What
does this have to do with automorphic
forms?

f’(x) is essentially the q-function

restricted to the real axis

which exists as an automorphic *distribution.
*

This automorphy explains the
replication.

In fact Gerver’s points x=2p/q are
the orbit of x=1/2 under G0(4)

and q is cuspidal in exactly this of the three
cusps (0,1/2,∞) of G0(4)\H