Tamely ramified Hida theory
[Théorie de Hida modérément ramifiée]
Annales de l'Institut Fourier, Tome 52 (2002) no. 1, pp. 1-45.

Soit J 1 la variété jacobienne de la courbe modulaire associée à Γ 1 (Np),(N,p)=1 et soit J 0 l’autre variété associée à Γ 1 (N)Γ 0 (p). Nous étudions J 1 [p-1] comme un module de Hecke et de Galois. On trouve une relation entre une matrice de périodes p-adiques et la variation infinitésimale de l’opérateur U p .

Let J 1 be the Jacobian of the modular curve associated with Γ 1 (Np),(p,N)=1 and J 0 the one associated with Γ 1 (N)Γ 0 (p). We study J 1 [p-1] as a Hecke and Galois-module. We relate a certain matrix of p-adic periods to the infinitesimal deformation of the U p -operator.

DOI : 10.5802/aif.1875
Classification : 11F85
Keywords: modular curve, $p$-adic periods, Hecke operators
Mot clés : courbe modulaire, périodes $p$-adiques, opérateurs de Hecke
Goldberger, Assaf 1 ; Shalit, Ehud de 2

1 University of Massachussetts, Department of Mathematics, Amherst MA (USA)
2 Hebrew University, Institute of Mathematics, Giv'at-Ram 91904 Jerusalem (Israël)
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Goldberger, Assaf; Shalit, Ehud de. Tamely ramified Hida theory. Annales de l'Institut Fourier, Tome 52 (2002) no. 1, pp. 1-45. doi : 10.5802/aif.1875. http://www.numdam.org/articles/10.5802/aif.1875/

[ALe] A.O.L. Atkin; J. Lehner Hecke operators on Γ 0 (m), Math. Annalen, Volume 185 (1970), pp. 134-160 | DOI | MR | Zbl

[ALi] A.O.L. Atkin; W. Li Twists of newforms and pseudo-eigenvalues of W-operators, Inv. Math., Volume 43 (1978), pp. 221-244 | DOI | MR | Zbl

[B] S. Bosch Abelian varieties from the rigid-analytic viewpoint, Barsotti Symposium in Algebraic Geometry (1994), pp. 51-63 | Zbl

[BLR] S. Bosch; W. Lütkebohmert; M. Raynaud Néron models, Ergebnisse der Math., 3 folge, 21, Springer, 1990 | MR | Zbl

[DR] P. Deligne; M. Rapoport Schémas de modules de courbes elliptiques, LNM, 349, Springer, 1973 | MR | Zbl

[dS1] E. de Shalit On certain Galois representations related to the modular curve X 1 (p), Compositio Math., Volume 95 (1995), pp. 69-100 | Numdam | MR | Zbl

[dS2] E. de Shalit p-adic periods and modular symbols of elliptic curves of prime conductor, Inv. Math., Volume 121 (1995), pp. 225-255 | DOI | MR | Zbl

[dS3] E. de Shalit Néron models and p-adic uniformization of generalized Jacobians (In preparation)

[E] B. Edixhoven L'action de l'algebre de Hecke sur les groupes de composantes des jacobiennes des courbes modulaires est ``Eisenstein'', Astérisque, Volume 196-197 (1991), pp. 59-70 | MR | Zbl

[GS] R. Greenberg; G. Stevens p-adic L-functions and p-adic periods of modular forms, Inv. Math., Volume 111 (1993), pp. 407-447 | DOI | MR | Zbl

[H] H. Hida Galois representations into GL 2 ( p [[X]]) attached to ordinary cusp forms, Inv. Math., Volume 85 (1986), pp. 545-613 | DOI | MR | Zbl

[KM] N. Katz; B. Mazur Arithmetic moduli of elliptic curves, Ann. Math. Studies, 108, Princeton, 1985 | MR | Zbl

[M] B. Mazur Modular curves and the Eisenstein ideal, Publ. Math. I.H.E.S, Volume 47 (1977), pp. 33-186 | Numdam | MR | Zbl

[MT] B. Mazur; J. Tate Refined conjectures of "Birch and Swinnerton-Dyer type", Duke Math. J., Volume 54 (1987), pp. 711-750 | MR | Zbl

[MTT] B. Mazur; J. Tate; J. Teitelbaum On p-adic analogues of the conjectures of Birch and Swinerton-Dyer, Inv. Math., Volume 84 (1986), pp. 1-48 | DOI | MR | Zbl

[MW1] B. Mazur; A. Wiles Class fields of abelian extensions of Q, Inv. Math., Volume 76 (1984), pp. 179-330 | DOI | MR | Zbl

[MW2] B. Mazur; A. Wiles On p-adic analytic families of Galois representations, Compositio Math., Volume 59 (1986), pp. 231-264 | Numdam | MR | Zbl

[Ri] K. Ribet Congruence relations between modular forms, Proc. International Congress of Math., Volume 17 (1983), pp. 503-514 | Zbl

[SGA7] A. Grothendieck Modéles de Néron et monodromie (exposé IX), SGA 71 (LNM), Volume 288 (1972) | Zbl

[W] A. Wiles Modular elliptic curves and Fermat's last theorem, Ann. of Math., Volume 141 (1995), pp. 443-551 | DOI | MR | Zbl

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