K3 surfaces over small number fields and Kummer constructions in families

2025/06/17 by Stefan Schröer, Schröer, Stefan
Computer Science · Mathematics · #11G05 #11R16 #14A20 #14D10 #14F20 #14J17 #14J28 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2506.14732

openalex publication_date 2025/06/17 · openalex created_date 2025/10/18 · openalex updated_date 2026/07/28

Abstract

We construct K3 surfaces over number fields that have good reduction everywhere. These do not exists over the rational numbers, by results of Abrashkin and Fontaine. Our surfaces exist for three quadratic number fields, and an infinite family of S₃-number fields. To this end we develop a theory of Kummer constructions in families, based on Romagny's notion of the effective models, here applied to sign involutions. This includes quotients of non-normal surfaces by infinitesimal group schemes in characteristic two, as developed by Kondo and myself. By the results of Brieskorn and Artin, the resulting families of normal K3 surfaces admit simultaneous resolutions of singularities, at least after suitable base-changes. These resolutions are constructed in two ways: First, by blowing-up families of one-dimensional centers that acquire embedded components. Second, by computing various l-adic local systems in terms of representation theory, and invoking Shepherd-Barrons results on the resolution functor, which is representable by a highly non-separated algebraic space.

Citations

• Generalizations of quasielliptic curves
• Supersingular reduction of Kummer surfaces in residue characteristic 2
• Loops on schemes and the algebraic fundamental group
• Elliptic curves over the rational numbers with semi-abelian reduction\n and two-division points
• There is no Enriques surface over the integers
• Kummer surfaces associated with group schemes
• Weyl group covers for Brieskorn's resolutions in all characteristics and the integral cohomology of G/P
• Enriques surfaces with normal K3-like coverings
• The p-radical closure of local noetherian rings
• Odd-dimensional cohomology with finite coefficients and roots of unity
• Elliptic Curves with Everywhere Good Reduction
• An error estimate for counting S₃-sextic number fields
• Effective models of group schemes
• Kummer surfaces for the selfproduct of the cuspidal rational curve
• Quotients by Groupoids
• Elliptic curves with good reduction away from 2: III

Related

• Kummer surfaces associated with group schemes
• Kummer surfaces for the selfproduct of the cuspidal rational curve
• Ordinary K3 surfaces over a finite field
• On K3 surfaces which dominate Kummer surfaces
• Inseparable Kummer surfaces