P-adic modular form: Difference between revisions

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{{DISPLAYTITLE:''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. {{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''', 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==

{{main|Overconvergent form}}

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==

*{{Citation | last1=Coleman | first1=Robert F. | title=Classical and overconvergent modular forms | url=http://dx.doi.org/10.1007/s002220050051 | doi=10.1007/s002220050051 | idmr={{MR|1369416}} | year=1996 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=124 | issue=1 | pages=215–241| bibcode=1996InMat.124..215C | s2cid=7995580 | url=http://www.numdam.org/item/JTNB_1995__7_1_333_0/ }}

*{{Citation | last1=Gouvêa | first1=Fernando Q.|author1-link= Fernando Q. Gouvêa | title=Arithmetic of p-adic modularModular formsForms | publisher=[[Springer-Verlag]] | ___location=Berlin, New York | series=Lecture Notes in Mathematics | isbn=978-3-540-18946-6 | doi=10.1007/BFb0082111 | idmr={{MR|1027593}} | year=1988 | volume=1304}}

*{{Citation | last1=Hida | first1=Haruzo | title=p-adic automorphic forms on Shimura varieties | publisher=[[Springer-Verlag]] | ___location=Berlin, New York | series=Springer Monographs in Mathematics | isbn=978-0-387-20711-7 |mr=2055355 | year=2004}}
*{{Citation | last1=Katz | first1=Nicholas M. | title=Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) | publisher=[[Springer-Verlag]] | ___location=Berlin, New York | series= Lecture Notes in Mathematics | isbn=978-3-540-06483-1 | doi=10.1007/978-3-540-37802-0_3 | idmr={{MR|0447119}} | year=1973 | volume=350 | chapter=p-adic properties of modular schemes and modular forms | pages=69–190}}

*{{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}}

[[Category:Modular forms]]

[[Category:p-adic numbers]]