Let p be prime. A function
is
said to be a modular function on the congruence subgroup
if it
satisfies the following conditions:
- (i)
- f is meromorphic on
,
- (ii)
-
for all
,
- (iii)
- f is meromorphic at the cusps
for
; i.e.
has the form
for some
where
.
As in the case of modular forms, condition (iii) amounts to the
following two conditions:

