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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie


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Volume 2018, Issue 740

Issues

Non-minimal modularity lifting in weight one

Frank Calegari
  • Department of Mathematics, University of Chicago, 5734 S. University Avenue, Room 208C, Chicago, Illinois 60637, USA
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Published Online: 2015-12-17 | DOI: https://doi.org/10.1515/crelle-2015-0071

Abstract

We prove an integral R=𝐓 theorem for odd two-dimensional p-adic representations of G𝐐 which are unramified at p, extending results of [5] to the non-minimal case. We prove, for any p, the existence of Katz modular forms modulo p of weight one which do not lift to characteristic zero.

References

  • [1]

    N. Bourbaki Éléments de mathématique. Chapitres 4 à 7, Masson, Paris 1981. Google Scholar

  • [2]

    K. Buzzard, Analytic continuation of overconvergent eigenforms, J. Amer. Math. Soc. 16 (2003), no. 1, 29–55, (electronic). CrossrefGoogle Scholar

  • [3]

    K. Buzzard, Computing weight 1 modular forms over 𝐂 and 𝐅¯p, preprint (2013), http://arxiv.org/abs/1205.5077.

  • [4]

    K. Buzzard and R. Taylor, Companion forms and weight one forms, Ann. of Math. (2) 149 (1999), no. 3, 905–919. CrossrefGoogle Scholar

  • [5]

    F. Calegari and D. Geraghty, Modularity lifting beyond the Taylor–Wiles method, preprint (2012), http://arxiv.org/abs/1207.4224.

  • [6]

    R. F. Coleman and B. Edixhoven, On the semi-simplicity of the Up-operator on modular forms, Math. Ann. 310 (1998), no. 1, 119–127. Google Scholar

  • [7]

    H. Darmon, F. Diamond and R. Taylor, Fermat’s last theorem, Elliptic curves, modular forms & Fermat’s last theorem (Hong Kong 1993), International Press, Cambridge (1997), 2–140. Google Scholar

  • [8]

    P. Deligne and M. Rapoport, Les schémas de modules de courbes elliptiques, Modular functions of one variable. II (Antwerp 1972), Lecture Notes in Math. 349, Springer, Berlin (1973), 143–316. Google Scholar

  • [9]

    F. Diamond, The Taylor–Wiles construction and multiplicity one, Invent. Math. 128 (1997), no. 2, 379–391. CrossrefGoogle Scholar

  • [10]

    B. Edixhoven, Comparison of integral structures on spaces of modular forms of weight two, and computation of spaces of forms mod 2 of weight one, J. Inst. Math. Jussieu 5 (2006), no. 1, 1–34. Google Scholar

  • [11]

    T. Gee, Automorphic lifts of prescribed types, Math. Ann. 350 (2011), no. 1, 107–144. CrossrefWeb of ScienceGoogle Scholar

  • [12]

    A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. I, Publ. Math. Inst. Hautes Études Sci. 20 (1964), Paper No. 259. Google Scholar

  • [13]

    N. M. Katz, p-adic properties of modular schemes and modular forms, Modular functions of one variable. III (Antwerp, 1972), Lecture Notes in Math. 350, Springer, Berlin (1973), 69–190. Google Scholar

  • [14]

    N. M. Katz, A result on modular forms in characteristic p, Modular functions of one variable. V (Bonn 1976), Lecture Notes in Math. 601, Springer, Berlin (1977), 53–61. Google Scholar

  • [15]

    C. Khare and J.-P. Wintenberger, Serre’s modularity conjecture. I, Invent. Math. 178 (2009), no. 3, 485–504. CrossrefGoogle Scholar

  • [16]

    C. Khare and J.-P. Wintenberger, Serre’s modularity conjecture. II, Invent. Math. 178 (2009), no. 3, 505–586. CrossrefGoogle Scholar

  • [17]

    M. Kisin, Moduli of finite flat group schemes, and modularity, Ann. of Math. (2) 170 (2009), no. 3, 1085–1180. CrossrefWeb of ScienceGoogle Scholar

  • [18]

    M. Kisin, The Fontaine–Mazur conjecture for GL2, J. Amer. Math. Soc. 22 (2009), no. 3, 641–690. Google Scholar

  • [19]

    B. Mazur, Modular curves and the Eisenstein ideal, Publ. Math. Inst. Hautes Études Sci. 47 (1977), 33–186. CrossrefGoogle Scholar

  • [20]

    G. J. Schaeffer, Hecke stability and weight 1 modular forms, preprint (2014), http://arxiv.org/abs/1406.0408.

  • [21]

    J. Shotton, Local deformation rings and a Breuil–Mézard conjecture when lp, preprint (2013), http://arxiv.org/abs/1309.1600.

  • [22]

    B. de Smit and H. W. Lenstra, Jr., Explicit construction of universal deformation rings, Modular forms and Fermat’s last theorem (Boston 1995), Springer, New York (1997), 313–326. Google Scholar

  • [23]

    A. Snowden, Singularities of ordinary deformation rings, preprint (2011), http://arxiv.org/abs/1111.3654.

  • [24]

    W. Stein, Modular forms, a computational approach, Grad. Stud. Math. 79, American Mathematical Society, Providence 2007. Google Scholar

  • [25]

    R. Taylor, Automorphy for some l-adic lifts of automorphic mod l Galois representations. II, Publ. Math. Inst. Hautes Études Sci. 108 (2008), 183–239. CrossrefGoogle Scholar

  • [26]

    R. Taylor and A. Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), no. 3, 553–572. CrossrefGoogle Scholar

  • [27]

    A. Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141 (1995), no. 3, 443–551. CrossrefGoogle Scholar

About the article

Received: 2014-07-08

Revised: 2015-01-07

Published Online: 2015-12-17

Published in Print: 2018-07-01


Funding Source: National Science Foundation

Award identifier / Grant number: DMS-1404620

The author was supported in part by NSF Grant DMS-1404620.


Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2018, Issue 740, Pages 41–62, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: https://doi.org/10.1515/crelle-2015-0071.

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