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Divisibility by $p$ for Markoff-like Surfaces

Matthew de Courcy-Ireland, Matthew Litman, Yuma Mizuno

Abstract

We study orbits in a family of Markoff-like surfaces with extra off-diagonal terms over prime fields $\mathbb{F}_p$. It is shown that, for a typical surface of this form, every non-trivial orbit has size divisible by $p$. This extends a theorem of W.Y. Chen from the Markoff surface itself to others in this family. The proof closely follows and elaborates on a recent argument of D.E. Martin. We expect that there is just one orbit generically. For some special parameters, we prove that there are at least two or four orbits. Cayley's cubic surface plays a role in parametrising the exceptional cases and dictating the number of solutions mod $p$.

Divisibility by $p$ for Markoff-like Surfaces

Abstract

We study orbits in a family of Markoff-like surfaces with extra off-diagonal terms over prime fields . It is shown that, for a typical surface of this form, every non-trivial orbit has size divisible by . This extends a theorem of W.Y. Chen from the Markoff surface itself to others in this family. The proof closely follows and elaborates on a recent argument of D.E. Martin. We expect that there is just one orbit generically. For some special parameters, we prove that there are at least two or four orbits. Cayley's cubic surface plays a role in parametrising the exceptional cases and dictating the number of solutions mod .

Paper Structure

This paper contains 16 sections, 10 theorems, 126 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Relation to generalised cluster algebras
  3. Interpretation as a character variety
  4. Graphs
  5. Divisibility of orbit sizes
  6. Vanishing coordinates
  7. Completing the proof
  8. Double Fixed Points
  9. Example
  10. Quadratic obstruction: proof of Theorem \ref{['thm:breakup']}
  11. Further splitting if $\alpha=\pm 2$
  12. Number of solutions mod $p$
  13. Completing the square
  14. Counterexamples
  15. Example: $(2,2,-2)$
  16. ...and 1 more sections

Key Result

Theorem 1.1

Assume $3+a_1+a_2+a_3 \neq 0$ and, for all $i=1,2,3$, $a_i^2 \neq 4$ in $\mathbb{F}_p$ for a prime $p \geq 5$. Then, except for the orbit of size $1$ containing $(0,0,0)$, any orbit under the three moves (eqn:move) has size divisible by $p$. If $a_i^2=4$ for some $i$ and then any orbit has size divisible by $p$.

Figures (8)

  • Figure 1: Left: the quiver for the cluster algebra associated with the classical Markoff equation. Right: the quiver for the generalised cluster algebra associated with the generalised Markoff equation with parameters $a_1$, $a_2$, $a_3$. The $f$ and $f_i$ are called exchange polynomials at the corresponding vertices.
  • Figure 2: The action of $m_{i-1}$ and $m_{i+1}$ on triples with $x_i=0$. For each value of $x_{i-1}$, they form a cycle whose length depends on the order of a solution to $r^2+a_i r + 1 = 0$. If $a_i=0$, then the cycle has length 4.
  • Figure 3: Values of $(\Delta_{i-1},\Delta_{i+1})$ in the Markoff case ($a_1=a_2=a_3=0$, all $\Delta_i=0$ on $\{x_i=0\}$, and $r=\sqrt{-1}$), where $\delta$ is an arbitrary value of $\Delta_{i-1}(x)$ to start the cycle.
  • Figure 4: The number of solutions to $x^2+Bxy+y^2+Dx+Ey+F=0$ in $\mathbb{F}_p$. Top: the number is $p-\chi(B^2-4)$ for smooth conics, in terms of the quadratic character $\chi \bmod p$. Bottom: if $D^2+E^2+FB^2-4F-BDE = 0$, there is a correction of $p$ times $\chi(B^2-4)$ or $\chi(D^2-4F)$. Blue: analogous conic sections over the reals.
  • Figure 5: Orbits of size 2 for $a=(2,2,-2)$ and $s \neq 0$.
  • ...and 3 more figures

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • proof
  • proof
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • ...and 10 more