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.

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