P-adic Modular Form

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. introduced ''p''-adic modular forms as limits of ordinary modular forms, and 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, which in turn include Hida's ordinary modular forms as special cases.

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_''p''×Z/(''p''–1)Z).

The ''p''-adic modular forms defined by Serre are special cases of those defined by Katz.

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^−''k''''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''₀ of a finite extension of the ''p''-adic numbers, such that ''E'' is not supersingular, in the sense that the Eisenstein series ''E''_''p''–1 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

Overconvergent ''p''-adic modular forms are similar to the modular forms defined by Katz, except that the form has to be defined on a larger collection of elliptic curves. Roughly speaking, the value of the Eisenstein series ''E''_''k''–1 on the form is no longer required to be invertible, but can be a smaller element of ''R''. Informally the series for the modular form converges on this larger collection of elliptic curves, hence the name "overconvergent".

References

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*{{Citation , last1=Serre , first1=Jean-Pierre , author1-link=Jean-Pierre Serre , title=Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, 1972) , publisher=Springer-Verlag , location=Berlin, New York , series=Lecture Notes in Math. , isbn=978-3-540-06483-1 , doi=10.1007/978-3-540-37802-0_4 , id=0404145 , year=1973 , volume=350 , chapter=Formes modulaires et fonctions zêta p-adiques , pages=191–268

Modular forms
p-adic numbers