Duke Mathematical Journal

On a conjecture of Conrad, Diamond, and Taylor

David Savitt

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We prove a conjecture of Conrad, Diamond, and Taylor on the size of certain deformation rings parametrizing potentially Barsotti-Tate Galois representations. To achieve this, we extend results of Breuil and Mézard (classifying Galois lattices in semistable representations in terms of "strongly divisible modules") to the potentially crystalline case in Hodge-Tate weights (0, 1). We then use these strongly divisible modules to compute the desired deformation rings. As a corollary, we obtain new results on the modularity of potentially Barsotti-Tate representations.

Article information

Duke Math. J., Volume 128, Number 1 (2005), 141-197.

First available in Project Euclid: 17 May 2005

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11F80: Galois representations
Secondary: 14L15: Group schemes


Savitt, David. On a conjecture of Conrad, Diamond, and Taylor. Duke Math. J. 128 (2005), no. 1, 141--197. doi:10.1215/S0012-7094-04-12816-7. https://projecteuclid.org/euclid.dmj/1116361230

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  • C. Breuil, Construction de représentations $p$-adiques semi-stables, Ann. Sci. École Norm. Sup. (4) 31 (1998), 281--327.
  • --. --. --. --., Représentations semi-stables et modules fortement divisibles, Invent. Math. 136 (1999), 89--122.
  • --. --. --. --., Groupes $p$-divisibles, groupes finis et modules filtrés, Ann. of Math. (2) 152 (2000), 489--549.
  • --. --. --. --., (p)-adic Hodge theory, deformations and local Langlands, notes from ``Advanced course on modular forms and (p)-adic Hodge theory,'' Centre de Recerca Matemàtica, Bellaterra, Spain, 2001, quadern no. 20(1), 7--82., available at http://www.crm.es
  • C. Breuil, B. Conrad, F. Diamond, and R. Taylor, On the modularity of elliptic curves over $\mathbf{Q}$: Wild (3)-adic exercises, J. Amer. Math. Soc. 14 (2001), 843--939.
  • C. Breuil and A. Mézard, Multiplicités modulaires et représentations de ${{\rm {{G}{L}}}}_2(\mathbf{Z}_p)$ et de $\mathrm{Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q}_p)$ en $l=p$, appendix by G. Henniart, Duke Math. J. 115 (2002), 205--310.
  • H. Carayol, Sur les représentations $l$-adiques associées aux formes modulaires de Hilbert, Ann. Sci. École Norm. Sup. (4) 19 (1986), 409--468.
  • R. Coleman and A. Iovita, Revealing hidden structures, preprint, 2003, http://www.mathstat.concordia.ca/faculty/iovita/research.html
  • P. Colmez and J.-M. Fontaine, Construction des représentations (p)-adiques semi-stables, Invent. Math. 140 (2000), 1--43.
  • B. Conrad, F. Diamond, and R. Taylor, Modularity of certain potentially Barsotti-Tate Galois representations, J. Amer. Math. Soc. 12 (1999), 521--567.
  • B. Edixhoven, The weight in Serre's conjectures on modular forms, Invent. Math. 109 (1992), 563--594.
  • G. Faltings, Hodge-Tate structures and modular forms, Math. Ann. 278 (1987), 133--149.
  • --. --. --. --., Crystalline cohomology of semistable curve --.-the $\mathbf{Q}_{p}$-theory, J. Algebraic Geom. 6 (1997), 1--18.
  • J.-M. Fontaine, ``Representations $p$-adiques semi-stables'' in Périodes $p$-adiques (Bures-sur-Yvette, France, 1988), Astérisque 223, Soc. Math. France, Montrouge, 1994, 113--184.
  • J.-M. Fontaine and B. Mazur, ``Geometric Galois representations'' in Elliptic Curves, Modular Forms, & Fermat's Last Theorem (Hong Kong, 1993), ed. J. Coates and S.-T. Yau, Ser. Number Theory 1, International Press, Cambridge, Mass., 1995, 41--78.
  • B. H. Gross, A tameness criterion for Galois representations associated to modular forms (mod $p$), Duke Math. J. 61 (1990), 445--517.
  • M. Kisin, Moduli of finite flat group schemes and modularity, preprint, 2004, http://www.math.uchicago.edu/~kisin/preprints.html
  • M. Raynaud, Schémas en groupes de type $(p,\ldots,p)$, Bull. Soc. Math. France 102 (1974), 241--280.
  • T. Saito, Modular forms and $p$-adic Hodge theory, Invent. Math. 129 (1997), 607--620.
  • D. Savitt, Modularity of some potentially Barsotti-Tate Galois representations, Compos. Math. 140 (2004), 31--63.
  • J. T. Tate, ``$p$-divisible groups'' in Proceedings of a Conference on Local Fields (Driebergen, Netherlands, 1966), ed. T. A. Springer, Springer, Berlin, 1967, 158--183.
  • --. --. --. --., ``Finite flat group schemes'' in Modular Forms and Fermat's Last Theorem (Boston, 1995), ed. G. Cornell, J. H. Silverman, and G. Stevens, Springer, New York, 1997, 121--154.