Classical and overconvergent modular forms of higher level
Journal de Théorie des Nombres de Bordeaux, Tome 9 (1997) no. 2, pp. 395-403.

Nous définissons la notion de forme modulaire surconvergente pour Γ 1 (Np n )p est un nombre premier, N et n sont des entiers et N est premier à p. Nous démontrons que toute forme primitive surconvergente pour Γ 1 (Np n ), de poids k et dont la valeur propre de U p associée est de valuation strictement inférieure à k-1 est une forme modulaire au sens classique.

We define the notion overconvergent modular forms on Γ 1 (Np n ) where p is a prime, N and n are positive integers and N is prime to p. We show that an overconvergent eigenform on Γ 1 (Np n ) of weight k whose U p -eigenvalue has valuation strictly less than k-1 is classical.

@article{JTNB_1997__9_2_395_0,
     author = {Coleman, Robert F.},
     title = {Classical and overconvergent modular forms of higher level},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {395--403},
     publisher = {Universit\'e Bordeaux I},
     volume = {9},
     number = {2},
     year = {1997},
     doi = {10.5802/jtnb.210},
     zbl = {0942.11025},
     mrnumber = {1617406},
     language = {en},
     url = {jtnb.centre-mersenne.org/item/JTNB_1997__9_2_395_0/}
}
Robert F. Coleman. Classical and overconvergent modular forms of higher level. Journal de Théorie des Nombres de Bordeaux, Tome 9 (1997) no. 2, pp. 395-403. doi : 10.5802/jtnb.210. https://jtnb.centre-mersenne.org/item/JTNB_1997__9_2_395_0/

[1] Coleman R., Reciprocity Laws on Curves, Compositio 72 (1989), 205-235. | Numdam | MR 1030142 | Zbl 0706.14013

[2] Coleman R., Classical and Overconvergent modular forms, Invent. Math. 124 (1996), 215-241. | MR 1369416 | Zbl 0851.11030

[3] Edixhoven B., Stable models of modular curves and applications, Thesis, University of Utrecht (unpublished).

[4] Katz N., P-adic properties of modular schemes and modular forms Modular Functions of one Variable III, Springer Lecture Notes 350 (197), 69-190. | MR 447119 | Zbl 0271.10033

[5] Katz N. and B. Mazur, Arithmetic Moduli of Elliptic Curves, Annals of Math. Stud. 108, Princeton University Press, 1985. | MR 772569 | Zbl 0576.14026

[6] Mazur B. and A. Wiles, "Class fields and abelian extensions of Q", Invent. Math. 76 (1984), 179-330. | MR 742853 | Zbl 0545.12005

[7] Li W., "Newforms and functional equations, ", Math. Ann. 212 (1975), 285-315. | MR 369263 | Zbl 0278.10026

[8] Mazur B. and A. Wiles, "On p-adic analytic families of Galois representations", Compositio Math. 59 (1986), 231-264. | Numdam | MR 860140 | Zbl 0654.12008

[9] Ogg A., "On the eigenvalues of Hecke operators", Math. Ann. 179 (1969), 101-108. | MR 269597 | Zbl 0169.10102

[10] Coleman R., p-adic Banach spaces and families of modular forms, Invent. math. 127 (1992), 917-979. | MR 1431135 | Zbl 0918.11026

[11] Coleman R., p-adic Shimura Isomorphism and p-adic Periods of modular forms, Contemp. Math. 165 (1997), 21-51. | MR 1279600 | Zbl 0838.11033