... Kempf’s-Rousseau’s Instability theory This appears to be very important in the arithmetic invariant theory In fact, from the very beginning of the modern geometric invariant theory, to study the closedness ... in the imperfect case In connection with Borel’s questions, in one of the early papers on geometric invariant theory over non- algebraically closed fields, D Birkes [Bi] introduced the following ... an algebraically closed field However, since the very beginning of modern geometric invariant theory, as presented in [Mu], [MFK], there has been a need to consider the relative case of the theory...