  In this paper, we study a class of linear-quadratic (LQ) mean field games of
controls with common noises and their corresponding $N$-player games. The
theory of mean field game of controls considers a class of mean field games
where the interaction is via the joint law of both the state and control. By
the stochastic maximum principle, we first analyze the limiting behavior of the
representative player and obtain his/her optimal control in a feedback form
with the given distributional flow of the population and its control. The mean
field equilibrium is determined by the Nash certainty equivalence (NCE) system.
Thanks to the common noise, we do not require any monotonicity conditions for
the solvability of the NCE system. We also study the master equation arising
from LQ mean field games of controls, which is a finite-dimensional
second-order parabolic equation. It can be shown that the master equation
admits a unique classical solution over an arbitrary time horizon without any
monotonicity conditions. Beyond that, we can solve the $N$-player games
directly by further assuming the non-degeneracy of the idiosyncratic noises. As
byproducts, we prove the quantitative convergence results from the $N$-player
game to the mean field game and the propagation of chaos property for the
related optimal trajectories.
