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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>−''k''</sup>''f''(''E'',ω), and satisfying some additional conditions such as being holomorphic in some sense.
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.
==Overconvergent forms==
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