Serre defined a ''p''-adic modular form to be a formal power series that is a limit of ordinary modular forms with integer coefficients. The weights of these ordinary modular forms need not be the same; in fact, if they are then the ''p''-adic modular form is just a linear combination of ordinary modular forms. In general the weight of a ''p''-adic modular form is a ''p''-adic number, given by the limit of the weights of the ordinary modular forms.
The ''p''-adic modular forms defined by Serre are special cases of those defined by Katz.
