  We provide a rigorous derivation of the precise late-time asymptotics for
solutions to the scalar wave equation on subextremal Kerr backgrounds,
including the asymptotics for projections to angular frequencies $\ell\geq 1$
and $\ell\geq 2$. The $\ell$-dependent asymptotics on Kerr spacetimes differ
significantly from the non-rotating Schwarzschild setting ("Price's law"). The
main differences with Schwarzschild are slower decay rates for higher angular
frequencies and oscillations along the null generators of the event horizon. We
introduce a physical space-based method that resolves the following two main
difficulties for establishing $\ell$-dependent asymptotics in the Kerr setting:
1) the coupling of angular modes and 2) a loss of ellipticity in the
ergoregion. Our mechanism identifies and exploits the existence of conserved
charges along null infinity via a time invertibility theory, which in turn
relies on new elliptic estimates in the full black hole exterior. This
framework is suitable for resolving the conflicting numerology in Kerr
late-time asymptotics that appears in the numerics literature.
