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Overconvergent modular forms and the Fontaine-Mazur conjecture

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Abstract

We prove a conjecture of Fontaine and Mazur on modularity of representations of G ℚ which are potentially semi-stable at p, for representations coming from finite slope, overconvergent eigenforms. We also give an application of our technique to a question of Gouvêa on overconvergence of certain modular forms. If ρ is a suitable representation of G ℚ,S on a two dimensional vector space over a finite field product of 𝔾 m and the S a finite set of primes – we construct a rigid analytic subspace of the universal deformation space of ρ, which is defined by a purely representation theoretic condition, and contains the eigencurve of Coleman-Mazur. Conjecturally, it is equal to this curve.

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References

  1. André, Y.: Filtrations de type Hasse-Arf et monodromie p-adique. Invent. Math. 148, 285–317 (2002)

    Article  Google Scholar 

  2. Buzzard, K., Dickinson, M., Shepherd-Barron, N., Taylor, R.: On icosohedral Artin representations. Duke Math. J. 109, 283–318 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Berger, L.: Représentations p-adiques et équations différentielles. Invent. Math. 148, 219–284 (2002)

    Article  MATH  Google Scholar 

  4. Bosch, S., Güntzer, U., Remmert, R.: Non-Archimedean analysis. A systematic approach to rigid analytic geometry. Grundlehren der Mathematischen Wissenschaften 261. Springer 1984

  5. Bloch, S., Kato, K.: L-functions and Tamagawa numbers of motives. The Grothendieck Festschrift, Vol. I., Progr. Math. 86, 333–400. Boston: Birkhäuser 1990

    Google Scholar 

  6. Boston, N.: Explicit Deformations of Galois Representations. Invent. Math. 103, 181–196 (1991)

    MathSciNet  MATH  Google Scholar 

  7. Böckle, G.: The generic fiber of the universal deformation space associated to a tame Galois representation. Manuscr. Math. 96, 231–246 (1998)

    Article  Google Scholar 

  8. Buzzard, K., Taylor, R.: Companion forms and weight one forms. Ann. Math. 149, 905–919 (1999)

    MathSciNet  MATH  Google Scholar 

  9. Coleman, R., Edixhoven, B.: On the semi-simplicity of the U p -operator on modular forms. Math. Ann. 310, 119–127 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  10. Colmez, P., Fontaine, J.M.: Construction des représentations p-adiques semi-stables. Invent. Math. 140, 1–43 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. Coleman, R., Mazur, B.: The Eigencurve. Galois Representations in Arithmetic Algebraic Geometry (Durham 1996). Lond. Math. Soc. Lect. Note Ser. 254, 1–113. Cambridge Univ. Press 1998

  12. Coleman, R.: Classical and overconvergent modular forms of higher level. J. Théor. Nombres Bordx. 9, 395–403 (1997)

    MATH  Google Scholar 

  13. Coleman, R.: p-adic Banach spaces, and families of modular forms. Invent. Math. 127, 417–479 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  14. Conrad, B.: Irreducible components of rigid spaces. Ann. Inst. Fourier 49, 473–541 (1999)

    MathSciNet  MATH  Google Scholar 

  15. Diamond, F., Flach, M., Guo, L.: Adjoint motives of modular forms and the Tamagawa number conjecture. Preprint, 109 pages (2001)

  16. Faltings, G.: Hodge-Tate structures and modular forms. Math. Ann. 278, 133–149 (1987)

    MathSciNet  MATH  Google Scholar 

  17. Fontaine, J.M., Mazur, B.: Geometric Galois Representations. Elliptic curves, modular forms, and Fermat’s last theorem (Hong Kong 1993), 41–78. Cambridge, MA: Internat. Press 1995

    Google Scholar 

  18. Fontaine, J.M.: Le corps des périodes p-adiques. Périodes p-adiques. Astérisque 223, 59–111 (1994)

    MATH  Google Scholar 

  19. Fontaine, J.M.: Représentations p-adiques semi-stables. Périodes p-adiques. Astérisque 223, 113–184 (1994)

    MATH  Google Scholar 

  20. Fontaine, J.M.: Représentations l-adiques potentiellement semi-stables. Périodes p-adiques. Astérisque 223, 321–347 (1994)

    MATH  Google Scholar 

  21. Fontaine, J.M.: Arithmétique des représentations galoisiennes p-adiques. Prépublication de l’Université d’Orsay. 2000–24. To appear in Asterisque

  22. Gouvêa, F.Q.: Arithmetic of p-adic Modular Forms. Lect. Notes Math. 1304. Springer 1988

  23. Hida, H.: Galois representations into GL2(ℤ p [[X]]) attached to ordinary cusp forms. Invent. Math. 85, 543–613 (1986)

    Google Scholar 

  24. Kedlaya, K.: Quasi-unipotence of overconvergent F-crystals. To appear in Ann. Math.

  25. Kiehl, R.: Der Endlichkeitssatz für eigentliche Abbildungen in der nichtarchimedischen Funktionentheorie. Invent. Math. 2, 191–214 (1967)

    MATH  Google Scholar 

  26. Kisin, M.: Geometric deformations of modular Galois representations. Preprint, 45 pages (2002)

  27. Mazur, B.: Deforming Galois representations. Galois groups over ℚ (Berkeley, CA, 1987). Math. Sci. Res. Inst. Publ. 16, 395–437. New York, Berlin: Springer 1989

  28. Mazur, B.: An introduction to the deformation theory of Galois representations. Modular forms and Fermat’s last theorem (Boston, MA, 1995), 243–311. New York: Springer 1997

    Google Scholar 

  29. Mebkhout, Z.: Analogue p-adique du théorème de Turrittin et le théorème de la monodromie p-adique. Invent. Math. 148, 319–351 (2002)

    Article  MATH  Google Scholar 

  30. Milne, J.: Arithmetic Duality Theorems. Perspect. Math. 1. Boston: Academic Press 1986

  31. Mazur, B., Wiles, A.: On p-adic analytic families of Galois representations. Compos. Math. 59, 231–264 (1986)

    MathSciNet  MATH  Google Scholar 

  32. Nekovář, J.: On p-adic height pairings. Séminaire de Théorie de Nombres, Paris 1990–91. Progr. Math. 108, 127–202. Boston: Birkhäuser 1993

    Google Scholar 

  33. Saito, T.: Modular forms and p-adic Hodge theory. Invent. Math. 129, 607–620 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  34. Sen, S.: The analytic variation of p-adic Hodge Structure. Ann. Math. 127, 647–661 (1988)

    Google Scholar 

  35. Sen, S.: An infinite dimensional Hodge-Tate theory. Bull. Soc. Math. Fr. 121, 13–34 (1993)

    MathSciNet  MATH  Google Scholar 

  36. Skinner, C., Wiles, A.: Residually reducible representations and modular forms. Publ. Math., Inst. Hautes Étud. Sci. 89, 5–126 (1999)

    Google Scholar 

  37. Skinner, C., Wiles, A.: Nearly ordinary deformations of residually irreducible representations. Ann. Fac. Sci. Toulouse, VI. Sér., Math. 10, 185–215 (2001)

    Google Scholar 

  38. Tate, J.: p-divisible Groups. Proceedings of a Conference on Local Fields, 158–183. Springer 1967

  39. Weston, T.: Algebraic Cycles, modular forms and Euler systems. J. Reine Angew. Math. 543, 103–145 (2002)

    MathSciNet  MATH  Google Scholar 

  40. Weston, T.: Geometric Euler systems for locally isotropic motives. To appear in Compos. Math.

  41. Wiles, A.: On ordinary λ-adic representations associated. to modular forms. Invent. Math. 94, 529–573 (1988)

    MATH  Google Scholar 

  42. Wiles, A.: Modular elliptic curves and Fermat’s last theorem. Ann. Math. 141, 443–551 (1995)

    MathSciNet  MATH  Google Scholar 

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  1. SFB 478, Westfälische Wilhelms Universität, Münster, Germany

    Mark Kisin

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Kisin, M. Overconvergent modular forms and the Fontaine-Mazur conjecture. Invent. math. 153, 373–454 (2003). https://doi.org/10.1007/s00222-003-0293-8

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