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Numerical experiments on families of $p$-adic modular forms. (English)
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 [{\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)