Existence or absence of superregular boson clouds around extremal Kerr black holes and its connection with number theory

We discuss the (non)existence of superregular scalar clouds (i.e., bound states of a massive and complex-valued scalar field Ψ) around exact extremal (a=M) Kerr black holes (BHs) possessing bounded radial derivatives at the horizon (in Boyer-Lindquist coordinates) as opposed to similar cloud solutions that exist but with unbounded derivatives in the same coordinate system. The latter solutions have been reported recently both analytically and numerically. The superregular clouds cannot be obtained from the regular clouds around subextremal Kerr BHs (|a|