Bianchi groups and automorphisms of rank-four $K3$ surfaces

Abstract

We relate the arithmetic of Bianchi groups to automorphism groups of Picard-rank-four $K3$ surfaces. Let $K$ be an imaginary quadratic field with ring of integers ${\mathcal{O}}_K$, and let $S_K={\mathrm{Herm}}_2({\mathcal{O}}_K)$ be the rank-four lattice of $2\times 2$ Hermitian matrices over ${\mathcal{O}}_K$, equipped with the quadratic form $2\det$. For an odd integer $N\ge 1$, we consider a very general $S_K(2N)$-polarized $K3$ surface $X_{K,2N}$. We prove that its automorphism group is commensurable with a level-$2N$ congruence subgroup of the Bianchi group. Furthermore, we also obtain exact realizations of congruence subgroups as full automorphism groups. Namely, if $K=\mathbb{Q}(i)$ or $K=\mathbb{Q}(\sqrt{-p})$, where $p$ is prime, then \[ \operatorname{Aut}(X_{K,2}) \cong PΓ_K(2). \] Thus, for every prime $p$, the projective principal congruence subgroup of level $2$ over ${\mathcal{O}}_{\mathbb{Q}(\sqrt{-p})}$ occurs as the full automorphism group of a Picard-rank-four $K3$ surface. At higher levels, the full automorphism group may be either the projective principal congruence subgroup or the strictly larger projective level subgroup ${\mathrm{Bi}}_K(2N)$, depending on the arithmetic of the primes dividing the level. We further explain these arithmetic groups geometrically. The surfaces $X_{K,2}$ arise as deformations of the Kummer surfaces $\mathrm{Km}(E_K\times E_K)$, yielding explicit double-cover models and genus-one fibrations. For $K=\mathbb{Q}(\sqrt{-2})$ and $K=\mathbb{Q}(\sqrt{-7})$, the automorphism group is generated by Mordell--Weil translations associated with genus-one fibrations coming from cusps, together with the covering involution. For $K=\mathbb{Q}(i)$ and $K=\mathbb{Q}(\sqrt{-3})$, we construct complete-intersection models in products of projective spaces and show that their automorphism groups are generated by deck involutions.

32 pages, 1 figure