Startseite Non-minimal modularity lifting in weight one
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

Non-minimal modularity lifting in weight one

  • Frank Calegari EMAIL logo
Veröffentlicht/Copyright: 17. Dezember 2015

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.

Award Identifier / Grant number: DMS-1404620

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

Acknowledgements

The debt this paper owes to [‘Modularity lifting beyond the Taylor–Wiles method’, preprint 2012, http://arxiv.org/abs/1207.4224] is clear, and the author thanks David Geraghty for many conversations. We thank Mark Kisin for the explaining a proof of Lemma 2.6, and we also thank Brian Conrad for a related proof of the same result in the context of rigid analytic geometry. We thank Toby Gee and Patrick Allen for several useful comments. We also thank Gabor Wiese for the original idea of proving modularity theorems in weight one by working in weight p.

References

[1] N. Bourbaki Éléments de mathématique. Chapitres 4 à 7, Masson, Paris 1981. Suche in Google Scholar

[2] K. Buzzard, Analytic continuation of overconvergent eigenforms, J. Amer. Math. Soc. 16 (2003), no. 1, 29–55, (electronic). 10.1090/S0894-0347-02-00405-8Suche in Google Scholar

[3] K. Buzzard, Computing weight 1 modular forms over 𝐂 and 𝐅¯p, preprint (2013), http://arxiv.org/abs/1205.5077. Suche in Google Scholar

[4] K. Buzzard and R. Taylor, Companion forms and weight one forms, Ann. of Math. (2) 149 (1999), no. 3, 905–919. 10.2307/121076Suche in Google Scholar

[5] F. Calegari and D. Geraghty, Modularity lifting beyond the Taylor–Wiles method, preprint (2012), http://arxiv.org/abs/1207.4224. Suche in Google Scholar

[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. 10.1007/s002080050140Suche in 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. 10.4310/CDM.1995.v1995.n1.a1Suche in 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. 10.1007/978-3-540-37855-6_4Suche in Google Scholar

[9] F. Diamond, The Taylor–Wiles construction and multiplicity one, Invent. Math. 128 (1997), no. 2, 379–391. 10.1007/s002220050144Suche in Google 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. 10.1017/S1474748005000113Suche in Google Scholar

[11] T. Gee, Automorphic lifts of prescribed types, Math. Ann. 350 (2011), no. 1, 107–144. 10.1007/s00208-010-0545-zSuche in Google 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. 10.1007/BF02684747Suche in 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. 10.1007/978-3-540-37802-0_3Suche in 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. 10.1007/BFb0063944Suche in Google Scholar

[15] C. Khare and J.-P. Wintenberger, Serre’s modularity conjecture. I, Invent. Math. 178 (2009), no. 3, 485–504. 10.1007/s00222-009-0205-7Suche in Google Scholar

[16] C. Khare and J.-P. Wintenberger, Serre’s modularity conjecture. II, Invent. Math. 178 (2009), no. 3, 505–586. 10.1007/s00222-009-0206-6Suche in Google Scholar

[17] M. Kisin, Moduli of finite flat group schemes, and modularity, Ann. of Math. (2) 170 (2009), no. 3, 1085–1180. 10.4007/annals.2009.170.1085Suche in Google Scholar

[18] M. Kisin, The Fontaine–Mazur conjecture for GL2, J. Amer. Math. Soc. 22 (2009), no. 3, 641–690. 10.1090/S0894-0347-09-00628-6Suche in Google Scholar

[19] B. Mazur, Modular curves and the Eisenstein ideal, Publ. Math. Inst. Hautes Études Sci. 47 (1977), 33–186. 10.1007/BF02684339Suche in Google Scholar

[20] G. J. Schaeffer, Hecke stability and weight 1 modular forms, preprint (2014), http://arxiv.org/abs/1406.0408. 10.1007/s00209-015-1477-9Suche in Google Scholar

[21] J. Shotton, Local deformation rings and a Breuil–Mézard conjecture when lp, preprint (2013), http://arxiv.org/abs/1309.1600. Suche in Google Scholar

[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. 10.1007/978-1-4612-1974-3_9Suche in Google Scholar

[23] A. Snowden, Singularities of ordinary deformation rings, preprint (2011), http://arxiv.org/abs/1111.3654. 10.1007/s00209-017-1912-1Suche in Google Scholar

[24] W. Stein, Modular forms, a computational approach, Grad. Stud. Math. 79, American Mathematical Society, Providence 2007. 10.1090/gsm/079Suche in 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. 10.1007/s10240-008-0015-2Suche in Google Scholar

[26] R. Taylor and A. Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), no. 3, 553–572. 10.2307/2118560Suche in Google Scholar

[27] A. Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141 (1995), no. 3, 443–551. 10.2307/2118559Suche in Google Scholar

Received: 2014-07-08
Revised: 2015-01-07
Published Online: 2015-12-17
Published in Print: 2018-07-01

© 2018 Walter de Gruyter GmbH, Berlin/Boston

Heruntergeladen am 10.10.2025 von https://www.degruyterbrill.com/document/doi/10.1515/crelle-2015-0071/html
Button zum nach oben scrollen