We investigate the stability of the magnetohydrodynamic (MHD) equations in the magnetostrophic (inertialess) regime, a singular limit relevant to rapidly rotating planetary cores such as the Earth's. In this regime, the Navier-Stokes equations reduce to a diagnostic force balance between the Coriolis force, buoyancy, and the Lorentz force. This talk focuses on deriving rigorous necessary conditions for the growth of the magnetic energy, thereby establishing nonlinear "antidynamo" theorems. By employing an additional poloidal-toroidal decomposition, we identify regions in parameter space, defined by the nondimensional Rayleigh, Reynolds and Peclet numbers, where individual components of magnetic field growth are impossible, corresponding to stability of trivial solutions. A key technical result is the identification of the L^p norms of temperature gradients and dissipation as critical to the dynamo threshold. The methodology is also applied to the linearised MHD equations, referred to as the Moffatt-Loper equations. Finally, we discuss the evolution of Ohmic dissipation and provide necessary conditions corresponding to a new class of nonlinear strong- and weak-field dynamo theory.