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In mathematics, a '''''p''-adic modular form''' is a ''p''-adic analog of a [[modular form]], with coefficients that are ''p''-adic numbers rather than complex numbers. {{harvtxt|Serre|1973}} introduced ''p''-adic modular forms as limits of ordinary modular forms, and {{harvtxt|Katz|1973}} shortly afterwards gave a geometric and more general definition. Katz's ''p''-adic modular forms include as special cases '''classical ''p''-adic modular forms''', which are more or less ''p''-adic linear combinations of the usual "classical" modular forms, and '''overconvergent ''p''-adic modular forms'''.
==Serre's definition==
==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>–''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 also has to satisfy some other conditions.
==Overconvergent forms==
==References==
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