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==Serre's definition==
Serre defined a ''p''-adic modular form to be a formal power series with ''p''-adic coefficients that is a ''p''-adic limit of classical modular forms with integer coefficients. The weights of these classical modular forms need not be the same; in fact, if they are then the ''p''-adic modular form is nothing more than a linear combination of classical 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 classical modular forms (in fact a slight refinement gives a weight in '''Z'''<sub>''p''</sub>×'''Z'''/(''p''–1)'''Z''').
The ''p''-adic modular forms defined by Serre are special cases of those defined by Katz.
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