Lifting laws and arithmetic invariant theory

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.

Keywords

arithmetic invariant theory, prehomogenous vector spaces, higher composition laws

2010 Mathematics Subject Classification

Primary 16W22. Secondary 17C50.

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This work partially supported by the NSF through grant DMS-1401858 and the Schmidt Fund at the IAS.

Received 1 August 2017

Published 23 October 2018