JSTeichR

November 2022

- Proofs from THE BOOK
- Convexity and Aigner’s Conjectures
- Can I prove these with one figure ?

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

\(x^2 + y^2 + z^2 - 3x y z = 0.\)

\((1,1,1),(1,1,2),(1,2,5),(1,5,13)\)

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

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

- \(x^2 + y^2 + z^2 - 3x y z = 0.\)

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

\((1,1,1),(1,1,2),(1,2,5),(1,5,13)\)

- 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 \(\mathbb{H}/\Gamma'\)

m = Markoff number

- Jack Button for m prime
- Zhang An elementary proof…
- Baragar m, 3m - 2, 3m + 2 prime
- Bugeaud, Reutenauer, Siksek
- Conclusion too hard!!!

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.

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 \(x,y\)

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.

Acting on \(\mathbb{F}_p^*\)

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

Acting on \(\mathbb{H}\)

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

- 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

\(G\) acting on \(X\) then

\(|G| |X/G| = \sum_{g} |X^g|\)

\(X^g\) = fixed points of the element \(g\)

\(X/G\) the orbit space.

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.

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

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

- identity \(|X^g| = p-1\)
- \(x \mapsto -x, |X^g| = 0\)

- \(x \mapsto 1/x, |X^g| = 2\)

- \(x \mapsto -1/x, |X^g| = \ldots\) ?

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

Acting on \(\mathbb{H}\)

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

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

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

\(\mathbb{Q}\cup \infty \subset\) 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

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

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

\(\mathrm{SL}(2,\mathbb{Z})\) 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 \(p/q\)

**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

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

- \(\Gamma = \mathrm{SL}(2,\mathbb{Z})\) 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

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

- 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.

- 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

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 \(i\).

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

- \(U'\) 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 \(=4\)

The automorphism \(V\) induced by \(V'\)

fixes two and exactly two arcs in \(X\).

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

- 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 = \(4p^2 >4\)

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

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

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

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