# Geometry of sums of squares

November 2022

Markov numbers are integers that appear in triples which are solutions of a Diophantine equation the so-called Markov cubic

$x^2 + y^2 + z^2 - 3x y z = 0.$

$(1,1,1),(1,1,2),(1,2,5),(1,5,13)$

### Theorem

$(x,y,z) \in \mathbb{R}_+,\,x^2 + y^2 + z^2 - x y z = 0.$

can be identified with the Teichmueller space of the punctured torus using Penner’s $\lambda$-lengths.

• $x^2 + y^2 + z^2 - 3x y z = 0.$

### Odd index Fibonacci numbers are Markoff numbers

$1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, ...$

$(1,1,1),(1,1,2),(1,2,5),(1,5,13)$

### Frobenius uniqueness conjecture

• The largest integer in a triple determines the two other numbers.
• For every Markoff number $m$ there are exactly 3 simple closed geodesics of length $2\cosh^{-1}(3m/2)$ on the modular torus $\mathbb{H}/\Gamma'$

### Partial results

m = Markoff number

## Button’s Theorem

If $z$ is a Markoff number which is prime
then there is a unique triple $z > y > x$

• $x^2 + y^2 + z^2 - 3x y z = 0.$
• $\bar{x}^2 + \bar{y}^2 = 0.$ in $\mathbb{F}_z$
• $(\bar{x}/\bar{y})^2 = -1$ in $\mathbb{F}_z$
• $\Rightarrow p = 2$ or $p − 1$ is a multiple of 4.

### Theorem 1.2

Let $p$ be a prime then $x^2 + y^2= p$ has a solution over $\mathbb{Z}$

• iff $p = 2$ or $p − 1$ is a multiple of 4.
• Button’s theorem follows from unicity of $x,y$

### Theorem 1.3

Let $p$ be a prime then $x^2 + xy + y^2= p$ has a solution over $\mathbb{Z}$

• iff $p = 3$ or $p − 1$ is a multiple of 6.

## two groups of order 4

Acting on $\mathbb{F}_p^*$

$\begin{array}{lll} x &\mapsto& -x \\ x &\mapsto& 1/x \end{array}$

Acting on $\mathbb{H}$

$\begin{array}{lll} z &\mapsto& -\bar{z} \\ z &\mapsto& 1/\bar{z} \end{array}$

## References etc

• Heath-Brown, Fermat’s two squares theorem. Invariant (1984)
• Zagier, A one-sentence proof that every prime p = 1 (mod 4) is a sum of two squares, 1990
• Elsholtz, Combinatorial Approach to Sums of Two Squares and Related Problems. (2010)
• Penner, The decorated Teichmueller space of punctured surfaces, Comm Math Phys (1987)
• Zagier text

## Let’s begin then…

### Burnside Lemma

• $G$ acting on $X$ then

$|G| |X/G| = \sum_{g} |X^g|$

• $X^g$ = fixed points of the element $g$

• $X/G$ the orbit space.

### Theorem 1.1

Let $p$ be a prime then

$x^2 = -1$

has a solution over $\mathbb{F}_p$

• iff $p = 2$ or $p − 1$ is a multiple of 4.

### Proof

Group acting on $X = \mathbb{F}_p^*$:

$\begin{array}{lll} x &\mapsto& x \\ x &\mapsto& -x \\ x &\mapsto& 1/x \\ x &\mapsto& -1/x \end{array}$

### Counting fixed points

• identity $|X^g| = p-1$
• $x \mapsto -x, |X^g| = 0$
• $x \mapsto 1/x, |X^g| = 2$
• $x \mapsto -1/x, |X^g| = \ldots$ ?

### Apply Burnside

• $|G| |X/G| = \sum_{g} |X^g|$
• $4 |X/G| = (p-1) + 2 + |X^{(x\mapsto -1/x)}|$
• $\Rightarrow |X^{(x\mapsto -1/x)}| = 2,\, \text{if }4 \not \mid (p+1)$
• $\Rightarrow \exists x,\, x^2 = -1,\, \text{if }4 \not \mid (p+1)$

## QED

### Theorem 1.2: sum of 2 squares

Acting on $\mathbb{H}$

$\begin{array}{lll} z &\mapsto& -\bar{z} \\ z &\mapsto& 1/\bar{z} \end{array}$

## Primitives

• $\mathbb{Z}^2$
• infinitely many primitive elements
• $(a,b)$ primitive iff $a,b \in \mathbb{Z}$ coprime
• $SL(2,\mathbb{Z})$ transitive on primitives

## Important

$\begin{eqnarray*} \{ \textit{primitives} \} &=& \mathbb{Q}\cup \infty\\ &\subset& \text{circle/projective line } \\ &=& \partial_\infty \mathbb{H} \end{eqnarray*}$

## Farey tessalation

$\mathbb{Q}\cup \infty \subset$ circle/projective line

• $(a,b)\text{ primitive } \mapsto a/b \in \mathbb{Q}\cup \infty$
• $\begin{pmatrix} a & c \\ b & d \end{pmatrix} \in \mathrm{SL}(2,\mathbb{Z})\mapsto$ arc joining $(a/b, c/d)$
• $(a/b, c/d)$ are Farey neighbors

source

source

## Definitions

• arc = Poincaré geodesic joining $a/b, c/d \in \mathbb{Q}\cup \infty$
• $\lambda$- length of arc $= |ad - bc|^2$

## Lemma

$\log \lambda$-length = length of the portion outside Ford circles tangent to the real line at its endpoints

## Ford circles

$\mathrm{SL}(2,\mathbb{Z})$ acts by Mobius transformations on $\mathbb{H}$

• $\begin{pmatrix} a & c \\ b & d \end{pmatrix}.z = \frac{az+b}{cz+d}$
• preserves the Poincaré (hyperbolic) metric
• the orbit of $F := \{ z, \mathrm{Im}\, z > 1\}$ are the Ford circles

• $p/q$ point of tangency with $\mathbb{R}$, diameter = $1/q^2$
• ie the diameter is the square of the inverse of the denominator of $p/q$

### Proof of lemma

• arc joining $a/b, c/d$ has $\lambda$-length $= |ad - bc|^2$
• $\log \lambda$-length = length of the portion outside Ford circles tangent to the real line at its endpoints

### Proof of lemma

• $\mathrm{SL}(2,\mathbb{Z})$ transitive,
• can suppose $a/b = \infty$ and $c/d = k/(ad - bc)$
• Ford circles $F$ tangent at $\infty$
• and another of diameter $1/(ad - bc)^2$

### Groups and quotients

• $\Gamma = \mathrm{SL}(2,\mathbb{Z})$ has torsion so $\mathbb{H}/\Gamma$ orbifold
• $\Gamma(2) = \ker (\mathrm{SL}(2,\mathbb{Z})\rightarrow \mathrm{SL}(2,\mathbb{F}_2))$
• $\Gamma' = [\Gamma,\Gamma]$
• $\mathbb{H}/\Gamma(2)$ three punctured sphere
• $\mathbb{H}/\Gamma'$ once punctured torus
• For Aigner’s conjectures the geometry of the simple geodesics on $\mathbb{H}/\Gamma'$ once punctured torus was important.
• For Fermat’s theorem it’s the automorphisms of $\mathbb{H}/\Gamma(2)$ = three punctured sphere

A three punctured sphere
can be cut up into 2 ideal triangles.

• Fundamental domain for $\Gamma(2)$

• $i, 1+i, \frac12 ( 1 + i)$ are midpoints

### reciprocals of sums of squares

• $i, 1+i, \frac12 ( 1 + i)$ are midpoints of arcs
• the lifts to $\mathbb{H}$ of the midpoints $=\Gamma.i$
• $\mathrm{Im} \frac{ai+b}{ci+d} = \frac{\mathrm{Im}\, i }{c^2 + d^2}$

## Lemma A

• Let $n$ be a positive integer.
• The number of ways of writing $n$ as a sum of squares $n = c^2 + d^2$ with $c,d$ coprime integers
• is equal to the number of integers $0 \leq k < n-1$ coprime to $n$ such that the line $\{ k/n + i t,\, t>0 \}$ contains a point in the $\Gamma$ orbit of $i$.

What is the group of automorphisms?

What is the subgroup of automorphisms
fixing the cusp labeled $\infty$?

• fixes the cusp and midpoint $\frac12(1+i)$
• dashed arca are invariant under the group
• one arc has $\lambda$-length 1, the other 2.

### the set $X$

• arcs joining cusps $\infty, 1$ with $\lambda$-length $p^2$
• lift to vertical lines with endpoints $k/p$ with $k$ odd
• $|X| = p - 1$ as before

### Lemma A

Let $n$ be a positive integer. The number of ways of writing $n$ as a sum of squares $n = c^2 + d^2$ with $c,d$ coprime integers is equal to the number of integers $0 \leq k < n-1$ coprime to $n$ such that the line $\{ k/n + i t,\, t>0 \}$ contains a point in the $\Gamma$ orbit of $i$.

### subgroup lifts to

• $U': z \mapsto 2-\bar{z},\, V' : z \mapsto \bar{z}/(\bar{z} - 1)$
• $U'\circ V' : z \mapsto z \mapsto (-z + 2) /( z + 1)$
• $U' \circ V'$ fixes $i+1$

## automorphisms

• $U'$ induces an automorphism no fixed points in $X,\, \text{if } p \geq 3$
• $V'$ is an inversion in a half circle with endpoints -1 and 1
• this arc’s projection to surface is simple arc of $\lambda$-length $=4$

## Lemma B

The automorphism $V$ induced by $V'$
fixes two and exactly two arcs in $X$.

• apply Burnside Lemma to prove Theorem 1.2
• $4 |X/G| = (p-1) + 2 + |X^{U\circ V}|$

## Proof

• If $\infty$ and $k/p$ are exchanged by an inversion swapping Ford circles
• Then the endpoints of the fixed circle are $(k-1)/p$ and $(k+1)/p$
• if $1 < k < p-1$ the arc joining these points has $\lambda$-length = $4p^2 >4$

## Button’s Theorem

If $z$ is a Markoff number which is prime
then there is a unique triple $z > y > x$

$x^2 + y^2 + z^2 - 3x y z = 0.$

• Button’s theorem follows from unicity in $z = c^2 + d^2$
• $\Leftrightarrow$ unique vertical geodesic in Lemma A.
• let’s look at that again
• Number of ways of writing $n$ as a sum of squares $c^2 + d^2$ with $c,d$ coprime integers
• = number of integers $0 \leq k < n-1$ coprime to $n$ such that the line $\{ k/n + i t,\, t>0 \}$ contains a point in the $\Gamma$ orbit of $i$.
• For every Markoff number $m>2$ there are exactly 6 simple closed geodesics of length $2\cosh^{-1}(3m/2)$ on the modular torus $\mathbb{H}/\Gamma'$
• $\Leftrightarrow$ exactly 6 simple arcs of $\lambda$-length $9m^2$ on $\mathbb{H}/\Gamma'$

pair of disjoint simple closed and arc geodesics

### Theorem

$(x,y,z) \in \mathbb{R}_+,\,x^2 + y^2 + z^2 - x y z = 0.$

can be identified with the Teichmueller space of the punctured torus.

exactly 6 simple arcs of $\lambda$-length $9m^2$ on $\mathbb{H}/\Gamma'$