  Let $\pi$ traverse a sequence of cuspidal automorphic representations of
GL(2) with large prime level, unramified central character and bounded infinity
type. For G either of the groups GL(1) or PGL(2), let H(G) denote the assertion
that subconvexity holds for G-twists of the adjoint $L$-function of $\pi$, with
polynomial dependence upon the conductor of the twist. We show that H(GL(1))
implies H(PGL(2)).
  In geometric terms, H(PGL(2)) corresponds roughly to an instance of
arithmetic quantum unique ergodicity with a power savings in the error term,
H(GL(1)) to the special case in which the relevant sequence of measures is
tested against an Eisenstein series.
