Automorphisms of certain Niemeier lattices and Elliptic Fibrations

TL;DR

This paper completes the classification of elliptic fibrations on a specific K3 surface by identifying a previously missing fibration, using lattice theory and Weierstrass models.

Contribution

It proves the completeness of the elliptic fibration list on a singular K3 surface, adding a new fibration based on lattice automorphisms.

Findings

Identification of a 53rd elliptic fibration

Theoretical characterization of the fibration

Explicit Weierstrass model provided

Abstract

Nishiyama introduced a lattice theoretic classification of the elliptic fibrations on a $K3$ surface. In a previous paper we used his method to exhibit $52$ elliptic fibrations, up to isomorphisms, of the singular $K3$ surface of discriminant $-12$. We prove here that the list is complete with a $53$th fibration, thanks to a remark of Elkies and Sch\"{u}tt. We characterize the fibration both theoretically and with a Weierstrass model.