Connectedness of special points in the Markoff mod $p$ graphs

TL;DR

The paper investigates strong approximation for the Markoff equation modulo primes by analyzing the Markoff mod $p$ graphs under the Vieta group action. It connects rotation orders to Pisano periods of the Fibonacci sequence, enabling explicit lower bounds on rotation orders in terms of the 2-adic valuation of $p+1$. The main result shows that for primes with large $\nu_2(p+1)$, certain elliptic coordinates force points to lie in the cage, yielding guaranteed integer lifts; in particular, Mersenne primes with $p\equiv \pm2\pmod{5}$ place the special point $(1,1,1)$ in the cage. The work thus extends connections between graph connectivity, number theory of Fibonacci periods, and Diophantine lifting, and discusses the density of such primes and potential generalizations.

Abstract

It is conjectured that the Markoff equation $X^2+Y^2+Z^2=3XYZ$ satisfies the special Diophantine property that every mod $p$ solution lifts to an integer solution. Progress toward this conjecture has been made by studying the connectedness of the graphs obtained from the action of the Vieta group on the nonzero mod $p$ solutions to the Markoff equation. In this paper, we use results on Pisano periods of the Fibonacci sequence to obtain explicit results on the connectedness of special points in this graph for primes $p$ where $p+1$ has large $2$-adic valuation. In particular, for Mersenne primes $p \equiv \pm 2 \,(\text{mod}\, 5)$, we show that the special point $(1, 1, 1)$ which is fixed under reduction modulo $p$ lies in a component of this graph which is known to be connected.

Figures (1)

• Figure 1: Percent of primes $p$ with $(1,1,1) \in \mathcal{C}(p)$

Theorems & Definitions (29)

• Theorem 1.1: Consequence of Theorem \ref{['main']}
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Lemma 2.4
• Proposition 2.5
• Definition 2.6
• Definition 2.7
• Proposition 2.8
• Remark 2.9