Orbits on K3 Surfaces of Markoff Type

Abstract

Let $\mathcal{W}\subset {\mathbb{P}}^1\times {\mathbb{P}}^1\times {\mathbb{P}}^1$ be a surface given by the vanishing of a (2, 2, 2)-form. These surfaces admit three involutions coming from the three projections $\mathcal{W}\to {\mathbb{P}}^1\times {\mathbb{P}}^1$, so we call them tri-involutive K3 (TIK3) surfaces. By analogy with the classical Markoff equation, we say that $\mathcal{W}$ is of Markoff type (MK3) if it is symmetric in its three coordinates and invariant under double sign changes. An MK3 surface admits a group of automorphisms $\mathcal{G}$ generated by the three involutions, coordinate permutations, and sign changes. In this paper we study the $\mathcal{G}$-orbit structure of points on TIK3 and MK3 surfaces. Over finite fields, we study fibral connectivity and the existence of large orbits, analogous to work of Bourgain, Gamburd, Sarnak and others for the classical Markoff equation. For a particular 1-parameter family of MK3 surfaces ${\mathcal{W}}_k$, we compute the full $\mathcal{G}$-orbit structure of ${\mathcal{W}}_k({\mathbb{F}}_p)$ for all primes $p\le 113$, and we use this data as a guide to find many finite $\mathcal{G}$-orbits in ${\mathcal{W}}_k(\mathbb{C})$, including a family of orbits of size 288 parameterized by a curve of genus 9.

KEYWORDS:

• K3 surface
• arithmetic dynamics
• finite orbits
• orbits over finite fields

2010 AMS SUBJECT CLASSIFICATION:

• Primary 37P55
• Secondary 14J28
• 37F80
• 37P25
• 37P35

Acknowledgments

The authors would like to thank Philip Boalch, Wei Ho, Ram Murty, and Igor Shparlinski for their helpful advice and Peter Sarnak for his encouragement. We also thank the referees for their very careful reading and many helpful suggestions that greatly improved the paper. Calculations in this article were done using Magma [8] and GP-PARI [31].

Notes

1 We remark that although the generic member of the family of surfaces given by the vanishing of a (2, 2, 2)-form is a K3 surface, there are special members that for not. For example, the classical Markoff equation defines a rational surface.

2 See Definition 3.1, but briefly, non-degeneracy means that the three involutions are well-defined.

3 For the Cayley cubic ${\mathcal{M}}_{1,4}$, the points $(2,t,t)$ for positive integers t generate distinct orbits, and their union is ${\mathcal{M}}_{1,4}(\mathbb{Z})$.

4 We recall that an algebraic K3 surface is a smooth projective geometrically connected surface with trivial canonical bundle and irregularity zero. In this paper we work directly with equations of the form (6) satisfying the non-degeneracy condition, so it not important for our purposes that our surfaces are K3. However, for completeness, we show in Section B that minimal regular models of generic surfaces in our families are K3 surfaces.

5 In general, an (a, b, c)-form is a global section to ${\mathcal{O}}_{{({\mathbb{P}}^1)}^3}(a,b,c)$, or more prosaically, an (a, b, c)-form is a polynomial f in $K[X_1,X_2;Y_1,Y_2;Z_1,Z_2]$ satisfying $f(u{X}_1,u{X}_2;v{Y}_1,v{Y}_2;w{Z}_1,Z{w}_2)=u^a{v}^b{w}^{c}f(X_1,X_2;Y_1,Y_2;Z_1,Z_2)$.

6 We note that ${\pi }_{12},{\pi }_{13},{\pi }_{23}$ are finite if and only if their fibers are 0-dimensional, in which case they are maps of degree 2.

7 The reason that we do not use $\{\phi \in \mathcal{G}:\phi (\mathcal{F})=\mathcal{F}\}$, which is the full subgroup that leaves $\mathcal{F}$ invariant, is because when using $\mathcal{G}$ to move around points in fibers of $\mathcal{W}$, we will want to apply one generator at a time.

8 We do not include the set of generators $\text{Gen}(\mathcal{G})$ in the notation for the fibral automorphism groups, since it will generally be clear from context. For example, for a generic TIK3 surface, we take $\text{Gen}(\mathcal{G})=\{{\sigma }_1,{\sigma }_2,{\sigma }_3\}$. If $\mathcal{W}$ has extra symmetries, for example if $\mathcal{W}$ is one of the MK3 surfaces described in Section 6, then $\text{Gen}(\mathcal{G})$ will also include some coordinate permutations and sign shifts.

9 We recall that although we write F using affine coordinates to ease notation, in our calculations it always represents a (2, 2, 2) form. In particular, the polynomial $F(x_0,y_0,z)$ denotes a degree 2 homogeneous form in the variables Z₁ and Z₂; cf. Definition 3.2.

10 Lemma 5.6(a) contains a proof that if $\mathcal{W}$ is a non-degenerate TIK3 surface, then these algebraic sets are one-dimensional, although they need not be irreducible.

11 See Remark 5.7 for examples where ${\mathcal{C}}_{y_0,z_0}^{(1)}$ is reducible.

12 We remark that ${({\mathit{\mu }}_2^3)}_1⋊{\mathfrak{S}}_3$ is isomorphic to ${\mathfrak{S}}_4$, but for our applications the group ${\mathcal{G}}^{°}$ appears more naturally as the semi-direct product.

13 We note that for MK3-surfaces, we take $\text{Gen}(\mathcal{G})$ as described in (16), so $\Gamma$-connectivity of fibers on MK3-surfaces may employ coordinate permutations and sign changes, as well as the usual ${\sigma }_i$ automorphisms.

14 If we also allow the $\delta$-inversion involutions described in Remark 8.7, then the 4 singular points form a single orbit.

15 Note that we’re really working in ${\mathbb{P}}^1$, so we formally set $0^{-1}=\infty$ and ${\infty }^{-1}=0$.

16 Indeed, this is true in the ring $\mathbb{Z}[2^{-1},x,y,z,k]$.

17 We note that $\beta =0$ gives the contradiction $1=0$, while $\beta =1$ yields k = – 4 and an orbit with fewer than 64 elements.

18 We use the convenient notation $\mathit{v}[j]$ to denote the jth coordinate of the vector $\mathit{v}$.

19 Somewhat surprisingly, for this example we find that ${\mathcal{G}}^{\sigma }·P_1=\mathcal{G}·P_1={\widehat{\mathcal{G}}}^{°}{\mathcal{G}}^{\sigma }·P_1$ in ${\mathcal{W}}_{13}({\mathbb{F}}_{71})$.

20 We have listed more generators than needed. For example, ${\sigma }_3={\tau }_{23}°{\sigma }_2°{\tau }_{23}$, so $\text{Aut}({\mathcal{W}}_{x_0}^{(1)})=〈{\sigma }_2,{\tau }_{23},{ϵ}_{23}〉$, and similarly for the other fibers.

Additional information

Funding

Silverman’s research supported by Simons Collaboration Grant #712332.