Experiments with the Markoff surface

Matthew de Courcy-Ireland; Seungjae Lee

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arXiv · 1812.07275

Experiments with the Markoff surface

Matthew de Courcy-Ireland·Seungjae Lee

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Abstract

We confirm, for the primes up to 3000, the conjecture of Bourgain, Gamburd, and Sarnak on strong approximation for the Markoff surface $x^2+y^2+z^2 = 3xyz$ modulo primes. For primes congruent to 3 modulo 4, we find data suggesting that some natural graphs constructed from this equation are asymptotically Ramanujan. For primes congruent to 1 modulo 4, the data suggest a weaker spectral gap. In both cases, there is close agreement with the Kesten-McKay law for the density of states for random 3-regular graphs. We also study the connectedness of other level sets $x^2+y^2+z^2-3xyz = k$. In the degenerate case of the Cayley cubic, we give a complete description of the orbits.

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BibTeX RIS

Matthew de Courcy-Ireland, Seungjae Lee. 2018-12-18. Experiments with the Markoff surface. https://arxiv.org/abs/1812.07275

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