P-adic modular form

Last updated April 30, 2019

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. Serre (1973) introduced p-adic modular forms as limits of ordinary modular forms, and 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.

In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections with number theory. Modular forms appear in other areas, such as algebraic topology, sphere packing, and string theory.

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^−kf(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 R₀ 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".

Eisenstein series, named after German mathematician Gotthold Eisenstein, are particular modular forms with infinite series expansions that may be written down directly. Originally defined for the modular group, Eisenstein series can be generalized in the theory of automorphic forms.

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References

Coleman, Robert F. (1996), "Classical and overconvergent modular forms", Inventiones Mathematicae , 124 (1): 215–241, doi:10.1007/s002220050051, ISSN 0020-9910, MR 1369416
Gouvêa, Fernando Q. (1988), Arithmetic of p-adic modular forms, Lecture Notes in Mathematics, 1304, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0082111, ISBN 978-3-540-18946-6, MR 1027593
Hida, Haruzo (2004), p-adic automorphic forms on Shimura varieties, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-0-387-20711-7, MR 2055355
Katz, Nicholas M. (1973), "p-adic properties of modular schemes and modular forms", Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Lecture Notes in Mathematics, 350, Berlin, New York: Springer-Verlag, pp. 69–190, doi:10.1007/978-3-540-37802-0_3, ISBN 978-3-540-06483-1, MR 0447119
Serre, Jean-Pierre (1973), "Formes modulaires et fonctions zêta p-adiques", Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, 1972), Lecture Notes in Math., 350, Berlin, New York: Springer-Verlag, pp. 191–268, doi:10.1007/978-3-540-37802-0_4, ISBN 978-3-540-06483-1, 0404145

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