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A universal deformation ring with unexpected Krull dimension

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A well known result of Mazur gives a lower bound for the Krull dimension of the universal deformation ring associated to an absolutely irreducible residual representation in terms of the group cohomology of the adjoint representation. The question about equality — at least in the Galois case — also goes back to Mazur. In the general case the question about equality is the subject of Gouvêa’s “Dimension conjecture”. In this note we provide a counterexample to this conjecture. More precisely, we construct an absolutely irreducible residual representation with smooth universal deformation ring of strict greater Krull dimension as expected.

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  1. Böckle, G.: The generic fiber of the universal deformation space associated to a tame Galois representation. Manuscripta Math. 96(2), 231–246 (1998)

    Google Scholar 

  2. Boston, N.: Explicit deformation of Galois representations. Invent. Math. 103(1), 181–196 (1991)

    Google Scholar 

  3. Gouvêa, F.Q.: Deforming Galois representations: a survey. In: Seminar on Fermat’s Last Theorem (Toronto, ON, 1993–1994), CMS Conf. Proc., vol. 17, Amer. Math. Soc., Providence, RI, pp. 179–207 (1995)

  4. Gouvêa, F.Q.: Deformations of Galois representations. In: Arithmetic algebraic geometry (Park City, UT, 1999), IAS/Park City Math. Ser., vol. 9, Amer. Math. Soc., Providence, RI,( Appendix 1 by Mark Dickinson, Appendix 2 by Tom Weston and Appendix 3 by Matthew Emerton), pp. 233–406 (2001)

  5. Hida, H.: Galois representations into \({\text{GL}}_{2}({\mathbf{Z}}_{p}[[X]])\) attached to ordinary cusp forms. Invent. Math. 85(3), 545–613 (1986)

    Google Scholar 

  6. Mazur, B.: Deforming Galois representations, Galois groups over \({Q}\) (Berkeley, CA, 1987), Math. Sci. Res. Inst. Publ., vol. 16, pp. 385–437, Springer, New York (1989)

  7. Zubkov, A.N.: Non-abelian free pro-p-groups cannot be represented by 2-by-2 matrices. Siberian Math. J. 28, 742–747 (1987).

    Google Scholar 

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The idea for this counterexample arose out of my Diploma thesis under the mentoring of Prof. N. Naumann. I would like to use this opportunity to thank Prof. N. Naumann for introducing me to the deformation theory of Galois representations and for all mentoring advice during the development of my thesis.

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Correspondence to Johannes Sprang.

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Sprang, J. A universal deformation ring with unexpected Krull dimension. Math. Z. 275, 647–652 (2013).

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Mathematics Subject Classification (2000)