Stable reduction and large Nielsen orbits on generalized Markoff surfaces
Abstract
We establish lower bounds on the orbit cardinalities of SL(2,ℤ) on the 𝔽p-points of the generalized Markoff surfaces given by x2 + y2 + z2 − xyz = τ+2. We show that outside of the reductions of a certain exceptional finite set of ℚ̅-points found by Dubrovin and Mazzocco, every orbit has cardinality ≥ p. Together with a recent result of Nathaniel Kingsbury-Neuschotz, this establishes, for all but finitely many primes p, the classification conjectures of McCullough-Wanderley (2013), and gives a complete classification of the connected components of the Hurwitz stack of SL(2,𝔽p)-covers of elliptic curves, only branched above the origin. This extends earlier work of Bourgain-Gamburd-Sarnak and the author in the Markoff case τ= −2. Our method studies certain noncongruence modular curves attached to a SL(2,ℤ)-orbit. These modular curves parametrize elliptic curves with a PSL(2,𝔽p)-cover. We use this moduli interpretation to show that the noncongruence modular curves involved have bad reduction, as a cover of the j-line, at some place above p. This implies that the degree of the cover is ≥ p, which translates into the desired bound on orbits. Key inputs include a result of Bouw-Wewers on bad reduction of three-point PSL(2,𝔽p)-covers of the projective line, and the general stable reduction theory of Henrio, Obus, Raynaud, and Wewers.
Related
- Campana strong approximation on generalised Markoff surfaces
- Brauer groups of generalised Markoff surfaces
- Universal torsors over quartic del Pezzo surfaces and stable rationality
- Links and singularities of stable maps from $3$-manifolds to surfaces
- Generalized Iskovskikh surfaces and unirationality of del Pezzo surfaces of degree $2$
BibTeX
@misc{chen2026stable,
title = {{Stable reduction and large Nielsen orbits on generalized Markoff surfaces}},
author = {William Y. Chen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16407},
eprint = {2610.16407},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16407},
}
RIS
TY - UNPB
TI - Stable reduction and large Nielsen orbits on generalized Markoff surfaces
AU - William Y. Chen
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16407
UR - https://arxiv.org/abs/2610.16407
AB - We establish lower bounds on the orbit cardinalities of SL(2,ℤ) on the 𝔽_p-points of the generalized Markoff surfaces given by x^2 + y^2 + z^2 − xyz = τ+2. We show that outside of the reductions of a certain exceptional finite set of ℚ̅-points found by Dubrovin and Mazzocco, every orbit has cardinality ≥ p. Together with a recent result of Nathaniel Kingsbury-Neuschotz, this establishes, for all but finitely many primes p, the classification conjectures of McCullough-Wanderley (2013), and gives a complete classification of the connected components of the Hurwitz stack of SL(2,𝔽_p)-covers of elliptic curves, only branched above the origin. This extends earlier work of Bourgain-Gamburd-Sarnak and the author in the Markoff case τ= −2. Our method studies certain noncongruence modular curves attached to a SL(2,ℤ)-orbit. These modular curves parametrize elliptic curves with a PSL(2,𝔽_p)-cover. We use this moduli interpretation to show that the noncongruence modular curves involved have bad reduction, as a cover of the j-line, at some place above p. This implies that the degree of the cover is ≥ p, which translates into the desired bound on orbits. Key inputs include a result of Bouw-Wewers on bad reduction of three-point PSL(2,𝔽_p)-covers of the projective line, and the general stable reduction theory of Henrio, Obus, Raynaud, and Wewers.
ER -