Idoneal genera and K3 surfaces covering an Enriques surface

TL;DR

The paper develops a lattice-theoretic framework to classify idoneal genera via the Smith--Minkowski--Siegel mass formula and to characterize transcendental lattices of K3 surfaces covering Enriques surfaces. It proves finiteness of idoneal genera, provides explicit rank-based enumeration (noting no idoneal genera for rank $r>13$ and a total of many genera up to rank $13$), and furnishes a complete co-idoneal-lattice enumeration using Nikulin’s discriminant-form theory. Central to the geometric application is Keum’s criterion, restated and analyzed through primitive embeddings of the transcendental lattice $T$ into $\mathbf{\Lambda}^- = \mathbf{U}\oplus \mathbf{U}(2) \oplus \mathbf{E}_8(-2)$, with a detailed treatment of discriminant forms and embedding possibilities. The authors provide practical algorithms and extensive computational data (including ancillary data files) to determine which $T$ embed primitively and which lattices are co-idoneal, thereby classifying Enriques-covering K3 surfaces in a precise lattice-theoretic manner. This work advances the understanding of Enriques–K3 geometry by translating geometric covering questions into finite, computable lattice problems governed by discriminant forms and mass considerations.

Abstract

Idoneal genera are a generalization of Euler's idoneal numbers. We enumerate all idoneal genera by means of the Smith--Minkowski--Siegel mass formula. As an application, we classify transcendental lattices of K3 surfaces covering an Enriques surface.

Theorems & Definitions (73)

• Definition 1.1
• Remark 1.2
• Example 1.3
• Theorem 1.4: see §\ref{['subsec:finiteness']}
• Remark 1.5
• Theorem 1.7: Keum's criterion Keum90
• Definition 1.8
• Theorem 1.9: see §\ref{['subsec:proof-transc-thm']}
• Theorem 1.10
• Definition 2.1