THE BEHAVIOR OF SINGULAR QUADRATIC FORMS UNDER PURELY INSEPARABLE EXTENSIONS

Ahmed Laghribi (1) , Diksha Mukhija (1)

1 LML - Laboratoire de Mathématiques de Lens

Let F be a field of characteristic 2 and K a purely inseparable modular extension of F. Our aim in this paper is to give a complete classification of anisotropic semisingular F-quadratic forms φ that have over K a maximal Witt index and a defect index at least equal to the half of the dimension of the quasilinear part. The case of totally singular quadratic forms will be also treated. Our method also allows us to classify the forms φ under the unique hypothesis of maximality of the Witt index over K. This extends a recent result of Sobiech studying the hyperbolicity of nonsingular F-quadratic forms over K [17]. Based on our classifications, we are able to give necessary and sufficient conditions under which an anisotropic semisingular F-quadratic form has a given Witt index over K. We also study the quasi-hyperbolicity of semisingular F-quadratic forms over function fields of certain irreducible polynomials and extend to such forms many results established by the first author in [11].

Domaines: Théorie des nombres [math.NT]

Ahmed Laghribi, Diksha Mukhija. THE BEHAVIOR OF SINGULAR QUADRATIC FORMS UNDER PURELY INSEPARABLE EXTENSIONS. 2022. ⟨hal-03698339⟩