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: