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 [{\it 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.
Reviewer: A.Dabrowski (Szczecin)