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.

References

• 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