Tamely ramified Hida theory

Assaf Goldberger[1]; Ehud de Shalit[2]

  • [1] University of Massachussetts, Department of Mathematics, Amherst MA (USA)
  • [2] Hebrew University, Institute of Mathematics, Giv'at-Ram 91904 Jerusalem (Israël)

Annales de l’institut Fourier (2002)

  • Volume: 52, Issue: 1, page 1-45
  • ISSN: 0373-0956

Abstract

top
Let J 1 be the Jacobian of the modular curve associated with Γ 1 ( N p ) , ( 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.

How to cite

top

Goldberger, Assaf, and Shalit, Ehud de. "Tamely ramified Hida theory." Annales de l’institut Fourier 52.1 (2002): 1-45. <http://eudml.org/doc/115973>.

@article{Goldberger2002,
abstract = {Let $J_1$ be the Jacobian of the modular curve associated with $\Gamma _1(Np),\; (p,N)=1$ and $J_0$ the one associated with $\Gamma _1(N)\cap \Gamma _0(p)$. 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.},
affiliation = {University of Massachussetts, Department of Mathematics, Amherst MA (USA); Hebrew University, Institute of Mathematics, Giv'at-Ram 91904 Jerusalem (Israël)},
author = {Goldberger, Assaf, Shalit, Ehud de},
journal = {Annales de l’institut Fourier},
keywords = {modular curve; $p$-adic periods; Hecke operators; p-adic periods; Hida theory; deformation theory},
language = {eng},
number = {1},
pages = {1-45},
publisher = {Association des Annales de l'Institut Fourier},
title = {Tamely ramified Hida theory},
url = {http://eudml.org/doc/115973},
volume = {52},
year = {2002},
}

TY - JOUR
AU - Goldberger, Assaf
AU - Shalit, Ehud de
TI - Tamely ramified Hida theory
JO - Annales de l’institut Fourier
PY - 2002
PB - Association des Annales de l'Institut Fourier
VL - 52
IS - 1
SP - 1
EP - 45
AB - Let $J_1$ be the Jacobian of the modular curve associated with $\Gamma _1(Np),\; (p,N)=1$ and $J_0$ the one associated with $\Gamma _1(N)\cap \Gamma _0(p)$. 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.
LA - eng
KW - modular curve; $p$-adic periods; Hecke operators; p-adic periods; Hida theory; deformation theory
UR - http://eudml.org/doc/115973
ER -

References

top
  1. A.O.L. Atkin, J. Lehner, Hecke operators on Γ 0 ( m ) , Math. Annalen 185 (1970), 134-160 Zbl0177.34901MR268123
  2. A.O.L. Atkin, W. Li, Twists of newforms and pseudo-eigenvalues of W-operators, Inv. Math. 43 (1978), 221-244 Zbl0369.10016MR508986
  3. S. Bosch, Abelian varieties from the rigid-analytic viewpoint, Barsotti Symposium in Algebraic Geometry (1994), 51-63, Academic Press Zbl0836.14028
  4. S. Bosch, W. Lütkebohmert, M. Raynaud, Néron models, 21 (1990), Springer Zbl0705.14001MR1045822
  5. P. Deligne, M. Rapoport, Schémas de modules de courbes elliptiques, 349 (1973), Springer Zbl0281.14010MR337993
  6. E. de Shalit, On certain Galois representations related to the modular curve X 1 ( p ) , Compositio Math. 95 (1995), 69-100 Zbl0853.11045MR1314697
  7. E. de Shalit, p -adic periods and modular symbols of elliptic curves of prime conductor, Inv. Math. 121 (1995), 225-255 Zbl1044.11576MR1346205
  8. E. de Shalit, Néron models and p-adic uniformization of generalized Jacobians 
  9. B. Edixhoven, L'action de l'algebre de Hecke sur les groupes de composantes des jacobiennes des courbes modulaires est ``Eisenstein'', Astérisque 196-197 (1991), 59-70 Zbl0781.14019MR1141457
  10. R. Greenberg, G. Stevens, p -adic L -functions and p -adic periods of modular forms, Inv. Math. 111 (1993), 407-447 Zbl0778.11034MR1198816
  11. H. Hida, Galois representations into G L 2 ( ℤ p [ [ X ] ] ) attached to ordinary cusp forms, Inv. Math. 85 (1986), 545-613 Zbl0612.10021MR848685
  12. N. Katz, B. Mazur, Arithmetic moduli of elliptic curves, 108 (1985), Princeton Zbl0576.14026MR772569
  13. B. Mazur, Modular curves and the Eisenstein ideal, Publ. Math. I.H.E.S 47 (1977), 33-186 Zbl0394.14008MR488287
  14. B. Mazur, J. Tate, Refined conjectures of "Birch and Swinnerton-Dyer type", Duke Math. J. 54 (1987), 711-750 Zbl0636.14004MR899413
  15. B. Mazur, J. Tate, J. Teitelbaum, On p-adic analogues of the conjectures of Birch and Swinerton-Dyer, Inv. Math. 84 (1986), 1-48 Zbl0699.14028MR830037
  16. B. Mazur, A. Wiles, Class fields of abelian extensions of Q , Inv. Math. 76 (1984), 179-330 Zbl0545.12005MR742853
  17. B. Mazur, A. Wiles, On p -adic analytic families of Galois representations, Compositio Math. 59 (1986), 231-264 Zbl0654.12008MR860140
  18. K. Ribet, Congruence relations between modular forms, Proc. International Congress of Math. 17 (1983), 503-514 Zbl0575.10024
  19. A. Grothendieck, Modéles de Néron et monodromie (exposé IX), SGA 71 288 (1972), Springer Zbl0248.14006
  20. A. Wiles, Modular elliptic curves and Fermat's last theorem, Ann. of Math. 141 (1995), 443-551 Zbl0823.11029MR1333035

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.