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

We define the notion overconvergent modular forms on ${\Gamma }_{1}\left(N{p}^{n}\right)$ where $p$ is a prime, $N$ and $n$ are positive integers and $N$ is prime to $p$. We show that an overconvergent eigenform on ${\Gamma }_{1}\left(N{p}^{n}\right)$ of weight $k$ whose ${U}_{p}$-eigenvalue has valuation strictly less than $k-1$ is classical.

Nous définissons la notion de forme modulaire surconvergente pour ${\Gamma }_{1}\left(N{p}^{n}\right)$$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 ${\Gamma }_{1}\left(N{p}^{n}\right)$, 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.

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author = {Robert F. Coleman},
title = {Classical and overconvergent modular forms of higher level},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {395--403},
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Robert F. Coleman. Classical and overconvergent modular forms of higher level. Journal de théorie des nombres de Bordeaux, Volume 9 (1997) no. 2, pp. 395-403. 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 | Zbl

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

[3] Edixhoven B., Stable models of modular curves and applications, Thesis, University of Utrecht (unpublished).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

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

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

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

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

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

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

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