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Geometric deformations of modular Galois representations

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Let f be a newform on Γ1(N), and V f the 2-dimensional p-adic Galois representation attached to f. Let S be a finite set of primes containing the primes divisors of Np, and denote by adV f the adjoint of V f . Under some mild conditions on f, we show that H1 g (Gℚ,S,adV f )=0.

Using this result, we show that the universal deformation space of the residual representation attached to f is smooth and 3-dimensional at the point corresponding to f. When f has finite slope, one can also use this result to give a deformation theoretic description of the “eigencurve” of Coleman-Mazur at the point corresponding to f.

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References

  1. Artin, M., Winters, G.: Degenerate fibres and stable reduction of curves. Topology 10, 373–383 (1971)

    Article  MathSciNet  Google Scholar 

  2. Carayol, H.: Sur les représentations galoisiennes modulo l attachés aux formes modulaires. Duke Math. J. 59, 785–801 (1989)

    Article  MathSciNet  Google Scholar 

  3. Conrad, B.: Modular forms, cohomology and the Ramanujan conjecture. To be published by Cambridge University Press

  4. Cline, E., Parshall, B., Scott, L.: Cohomology of finite groups of Lie type I. Publ. Math., Inst. Hautes Étud. Sci. 45, 169–191 (1975)

    Article  MathSciNet  Google Scholar 

  5. Coates, J., Sujatha, R., Wintenberger, J.-P.: On the Euler-Poincaré characteristics of finite dimensional p-adic Galois representations. Publ. Math., Inst. Hautes Étud. Sci. 93, 107–143 (2001)

    Article  MathSciNet  Google Scholar 

  6. De Shalit, E.: Hecke rings, and universal deformation rings. In: Modular forms and Fermat’s last theorem, pp. 421–445. Boston: Springer 1997

  7. Diamond, F., Im, J.: Modular forms and modular curves. Seminar on Fermat’s Last Theorem (Toronto, ON, 1993–94). Canadian Math. Soc. Conference Proceedings 17, pp. 39–131. Providence, RI: Am. Math. Soc. 1995

  8. Deligne, P., Rapoport, M.: Les schémas de modules de courbes elliptiques. In: Modular functions of one variable II, Antwerp. Lect. Notes Math. 349. Berlin, Heidelberg: Springer 1973

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

    Article  MathSciNet  Google Scholar 

  10. Faltings, G.: p-adic Hodge Theory. J. Am. Math. Soc. 1, 255–299 (1988)

    MathSciNet  Google Scholar 

  11. Faltings, G.: Crystalline cohomology of semi-stable curves, the ℚ p theory. J. Algebr. Geom. 6, 1–18 (1997)

    MathSciNet  Google Scholar 

  12. Faltings, G.: Almost étale extensions. In: Cohomologies p-adiques et applications arithmétiques, II. Astérisque 279, 185–270 (2002)

    MathSciNet  Google Scholar 

  13. Flach, M.: A finiteness theorem for the symmetric square of an elliptic curve. Invent. Math. 109, 307–327 (1992)

    Article  MathSciNet  Google Scholar 

  14. Flach, M.: Annihilation of Selmer groups for the adjoint representation of a modular form. In: Seminar on Fermat’s Last Theorem. CMS Conf. Proc. 17, 249–265 (1995)

    MathSciNet  Google Scholar 

  15. Fontaine, J.M.: Le corps des périodes p-adiques. In: Société Mathématique de France. Periodes p-adiques. Astérisque 223, 59–111 (1994)

    MathSciNet  Google Scholar 

  16. Gouvêa, F.: Deformations of Galois representations. In: Am. Math. Soc., Arithmetic algebraic geometry (Park City, UT, 1999). IAS/Park City Math. Ser. 9, 233–406. Princeton, NJ: AMS 2001

  17. Katz, N.: p-adic properties of modular schemes and modular forms. In: Modular functions of one variable III. Lect. Notes Math. 350, 70–190 (1973)

    Google Scholar 

  18. Kisin, M.: Overconvergent modular forms and the Fontaine-Mazur conjecture. Invent. Math. 153, 373–454 (2003)

    Article  MathSciNet  Google Scholar 

  19. Kisin, M.: Potential semi-stability of p-adic étale cohomology. Isr. J. Math. 129 157–173 (2002)

  20. Katz, N., Mazur, B.: Arithmetic moduli of elliptic curves. Princeton: Princeton University Press 1985

  21. Langlands, R.: Modular forms and ℓ-adic representations. In: Modular functions of one variable II. Lect. Notes Math. 349, 361–500 (1973)

    MathSciNet  Google Scholar 

  22. Nekovář, J.: On p-adic height pairings. Seminaire de Théorie des Nombres, Paris, 1990–91. Prog. Math. 108, 127–202 (1993)

  23. Nekovář, J.: p-adic Abel-Jacobi maps and p-adic heights. In: Algebraic Cycles (Banff, 1998). CRM Proc. Lect. Notes 24, 367–379 (2000)

    Google Scholar 

  24. Niziol, W.: On the image of p-adic regulators. Invent. Math. 127, 375–400 (1997)

    Article  MathSciNet  Google Scholar 

  25. Ribet, K.: Galois representations attached to eigenforms with Nebentypus. In: Modular functions of one variable V (Bonn 1976). Lect. Notes Math. 601, 17–51 (1977)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  27. Serre, J.-P.: Sur les groupes de congruence des variétés abeliennes II. Izv. Akad. Nauk SSSR 35, 731–735 (1971)

    Google Scholar 

  28. Tate, J.: p-divisible Groups. In: Proceedings of a Conference on Local Fields, pp. 158–183. Berlin, Heidelberg: Springer 1967

  29. Taylor, R., Wiles, A.: Ring-theoretic properties of certain Hecke algebras. Ann. Math. 141, 553–572 (1995)

    Article  Google Scholar 

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

    MathSciNet  Google Scholar 

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

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

    Article  MathSciNet  Google Scholar 

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Kisin, M. Geometric deformations of modular Galois representations. Invent. math. 157, 275–328 (2004). https://doi.org/10.1007/s00222-003-0351-2

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