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. (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.

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".

References

• Coleman, Robert F. (1996), "Classical and overconvergent modular forms", Inventiones Mathematicae 124 (1): 215–241, doi:10.1007/s002220050051, ISSN 0020-9910, http://www.numdam.org/item/JTNB_1995__7_1_333_0/ 
• 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 
• 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 
• 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 
• 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, 0404145, ISBN 978-3-540-06483-1