Numerical experiments on families of $$p$$-adic modular forms.(English)Zbl 0990.11029

Buell, D. A. (ed.) et al., Computational perspectives on number theory. Proceedings of a conference in honor of A. O. L. Atkin, Chicago, IL, USA, September 1995. Providence, RI: American Mathematical Society. AMS/IP Stud. Adv. Math. 7, 143-158 (1998).
The authors prove several special cases of the Gouvea-Mazur conjectures concerning the existence of congruences between over-convergent $$p$$-adic modular forms. The basic ingredients are: a numerical analysis of the geometry of the zero locus of the characteristic series of Atkin’s $$U$$ operator, and Koike’s formula [M. Koike, Nagoya Math. J. 56, 45-52 (1975; Zbl 0301.10026)]. They obtain lower bounds for the corresponding Newton polygon. They apply Koike’s formula to the problem of computing $$p$$-adic periods of modular forms and provide some examples.
For the entire collection see [Zbl 0881.00035].

MSC:

 11F33 Congruences for modular and $$p$$-adic modular forms 11Y35 Analytic computations

Zbl 0301.10026