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The Geometric Disposition of Diophantine Equations

Anthony Várilly-Alvarado

Communicated by Notices Associate Editor William McCallum

Article cover

In 2005, writer David Foster Wallace delivered a remarkable commencement speech at Kenyon College. He began by begrudgingly offering “a standard requirement of US commencement speeches,” the parable-ish story:

There are these two young fish swimming along and they happen to meet an older fish swimming the other way, who nods at them and says “Morning, boys. How’s the water?” And the two young fish swim on for a bit, and then eventually one of them looks over at the other and goes “What the hell is water?”

This article is a story about water. It is a story about trying to understand the natural habitat of certain problems that, on their face, look like problems about whole numbers. To be sure, the problems we discuss are number theoretic in character, but the way to access them and to think about them is informed by a different part of mathematics: geometry.

1. Three Problems

(1)

Which whole numbers can be expressed as a sum of three cubes?

(2)

Is there a box such that the distance between any two of its corners is a positive whole number?

(3)

Is there a magic square whose entries are distinct nonzero squares?

Recall an magic square is an grid, filled with distinct positive integers, whose rows, columns, and diagonals add up to the same number. For example, in 1514 the German artist Albrecht Dürer included the following magic square in Melencolia I (see Figure 1):

\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \begin{tikzpicture}
\draw[step=0.8cm] (-0.805,-0.805) grid (2.4,2.4);
\node at (-0.4,+2.0) {\Large{16}};
\node at (+0.4,+2.0) {\Large{3}};
\node at (+1.2,+2.0) {\Large{2}};
\node at (+2.0,+2.0) {\Large{13}};
\node at (-0.4,+1.2) {\Large{5}};
\node at (+0.4,+1.2) {\Large{10}};
\node at (+1.2,+1.2) {\Large{11}};
\node at (+2.0,+1.2) {\Large{8}};
\node at (-0.4,+0.4) {\Large{9}};
\node at (+0.4,+0.4) {\Large{6}};
\node at (+1.2,+0.4) {\Large{7}};
\node at (+2.0,+0.4) {\Large{12}};
\node at (-0.4,-0.4) {\Large{4}};
\node at (+0.4,-0.4) {\Large{15}};
\node at (+1.2,-0.4) {\Large{14}};
\node at (+2.0,-0.4) {\Large{1}};
\end{tikzpicture}
Figure 1.

Melencolia I, Albrecht Dürer (1514). There is a magic square in the top right of the engraving.

Graphic for Figure 1.  without alt text

Although seemingly unrelated, the three problems above share many features. They all ask questions about algebraic relations between whole numbers. They also all have avatars as problems about rational points on algebraic surfaces. The most vexing commonality of these three problems is their current status: they are all open.

1.1. Historical remarks

Problem (1) asks: for which integers do there exist integers , , and such that

In 1825, Samuel Ryley showed that every integer (indeed, every rational number) is the sum of three rational cubes. Further progress through 2007 is nicely documented in BPTYJ07, §2, where the first (and smallest!) solution to is given:

This solution was found in 1999; Daniel Bernstein found the same solution independently and contemporaneously, based on ideas suggested by Noam Elkies.⁠Footnote1 At the beginning of 2019, the only not known to be expressible (or not!) as a sum of three integer cubes were and . Shortly thereafter, Andrew Booker Boo19 showed that

and this is the smallest solution to the problem! A few months later, Booker joined forces with Andrew Sutherland to find the smallest solution to :

For some integers , the diophantine equation 1.1 admits no integral solutions: indeed, the set of cubes modulo is , and hence three cubes cannot add up to or modulo . Based on analytic arguments predicting the distribution of solutions to 1.1, Heath-Brown HB92, p. 623 proposed the following conjecture.

Conjecture 1.1.

For integers there exist integers , , and such that 1.1 holds.

A box witnessing a positive solution to problem (2) is called a perfect cuboid (Figure 2). Euler studied the closely related problem of finding boxes whose sides and face diagonals are positive integers; it seems likely he considered the problem of the existence of a perfect cuboid, though no written record of such an exploration appears to exist. The literature surrounding this problem is nicely summarized in van Luijk’s undergraduate thesis vL00.

Figure 2.

Can , , , , , , and be integers?

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\coordinate(B) at (1.6,1.3,-1.3) ;
\coordinate(C) at (-1.7,1.4,1.3) ;
\coordinate(D) at (1.7,-1.4,1.3) ;
\coordinate(E) at (-1.6,1.3,-1.3) ;
\coordinate(F) at (-1.7,-1.4,1.3) ;
\coordinate(G) at (1.6,-1.4,-1.3) ;
\coordinate(H) at (-1.6,-1.4,-1.3) ;
\begin{scope}[every node/.style={midway}]
\draw(D) -- node[right=2pt] {$x_3$} (G) -- (B) -- (E) -- (C)
-- node[above=3pt, right] {$x_2$} (F) -- node[above] {$x_1$} (D) ;
\draw(A) -- (D) ;
\draw(B) -- (A) -- (C) ;
\draw[dashed] (G) -- (H) ;
\draw[dashed] (E) -- (H) -- (F) ;
\draw[blue] (B) -- node[above=5pt, left] {$y_2$} (C)
-- node[above] {$y_3$} (D) -- node[below=2pt, right=-2pt] {$y_1$} (B) ;
\draw[dashed, thick, red] (E) -- node[above right] {$z$} (D) ;
\end{scope}
\foreach\point in {A,B,C,D,E,F,G,H} \draw(\point) node {$\bull$} ;
\end{tikzpicture}

Magic squares have a long history. Tradition has it that the Lo Shu, the earliest recorded magic square, was first observed by Emperor Yu upon the back of a turtle (ca. 2,200 BC). The search for a magic square of squares was popularized by Martin Gardner in 1996 Gar96; he attributed the problem to Martin LaBar (1984), though it had been studied by Euler in 1770 and Lucas in 1876 Boy05. Andrew Bremner has used the arithmetic of elliptic curves and K3 surfaces to study two related problems: finding squares with distinct square entries such that as many as possible of the eight row, columns, and diagonals are equal Bre99, and finding magic squares with distinct entries, with as many entries as possible being squares Bre01. Our own investigations BTVA into the problem of finding magic squares of squares are inspired by a similar geometric point of view, although we work with surfaces of general type, as explained below.

1.2. Geometry determines arithmetic

The three problems above can all be phrased as questions involving the rational or integral points on certain algebraic surfaces. We aim to show how our current understanding of the arithmetic of algebraic surfaces informs the expectations many arithmetic geometers harbor for the answers to our three problems. Geometry determines arithmetic shall be our mantra. To develop a feel for this mantra, we turn to a lower-dimensional situation: the arithmetic geometry of curves.

1.3. Fermat’s Last Theorem: A geometric restatement

Fermat’s Last Theorem, i.e., the statement that for every solution to the equation

satisfies , is a statement about rational points on a smooth, projective plane curve. Recall that the set of rational points on the projective plane is

We write for the equivalence class of . The projective plane can be thought of as a compactification of the Cartesian plane ; here we identify with the point . The subset of whose -coordinate is zero gives the set of rational points on the “line at infinity” that is used to compactify to . We define analogously:

A trivial but powerful observation is that every point has a representative (unique up to a global sign) with relatively prime integers, obtained by clearing denominators and removing common factors from any given representative. This representation of a rational point is almost unique: its only ambiguity is a global sign. For example,

The zero-set of the Fermat expression defines a curve in the projective plane , whose rational points are

The coordinate axes in define a reducible curve whose rational points are given by

Fermat’s Last Theorem can be restated as follows: given an integer , we have

We describe below some fundamental results on the arithmetic of curves, and apply them to study the set . This will not give a proof of Fermat’s Last Theorem, as we will not use anything special about the Fermat curve, other than its smoothness, its degree, and the fact that it contains rational points (e.g., ). Our goal is thus not a description of the fundamental breakthroughs of Wiles and Taylor–Wiles; rather we use the Fermat curve as an excuse for a tour of the arithmetic of curves.

2. Arithmetic of Curves

2.1. The genus of a nice curve

By a nice variety we mean an algebraic variety over a field that satisfies a few technical hypotheses: should be smooth, projective, and geometrically integral. A nice curve is a -dimensional nice variety. For example, suppose that is given by the zero-locus of a homogeneous degree polynomial in the projective plane . Let be the ideal

considered in the ring , where denotes a fixed algebraic closure of ; the generator in is redundant if . The Jacobian criterion and the projective Nullstellensatz together imply that is nice if some power of the “irrelevant ideal” is contained in . For the Fermat curve, we take and (); the ideal is

and we can check that , so the Fermat curve is nice.

Nice curves have one fundamental discrete invariant: their genus. It is the dimension of the vector space of global -forms; when is a nice plane curve, defined by a homogeneous polynomial of degree , this dimension coincides with the quantity

If , then the set of complex points can be given the structure of a compact Riemann surface , and the genus above coincides with the number of handles on .

2.2. Kodaira dimension of a nice curve

Two nice varieties and defined over a field are said to be -birational if there exist open sets and (for the Zariski topology) such that and are isomorphic as varieties over ; informally, the isomorphism should be given by rational functions with coefficients in . This is a very strong condition: nonempty open subsets in the Zariski topology of a nice variety are dense! For example, the proper Zariski-closed subsets on a nice curve over are finite sets of points. Many arithmetic questions about varieties have answers that depend only on the birational class of the variety; for example, if and are nice -varieties that are -birational to each other, then has a -point if and only if has a -point (this follows from the Lang–Nishimura lemma Poo17, §3.6.4).

Birational invariants and birational classification theorems thus guide our expectations for the properties of the set of rational points on an algebraic variety. The genus of a nice curve is a birational invariant. A related birational invariant is the Kodaira dimension , whose precise definition is given below in §3.3. For nice curves, suffice it to say for now that

The Kodaira dimension of a nice curve indicates curvature. For example, if , the Riemann surface has positive curvature if , it has flat curvature if , and it is negatively curved if .

2.3. Rational points on curves vis-à-vis Kodaira dimension

Let be a nice curve over .

: In this case, if , one can show that is isomorphic over to the projective line . This is done using a stereographic projection; we shall see a concrete example below (§2.4).

: In this case, if , then is an elliptic curve (by definition!), and it is well known that the rational points of can be endowed with the structure of an abelian group. This group is finitely generated by a theorem of Mordell from 1922 Sil09, VIII.4. The structure theorem for finitely generated abelian groups then implies that

as abelian groups, where is the subgroup of consisting of points of finite order. The integer is called the rank of and it plays a major rôle in the Birch–Swinnerton-Dyer conjecture. A spectacular theorem of Mazur says that there are only possibilities for the isomorphism class of the group .

: In this case, we say is of general type. Faltings showed in 1983 that for curves of general type, the set is finite. His work, which simultaneously solved several major open problems in arithmetic geometry, earned him a Fields Medal.

2.4. Fermat curves

What can the general theory of the arithmetic of curves tell us about Fermat’s Last Theorem? Let

be the Fermat curve of exponent , considered as a nice curve in the projective plane .

2.4.1. A rational curve: . The curve has genus by 2.1, and hence . Since , general theory predicts that is -isomorphic to the projective line . We construct an isomorphism using the (inverse of) stereographic projection. Recall we identified with ; this identification is valid in the locus of where , where we have set and . The part of in this affine “patch” is the circle

and the point is identified with the point . We use this point to create the isomorphism , as follows. Define the map of algebraic varieties

This is the inverse map to a stereographic projection, as Figure 3 shows.

Figure 3.

Stereographic projection.

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\coordinate(O) at (0,0) ;
\coordinate(A) at (0,2) ;
\coordinate(B) at (2.8,0) ;
\coordinate(X) at (3.8,0) ;
\coordinate(Xm) at (-2.3,0) ;
\coordinate(Y) at (0,3) ;
\coordinate(Ym) at (0,-2.3) ;
\draw[->, line width=.6] (Xm) -- (X) node[right] {$X$} ;
\draw[->, line width=.6] (Ym) -- (Y) node[above] {$Y$} ;
\node[draw] (nK) at (O) [circle, minimum size=4cm, line width=.6] {} ;
\coordinate(C) at (intersection 1 of A--B and nK) ;
\draw[red, line width=.6] ($ (A)!-0.4!(B) $) -- (A) node[above right] {$(0,1)$}
-- (B) node[below=4pt] {$(s,0)$} -- ($ (A)!1.4!(B) $) ;
\draw(C) node[above right]
{$\bigl( \frac{2s}{s^2+1}, \frac{s^2-1}{s^2+1} \bigr)$} ;
\draw(Ym) node[below=3pt] {$X^2 + Y^2 = 1$} ;
\foreach\point in {A,B,C} \draw[red] (\point) node {$\bull$} ;
\end{tikzpicture}

We want to extend our construction to a map in such a way that

Let and be homogeneous coordinates of . Set ; this identifies with . The map above becomes

which can be rewritten more pleasantly as

This is the map we are looking for! It makes sense even for points like . By inspection, the map

is an inverse for , at least for . Since is an isomorphism, and since rational points are determined only up to a scalar multiple, we conclude that every rational point on the curve has a representative of the form

where and are relatively prime integers. The point is also determined only up to a scalar multiple. We conclude that every Pythagorean triple, i.e., every integral solution to , has the form

where and , are relatively prime integers. The reason we allow to possibly be a half-integer is that if and are both odd, then .

This account of the shape of Pythagorean triples reflects our mantra: geometry determines arithmetic! Many of us learned a proof of the shape of Pythagorean triples that uses only basic algebra and divisibility relations; its “elementary nature” undercuts both the beauty of the geometric proof and a feeling of genuine understanding.

2.4.2. An elliptic curve: . The curve has genus by 2.1, and hence . Since , the curve is an elliptic curve, so by Mordell’s theorem, we have . We claim that

This can be shown with an important tool: reduction modulo a prime . Let be the Fermat curve with , defined over the finite field with elements. We define a reduction map by sending to , where we have chosen a representative with , , and relatively prime integers. As long as the curve is nice, and crucially we have an injection of groups

via the reduction map we just described (see Sil09, §VII.3). Because there are only finitely many solutions to the equation in , it is straightforward to compute that and . This shows that is either the trivial group, or it is .

To show that the rank of is zero, we may apply deep results of Gross–Zagier and Kolyvagin on the -series of elliptic curves with complex multiplication GZ86Kol88. Loosely speaking, an elliptic curve has complex multiplication if it has an unusually large endomorphism ring. The -series of is defined by an Euler product over the primes, whose individual factors record information about the modulo reduction of . The function converges for ; even before Wiles’ work on Fermat’s Last Theorem, we knew that if has complex multiplication, then has an analytic continuation to the entire complex plane, thanks to work of Deuring and Weil. The Birch–Swinnerton-Dyer conjecture predicts that the order of vanish of at is equal to the rank of ; the results in GZ86Kol88 together imply that if , then , i.e., is finite.

The curve is isomorphic to the curve,

considered in the usual affine plane . This curve has -invariant (see Sil09, III.1), and hence has complex multiplication. Using computer software, we can check that in this case

and hence is finite, equal to the set given in 2.2.

2.4.3. Curves of general type: . The curves have genus whenever by 2.1, and hence . By Faltings’ theorem, we know that is finite; it is however nonempty, as . Faltings’ proof of his theorem is not effective. This is as far as the general qualitative theory of curves will take us. The methods of Wiles and Taylor–Wiles opened up whole new research programs in number theory, but the connection to Fermat’s Last Theorem uses the explicit shape of Fermat’s equation to construct, from a putative nontrivial solution, an elliptic curve too exotic to exist.

3. Arithmetic of Higher-dimensional Varieties

We now leave the realm of algebraic curves to explore higher-dimensional spaces. From here on out, unless otherwise stated, all varieties are assumed to be nice; by a surface, we mean a nice variety of dimension . A concrete example of a surface in projective -space , with coordinates , , , and , is given by

This is an example of a K3 surface; it is simply connected and carries a nowhere-vanishing holomorphic -form. All smooth quartic surfaces in have these two properties; the one above was considered by Swinnerton-Dyer. It has rational points, e.g., is in . It is unknown if is finite or not.

3.1. Local obstructions

Because the varieties we study are projective, and hence both the denominators of the coordinates of a rational point as well as the denominators in the defining equations of can be cleared out, the sets of integral solutions and rational solutions coincide; see §1.3. This incidental reframing affords an important tool: reduction modulo for any prime and any exponent . In order to have , i.e., a nontrivial integral solution to the set of equations defining , the same set of equations must have solutions modulo for every prime and every exponent ; we call these solutions points modulo . Our set of equations must also have solutions in . If fails to have points modulo for some , or if , we say there is a local obstruction to the existence of rational points.⁠Footnote2

2

Motivating the terminology here is the statement that for a locally compact field that contains , namely , or , the field of -adic numbers.

We have all experienced local obstructions: one of the first proofs many of us are exposed to is the irrationality of . Equivalently, the variety in has no -points: there are no nontrivial solutions⁠Footnote3 to its defining equation modulo .

3

More precisely, if the variety has a rational point , then we may assume that and are coprime integers, because . On the other hand, the only solution to requires that both and are divisible by .

For a given prime and exponent there are only finitely many possible solutions modulo to the set of equations defining . But there are infinitely many primes and exponents . The necessity of local solutions asks us to trade, with no assurance of success, one hard problem for infinitely many easier problems. Is this a good trade-off? Most definitely, thanks to the Weil conjectures, now theorems after the revolutionary efforts and insights of Dwork, Grothendieck, and Deligne (see Poo17, Ch. 7 for an introduction to the subject). In short, the Weil conjectures give a precise bound , in terms of the geometry of , such that has solutions modulo for all , as long as the reductions of the equations of modulo define a smooth projective variety over . By means of Hensel’s lemma, a -adic analogue of the Newton–Raphson method, smooth solutions modulo can be leveraged to construct solutions modulo for all ; see Poo17, Theorem 3.5.63. This leaves a finite set of primes to check: those , and those primes for which does not have smooth reduction modulo . The later can be calculated explicitly with a Gröbner basis computation. It then remains to find solutions modulo for and some small that are liftable to solutions modulo for all using Hensel’s lemma, whenever possible. Checking that often comes down to a Lagrange multipliers problem.

3.2. Local obstructions are not enough

Sadly, there are nice varieties that have points modulo for all primes and all exponents , as well as -points, for which