Tracking the symmetries of $\mathbb Z_3$-orbifold K3s within the Mathieu groups

TL;DR

This work characterizes the symmetries of $Z_3$-orbifold limits of K3 surfaces by determining their holomorphic symplectic automorphism group as $$(\mathbb{Z}_3)^2\rtimes \mathbb{Z}_4$$ and embedding this group into the Mathieu groups $M_{12}$ and $M_{24}$ through a lattice-theoretic framework. Central to the method are two complementary constructions of the $Z_3$-orbifold K3 X from a $\mathbb{Z}_3$-symmetric torus, a detailed analysis of the integral cohomology via the Kummer-like lattice $P$ and its complement $K$, and Nikulin gluing to reconstruct $H^2(X,\mathbb{Z})$. The key technical novelty is a primitive embedding of $P(-1)$ into the Niemeier lattice $N$ of type $A_2^{12}$, enabling a faithful representation of the symmetries as automorphisms of $N$ and thus as permutations in $M_{12}$ and $M_{24}$. The results provide a concrete geometric realization that links K3 orbifold geometry, lattice theory, and the Mathieu groups, and they lay groundwork toward a geometric/CF-theoretic explanation of how Kummer and $Z_3$-orbifold symmetries combine to generate $M_{24}$. The approach advances the symmetry-surfeing program by extending Kummer-type techniques to new orbifold limits and by clarifying the lattice-embedding landscape behind Mathieu Moonshine.

Abstract

For $\mathbb Z_3$-orbifold limits of K3, we provide a counterpart to the extensive studies by Nikulin and others of the geometry and symmetries of classical Kummer surfaces. In particular, we determine the group of holomorphic symplectic automorphisms of $\mathbb Z_3$-orbifold limits of K3. We moreover track this group within two of the Mathieu groups, which involves a variation of Kondo's lattice techniques that Taormina and Wendland introduced earlier in their study of the symmetries of Kummer surfaces and the genesis of their symmetry surfing programme. Specifically, we realise the finite group of symplectic automorphisms of this class of K3 surfaces as a subgroup of the sporadic groups Mathieu 12 and Mathieu 24 in terms of permutations of 12, resp. 24 elements. As a proof of concept, we construct an embedding that yields the largest Mathieu group when the symmetry group of $\mathbb Z_3$-orbifold K3s is combined with all symmetries of Kummer surfaces.

Figures (7)

• Figure 1: The lattice $\hbox{span}_\mathbb Z\{1,\xi\}$ with generators $1$ and $\xi=\exp(2\pi i/3)$, lattice points $\bullet$, a fundamental cell (blue), and the three representatives $\bullet$ of fixed points under the $\mathbb Z_3$-action generated by $[z]\mapsto [\xi z]$ on $\mathbb C/\hbox{span}_\mathbb Z\{1,\xi\}$ within that cell. We include the half-open segments $[0,1)$ from $0$ to $1$ and $[0,\xi+1)$ from $0$ to $\xi+1$ in the fundamental cell, but not the rest of the boundary.
• Figure 2: Blow-up of $\mathbb C^2$ in the origin, followed by blow-ups in $P_1$, $P_2$.
• Figure 3: The affine plane $\mathbb F_3^2$ with two pairs of affine parallel lines highlighted.
• Figure 4: (a) Two copies $\mathcal{C}_{00}^{(1)}$, $\mathcal{C}_{00}^{(2)}$ of $\mathbb P^1$, depicted as (complex) lines, which intersect transversally in one point. (b) The Dynkin diagram of type $A_2$.
• Figure 5: A small window on the even Lorentzian self-dual lattice $\Gamma^{1,1}$ constructed in terms of two orthogonal copies of the one-dimensional lattice $\Lambda_{D_1}$. The lattice points represented by dots, filled and empty triangles, and circles correspond to vectors $(v_1,v_2)\in ([i],[i])$ for $[i]=[0], [i]=[v], [i]=[s]$ and $[i]=[c]$ respectively. The vectors $\upsilon_0$ and $\upsilon$ generate $\Gamma^{1,1}$.

Theorems & Definitions (24)

• Proposition 2.1.1
• Lemma 2.3.1
• Lemma 2.3.2
• Proposition 2.3.3
• Lemma 2.3.4
• Proposition 2.3.5
• Proposition 2.3.6
• Proposition 2.3.7
• Lemma 2.4.1
• Lemma 2.4.2