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==Katz's definition==
A classical modular form of weight ''k'' can be thought of roughly as a function ''f'' from pairs (''E'',ω) of a complex elliptic curve with a holomorphic 1-form ω to complex numbers, such that ''f''(''E'',λω) = λ<sup>
Katz's definition of a ''p''-adic modular form is similar, except that ''E'' is now an elliptic curve over some algebra ''R'' (with ''p'' nilpotent) over the ring of integers ''R''<sub>0</sub> of a finite extension of the ''p''-adic numbers, such that ''E'' is not supersingular, in the sense that the Eisenstein series ''E''<sub>''p''–1</sub> is invertible at (''E'',ω). The ''p''-adic modular form ''f'' now takes values in ''R'' rather than in the complex numbers. The ''p''adic modular form also has to satisfy some other conditions analogous to the condition that a classical modular form should be holomorphic.
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