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.
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–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 Ep–1 is invertible at (E,ω). The p-adic modular form f now takes values in R rather then in the complex numbers. The padic 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 Ek–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, MR1369416
- Gouvêa, Fernando Q. (1988), Arithmetic of p-adic modular forms, Lecture Notes in Mathematics, vol. 1304, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0082111, ISBN 978-3-540-18946-6, MR1027593
- 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, vol. 350, Berlin, New York: Springer-Verlag, pp. 69–190, doi:10.1007/978-3-540-37802-0_3, ISBN 978-3-540-06483-1, MR0447119
- 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., vol. 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