ on the negative side, so if the value $1$ is to occur it must be on the separator line itself. Moreover, we see from the initial data used in the construction, that the value $1$ occurs on the line so that there is a region in the topograph carrying the value $1$, and by periodicity there must be other regions which do. These correspond to interesting solutions to Pell’s equation. One way of finding such solutions is to construct the fractional linear translation which stabilizes the separator line. It is not difficult to see that this transformation must have infinite order, so that there are infinitely many solutions to Pell’s equation.

As always, there comes a point where it needs to be decided what it means for two objects under consideration to be equivalent. In this setting, there is a very natural definition, namely, two quadratic forms $Q_1$ and $Q_2$ are equivalent if there is a fractional linear transformation throwing the topograph of one to the topograph of the other. Given the fact that topographs are locally determined by their values around any vertex, an alternative equivalent definition is that there are vertices $v_1$ and $v_2$ so that the values of the $Q_1$-topograph around $v_1$ are the same as those in the $Q_2$-topograph around $v_2$. With this equivalence decided, one ask about obstructions to forms being equivalent and again there is a simple obstruction: given a quadratic form $Q(x,y) = a x^2 + b xy + c y^2$, one defines its discriminant to be $b^2 - 4ac$. Notice that any discriminant is congruent to a square modulo $4$, so it must be either $0$ or $1$ mod $4$. Discriminants play a central role in the theory of quadratic forms and it’s not difficult to see that equivalent forms have equal discriminant. Moreover, it can be shown that there are only a finite number of equivalence classes of forms with a given nonzero discriminant.

However, it’s natural to refine this situation a little further. If the fractional linear transformation can be chosen to be orientation preserving, then one says the forms are properly equivalent. Also, if you multiply a form by some constant $d > 0$ its essential features are not significantly changed, and so it’s reasonable to restrict attention to primitive forms, which is to say forms for which $a, b, c$ have no common divisor. Equivalence preserves primitivity. The number of proper equivalence classes of primitive forms of a given discriminant $\Delta$ is called the class number, $h_{\Delta }$, for that discriminant. This is a number which can be computed algorithmically for any given discriminant $\Delta$. Despite this, the dependence of $h_{\Delta }$ upon $\Delta$ is mysterious and not well understood.

It is of especial interest to study the discriminants for which all forms are primitive, these are the so-called fundamental discriminants. Every nonsquare discriminant $\Delta$ can be written $\Delta = d^2 \Delta '$ where $\Delta '$ is fundamental, and one can relate $h_{\Delta }$ to $h_{\Delta '}$, so that the study of class numbers is largely about the case of fundamental discriminants.

The question of which discriminants have class number $1$ has been a magnet for research for a long time; this amounts to finding the discriminants for which all primitive forms are equivalent. For example, the nine fundamental discriminants $\Delta = -3, -4, -7, -8, -11, -19, -43$, $-67, -163$ have class number $1$. A conjecture of Gauss from around $1800$ was that there were no other negative discriminants of class number $1$, this was finally resolved in the affirmative in the sixties. This is to be contrasted with the situation for positive discriminants with class number $1$ which is not at all understood. It seems likely that there are in fact infinitely many, however, this is still unsolved and remains one of the most basic unsolved problems about quadratic forms.

Class numbers and the representation of integers by quadratic forms are the beginning of a rich and interesting theory, which is developed further in the chapters which follow. While the point of view might be a little idiosyncratic, both in terms of the material that is addressed, as well as the manner in which the book chooses to address it, many will find this geometric viewpoint hugely appealing. Certainly this reviewer did. It is a beautiful book.

D. D. Long is a professor of mathematics at the University of California, Santa Barbara. His email address is long@math.ucsb.edu.

Article DOI: 10.1090/noti2739

Credits

Figures 1–4 are courtesy of Allen E. Hatcher.

Photo of D. D. Long is courtesy of Chris Leininger.