Persistence of Galois property of hypersurfaces over algebraic integers across other characteristics

TL;DR

This paper studies how the Galois property of hypersurfaces over algebraic integers remains stable under reduction modulo primes, and characterizes Galois groups for quartic hypersurfaces based on field characteristics.

Contribution

It demonstrates the persistence of the Galois property across various characteristics and provides criteria for Galois groups of quartic hypersurfaces.

Findings

Galois property persists under reduction for almost all primes.

Necessary and sufficient conditions for Galois groups of quartic hypersurfaces.

Characterization of Galois groups as projective linear groups depending on characteristic.

Abstract

In this paper, we investigate hypersurfaces defined over a ring of algebraic integers, and show that if the projection from a point induces a Galois extension over either a number field or the residue field associated with a prime ideal satisfying certain conditions, then the Galois property persists under reduction modulo the residue field associated with all but finitely many such prime ideals. Furthermore, for quartic hypersurfaces, we provide necessary and sufficient conditions for the Galois group to be given by a projective linear group, depending on the characteristic of the base field.