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Generalized Kummer surfaces over finite fields

Sergey Rybakov

TL;DR

This work refines Katsura’s insights on finite groups acting on abelian surfaces whose quotients are birational to K3 surfaces, focusing on generalized Kummer surfaces over finite fields. It develops a rigid-action framework via the rigid quotient Q[G]^{rig}, classifies possible group actions on abelian surfaces, and analyzes the resulting singularities and Néron–Severi lattices of the resolutions. The paper then connects these geometric data to arithmetic, deriving zeta functions and explicit Frobenius traces for supersingular generalized Kummer surfaces, and provides conditions for the existence of rigid and symplectic actions in various characteristics. The results yield concrete, nonnegative, even Frobenius traces and a detailed catalog of group-field combinations that yield supersingular generalized Kummer surfaces, advancing the arithmetic-geometric classification of these objects.

Abstract

In this paper, we prove a refinement of the Katsura theorem on finite groups acting on abelian surfaces such that the quotient is birational to a $K3$ surface. As an application, we compute traces of Frobenius on the Neron--Severi groups of supersingular generalized Kummer surfaces over finite fields.

Generalized Kummer surfaces over finite fields

TL;DR

This work refines Katsura’s insights on finite groups acting on abelian surfaces whose quotients are birational to K3 surfaces, focusing on generalized Kummer surfaces over finite fields. It develops a rigid-action framework via the rigid quotient Q[G]^{rig}, classifies possible group actions on abelian surfaces, and analyzes the resulting singularities and Néron–Severi lattices of the resolutions. The paper then connects these geometric data to arithmetic, deriving zeta functions and explicit Frobenius traces for supersingular generalized Kummer surfaces, and provides conditions for the existence of rigid and symplectic actions in various characteristics. The results yield concrete, nonnegative, even Frobenius traces and a detailed catalog of group-field combinations that yield supersingular generalized Kummer surfaces, advancing the arithmetic-geometric classification of these objects.

Abstract

In this paper, we prove a refinement of the Katsura theorem on finite groups acting on abelian surfaces such that the quotient is birational to a surface. As an application, we compute traces of Frobenius on the Neron--Severi groups of supersingular generalized Kummer surfaces over finite fields.
Paper Structure (17 sections, 72 equations)

This paper contains 17 sections, 72 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Central simple algebras over number fields
  5. Abelian varieties over finite fields
  6. Diedonné modules of abelian varieties
  7. Finite group actions on abelian varieties
  8. Rigid actions on abelian varieties
  9. An equivalence of categories.
  10. Finite subgroups of endomorphism algebras.
  11. Rigid and symplectic actions on abelian surfaces
  12. Singularities of $A/G$
  13. A refinement of the Katsura theorem.
  14. Traces of Frobenius
  15. Zeta functions of abelian varieties and generalized Kummer surfaces
  16. ...and 2 more sections

Theorems & Definitions (38)

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  • ...and 28 more