Stable reduction and large Nielsen orbits on generalized Markoff surfaces

Source abstract

We establish lower bounds on the orbit cardinalities of $\text{SL}(2,\mathbb{Z})$ on the $\mathbb{F}_p$-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 $\overline{\mathbb{Q}}$-points found by Dubrovin and Mazzocco, every orbit has cardinality $\ge p$. 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 $\text{SL}(2,\mathbb{F}_p)$-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 $\text{SL}(2,\mathbb{Z})$-orbit. These modular curves parametrize elliptic curves with a $\text{PSL}(2,\mathbb{F}_p)$-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 $\ge p$, which translates into the desired bound on orbits. Key inputs include a result of Bouw-Wewers on bad reduction of three-point $\text{PSL}(2,\mathbb{F}_p)$-covers of the projective line, and the general stable reduction theory of Henrio, Obus, Raynaud, and Wewers.