On the characteristic power series of the U operator
Annales de l'Institut Fourier, Volume 43 (1993) no. 2, p. 301-312

We show that the coefficients of the characteristic power series of Atkin’s U operator acting on overconvergent p-adic modular forms of weight k vary p-adically continuously as functions of k. Are they in fact Iwasawa functions of k ?

Soit A m (k) le coefficient m-ième de la série puissance caractéristique de l’opérateur U de Atkin agissant sur l’espace de formes modulaires p-adiques surconvergentes de poids k (k entier). Nous montrons que les fonctions a m (k) sont p-adiquement continues en k. Sont-elles des fonctions de Type Iwasawa en k ?

@article{AIF_1993__43_2_301_0,
     author = {Gouv\^ea, Fernando Q. and Mazur, Barry},
     title = {On the characteristic power series of the U operator},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {43},
     number = {2},
     year = {1993},
     pages = {301-312},
     doi = {10.5802/aif.1332},
     mrnumber = {1220270},
     zbl = {0779.11022},
     mrnumber = {94c:11039},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1993__43_2_301_0}
}
Gouvêa, Fernando Q.; Mazur, Barry. On the characteristic power series of the U operator. Annales de l'Institut Fourier, Volume 43 (1993) no. 2, pp. 301-312. doi : 10.5802/aif.1332. http://www.numdam.org/item/AIF_1993__43_2_301_0/

[Dwo] B. Dwork, On the zeta function of a hypersurface, Publ. Math. I. H. E. S., 12 (1962), 5-68. | Numdam | MR 28 #3039 | Zbl 0173.48601

[GM] F. Q. Gouvêa and B. Mazur, Families of modular eigenforms, Mathematics of Computation, 58 (1992), 793-806. | MR 93d:11049 | Zbl 0773.11030

[Gou] F. Q. Gouvêa, Continuity properties of p-adic modular forms, to appear in the Proceedings of the Workshop on Elliptic Curves and Related Topics held in St. Adèle, Québec, February, 1992. | Zbl 0829.11026

[Gou2] F. Q. Gouvêa, Arithmetic of p-adic modular forms, Lecture Notes in Mathematics, vol. 1304, Springer-Verlag, Berlin, Heidelberg, New York, 1988. | MR 91e:11056 | Zbl 0641.10024

[Kat] N. M. Katz, p-adic properties of modular schemes and modular forms, Modular Forms in One Variable III (SLN 350) (Berlin, Heidelberg, New York) (W. Kuijk and Jean-Pierre Serre, eds.), Springer-Verlag, 1973. | MR 56 #5434 | Zbl 0271.10033

[Lan] S. Lang, Cyclotomic fields I and II, Springer-Verlag, Berlin, Heidelberg, New York, 1989.

[Mon] P. Monsky, Formal cohomology : III : Fixed point theorems, Ann. of Math., (2), 93 (1971), 315-343. | MR 48 #296 | Zbl 0213.47501

[Ser] J.-P. Serre, Endomorphismes complètement continus des espaces de Banach p-adiques, Publ. Math. I.H.E.S., 12 (1962), 69-85. | Numdam | MR 26 #1733 | Zbl 0104.33601

[Ser] J.-P. Serre, Formes modulaires et fonctions zêta p-adiques, Modular Forms in One Variable III (SLN 350) (Berlin, Heidelberg, New York) (W. Kuijk and Jean-Pierre Serre, eds.), Springer-Verlag, 1973. | MR 53 #7949a | Zbl 0277.12014