Lifting laws and arithmetic invariant theory

TL;DR

This paper explores lifting laws in prehomogeneous vector spaces, demonstrating their utility in solving problems in arithmetic invariant theory and extending Bhargava's parametrization theorems.

Contribution

It introduces new lifting laws and applies them to advance understanding in arithmetic invariant theory, including twisted versions of Bhargava's parametrizations.

Findings

Proved several new lifting laws for prehomogeneous vector spaces.

Applied lifting laws to solve specific problems in arithmetic invariant theory.

Extended Bhargava's parametrization theorems with twisted versions.

Abstract

In this paper we discuss lifting laws which, roughly, are ways of "lifting" elements of the open orbit of one prehomogeneous vector space to elements of the minimal nonzero orbit of another prehomogeneous vector space. We prove a handful of these lifting laws, and show how they can be used to help solve certain problems in arithmetic invariant theory. Of the results contained in this article are twisted versions of certain parametrization theorems of Bhargava.