Skip to Main Content

Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Modularity of certain potentially Barsotti-Tate Galois representations
HTML articles powered by AMS MathViewer

by Brian Conrad, Fred Diamond and Richard Taylor
J. Amer. Math. Soc. 12 (1999), 521-567


We show that certain potentially semistable lifts of modular mod $l$ representations are themselves modular. As a result we show that any elliptic curve over the rational numbers with conductor not divisible by 27 is modular.
  • Siegfried Bosch, Werner Lütkebohmert, and Michel Raynaud, Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 21, Springer-Verlag, Berlin, 1990. MR 1045822, DOI 10.1007/978-3-642-51438-8
  • Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982. MR 672956, DOI 10.1007/978-1-4684-9327-6
  • Henri Carayol, Sur les représentations $l$-adiques associées aux formes modulaires de Hilbert, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 3, 409–468 (French). MR 870690, DOI 10.24033/asens.1512
  • Henri Carayol, Sur les représentations galoisiennes modulo $l$ attachées aux formes modulaires, Duke Math. J. 59 (1989), no. 3, 785–801 (French). MR 1046750, DOI 10.1215/S0012-7094-89-05937-1
  • B. Conrad, Finite group schemes over bases with low ramification, to appear in Compositio Mathematica.
  • B. Conrad, Ramified deformation problems, to appear in Duke Math. Journal.
  • J. E. Cremona, Algorithms for modular elliptic curves, Cambridge University Press, Cambridge, 1992. MR 1201151
  • Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. I, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1981. With applications to finite groups and orders. MR 632548
  • H. Darmon, F. Diamond, R. Taylor, Fermat’s Last Theorem, in Current Developments in Mathematics, 1995, International Press, 1996, pp. 1–154.
  • B. de Smit, H. Lenstra, Explicit construction of universal deformation rings, in Modular Forms and Fermat’s Last Theorem (Boston, 1995), Springer-Verlag, 1997, pp. 313–326.
  • Fred Diamond, The refined conjecture of Serre, Elliptic curves, modular forms, & Fermat’s last theorem (Hong Kong, 1993) Ser. Number Theory, I, Int. Press, Cambridge, MA, 1995, pp. 22–37. MR 1363493
  • Fred Diamond, On deformation rings and Hecke rings, Ann. of Math. (2) 144 (1996), no. 1, 137–166. MR 1405946, DOI 10.2307/2118586
  • Fred Diamond, The Taylor-Wiles construction and multiplicity one, Invent. Math. 128 (1997), no. 2, 379–391. MR 1440309, DOI 10.1007/s002220050144
  • Fred Diamond and Richard Taylor, Nonoptimal levels of mod $l$ modular representations, Invent. Math. 115 (1994), no. 3, 435–462. MR 1262939, DOI 10.1007/BF01231768
  • Fred Diamond and Richard Taylor, Lifting modular mod $l$ representations, Duke Math. J. 74 (1994), no. 2, 253–269. MR 1272977, DOI 10.1215/S0012-7094-94-07413-9
  • Bas Edixhoven, The weight in Serre’s conjectures on modular forms, Invent. Math. 109 (1992), no. 3, 563–594. MR 1176206, DOI 10.1007/BF01232041
  • N. Elkies, Elliptic and modular curves over finite fields, and related computational issues, to appear in Computational Perspectives on Number Theory (J. Teitelbaum, ed.).
  • Jean-Marc Fontaine, Groupes $p$-divisibles sur les corps locaux, Astérisque, No. 47-48, Société Mathématique de France, Paris, 1977 (French). MR 0498610
  • Jean-Marc Fontaine, Le corps des périodes $p$-adiques, Astérisque 223 (1994), 59–111 (French). With an appendix by Pierre Colmez; Périodes $p$-adiques (Bures-sur-Yvette, 1988). MR 1293971
  • Jean-Marc Fontaine, Représentations $p$-adiques semi-stables, Astérisque 223 (1994), 113–184 (French). With an appendix by Pierre Colmez; Périodes $p$-adiques (Bures-sur-Yvette, 1988). MR 1293972
  • Jean-Marc Fontaine, Sur certains types de représentations $p$-adiques du groupe de Galois d’un corps local; construction d’un anneau de Barsotti-Tate, Ann. of Math. (2) 115 (1982), no. 3, 529–577 (French). MR 657238, DOI 10.2307/2007012
  • Jean-Marc Fontaine and Barry Mazur, Geometric Galois representations, Elliptic curves, modular forms, & Fermat’s last theorem (Hong Kong, 1993) Ser. Number Theory, I, Int. Press, Cambridge, MA, 1995, pp. 41–78. MR 1363495
  • K. Fujiwara, Deformation rings and Hecke algebras in the totally real case, preprint.
  • Paul Gérardin, Facteurs locaux des algèbres simples de rang $4$. I, Reductive groups and automorphic forms, I (Paris, 1976/1977) Publ. Math. Univ. Paris VII, vol. 1, Univ. Paris VII, Paris, 1978, pp. 37–77 (French). MR 680785
  • Groupes de monodromie en géométrie algébrique. I, Lecture Notes in Mathematics, Vol. 288, Springer-Verlag, Berlin-New York, 1972 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 I); Dirigé par A. Grothendieck. Avec la collaboration de M. Raynaud et D. S. Rim. MR 0354656
  • Schémas en groupes. I: Propriétés générales des schémas en groupes, Lecture Notes in Mathematics, Vol. 151, Springer-Verlag, Berlin-New York, 1970 (French). Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3); Dirigé par M. Demazure et A. Grothendieck. MR 0274458
  • Yasutaka Ihara, On modular curves over finite fields, Discrete subgroups of Lie groups and applications to moduli (Internat. Colloq., Bombay, 1973) Oxford Univ. Press, Bombay, 1975, pp. 161–202. MR 0399105
  • Nicholas M. Katz and Barry Mazur, Arithmetic moduli of elliptic curves, Annals of Mathematics Studies, vol. 108, Princeton University Press, Princeton, NJ, 1985. MR 772569, DOI 10.1515/9781400881710
  • Nicholas M. Katz and William Messing, Some consequences of the Riemann hypothesis for varieties over finite fields, Invent. Math. 23 (1974), 73–77. MR 332791, DOI 10.1007/BF01405203
  • B. Mazur, Deforming Galois representations, Galois groups over $\textbf {Q}$ (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 16, Springer, New York, 1989, pp. 385–437. MR 1012172, DOI 10.1007/978-1-4613-9649-9_{7}
  • David Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Published for the Tata Institute of Fundamental Research, Bombay by Oxford University Press, London, 1970. MR 0282985
  • Ravi Ramakrishna, On a variation of Mazur’s deformation functor, Compositio Math. 87 (1993), no. 3, 269–286. MR 1227448
  • Michel Raynaud, Schémas en groupes de type $(p,\dots , p)$, Bull. Soc. Math. France 102 (1974), 241–280 (French). MR 419467, DOI 10.24033/bsmf.1779
  • Takeshi Saito, Modular forms and $p$-adic Hodge theory, Invent. Math. 129 (1997), no. 3, 607–620. MR 1465337, DOI 10.1007/s002220050175
  • Jean-Pierre Serre, Le problème des groupes de congruence pour SL2, Ann. of Math. (2) 92 (1970), 489–527 (French). MR 272790, DOI 10.2307/1970630
  • Jean-Pierre Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), no. 4, 259–331 (French). MR 387283, DOI 10.1007/BF01405086
  • Jean-Pierre Serre, Cohomologie Galoisienne, Lecture Notes in Mathematics, Vol. 5, Springer-Verlag, Berlin-New York, 1973. Cours au Collège de France, Paris, 1962–1963; Avec des textes inédits de J. Tate et de Jean-Louis Verdier; Quatrième édition. MR 0404227, DOI 10.1007/978-3-662-21553-1
  • Jean-Pierre Serre, Linear representations of finite groups, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott. MR 0450380, DOI 10.1007/978-1-4684-9458-7
  • Jean-Pierre Serre, Trees, Springer-Verlag, Berlin-New York, 1980. Translated from the French by John Stillwell. MR 607504, DOI 10.1007/978-3-642-61856-7
  • Jean-Pierre Serre, Sur les représentations modulaires de degré $2$ de $\mathrm {Gal}(\overline {\mathbf {Q}}/\mathbf {Q})$, Duke Math. J. 54 (1987), no. 1, 179–230 (French). MR 885783, DOI 10.1215/S0012-7094-87-05413-5
  • Jean-Pierre Serre and John Tate, Good reduction of abelian varieties, Ann. of Math. (2) 88 (1968), 492–517. MR 236190, DOI 10.2307/1970722
  • Goro Shimura, Algebraic number fields and symplectic discontinuous groups, Ann. of Math. (2) 86 (1967), 503–592. MR 222048, DOI 10.2307/1970613
  • Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Kanô Memorial Lectures, No. 1, Iwanami Shoten Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. MR 0314766
  • J. T. Tate, $p$-divisible groups, Proc. Conf. Local Fields (Driebergen, 1966) Springer, Berlin, 1967, pp. 158–183. MR 0231827
  • Richard Taylor and Andrew Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), no. 3, 553–572. MR 1333036, DOI 10.2307/2118560
  • Andrew Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141 (1995), no. 3, 443–551. MR 1333035, DOI 10.2307/2118559
Similar Articles
  • Retrieve articles in Journal of the American Mathematical Society with MSC (1991): 11F80, 11G18
  • Retrieve articles in all journals with MSC (1991): 11F80, 11G18
Bibliographic Information
  • Brian Conrad
  • Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
  • MR Author ID: 637175
  • Email:
  • Fred Diamond
  • Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08854
  • Email:
  • Richard Taylor
  • Email:
  • Received by editor(s): April 1, 1998
  • Received by editor(s) in revised form: September 1, 1998
  • Additional Notes: The first author was supported by an N.S.F. Postdoctoral Fellowship, and would like to thank the Institute for Advanced Study for its hospitality. The second author was at M.I.T. during part of the research, and for another part was visiting Université de Paris-Sud supported by the C.N.R.S. The third author was supported by a grant from the N.S.F. All of the authors are grateful to Centre Émile Borel at the Institut Henri Poincaré for its hospitality at the $p$-adic semester.
  • © Copyright 1999 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 12 (1999), 521-567
  • MSC (1991): Primary 11F80; Secondary 11G18
  • DOI:
  • MathSciNet review: 1639612