The inverse problem for universal deformation rings and the special linear group
Author:
Krzysztof Dorobisz
Journal:
Trans. Amer. Math. Soc. 368 (2016), 8597-8613
MSC (2010):
Primary 11G99; Secondary 11F80, 20C20
DOI:
https://doi.org/10.1090/tran6644
Published electronically:
March 1, 2016
MathSciNet review:
3551582
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: We present a solution to the inverse problem for universal deformation rings of group representations. Namely, we show that every complete noetherian local commutative ring with a finite residue field
can be realized as the universal deformation ring of a continuous linear representation of a profinite group. More specifically,
is the universal deformation ring of the natural representation of
in
, provided that
. We also check for which
an analogous result is true in case
and
.
- [Alp] J. L. Alperin, Local representation theory, Cambridge Studies in Advanced Mathematics, vol. 11, Cambridge University Press, Cambridge, 1986. Modular representations as an introduction to the local representation theory of finite groups. MR 860771 (87i:20002)
- [ATL]
R. Wilson et al., ATLAS of finite group representations v.3.
http://brauer.maths.qmul.ac.uk/Atlas/lin/L27/gap0/L27G1-Ar3bB0.g - [BCdS] Frauke M. Bleher, Ted Chinburg, and Bart de Smit, Inverse problems for deformation rings, Trans. Amer. Math. Soc. 365 (2013), no. 11, 6149-6165. MR 3091278, https://doi.org/10.1090/S0002-9947-2013-05848-5
- [BC1] Frauke M. Bleher and Ted Chinburg, Universal deformation rings need not be complete intersections, C. R. Math. Acad. Sci. Paris 342 (2006), no. 4, 229-232 (English, with English and French summaries). MR 2196003 (2007b:20053), https://doi.org/10.1016/j.crma.2005.12.006
- [BC2] Frauke M. Bleher and Ted Chinburg, Universal deformation rings need not be complete intersections, Math. Ann. 337 (2007), no. 4, 739-767. MR 2285736 (2008g:11093), https://doi.org/10.1007/s00208-006-0054-2
- [BC3] Frauke M. Bleher and Ted Chinburg, Universal deformation rings and cyclic blocks, Math. Ann. 318 (2000), no. 4, 805-836. MR 1802512 (2001m:20013), https://doi.org/10.1007/s002080000148
- [By] Jakub Byszewski, A universal deformation ring which is not a complete intersection ring, C. R. Math. Acad. Sci. Paris 343 (2006), no. 9, 565-568 (English, with English and French summaries). MR 2269865 (2007i:20051), https://doi.org/10.1016/j.crma.2006.09.015
- [Co] H. S. M. Coxeter, Regular complex polytopes, Cambridge University Press, London-New York, 1974. MR 0370328 (51 #6555)
- [dSL] Bart de Smit and Hendrik W. Lenstra Jr., Explicit construction of universal deformation rings, Modular forms and Fermat's last theorem (Boston, MA, 1995) Springer, New York, 1997, pp. 313-326. MR 1638482
- [EM] T. Eardley and J. Manoharmayum, The inverse deformation problem, ArXiv:1307.8356.
- [Go] Fernando Q. Gouvêa, Deformations of Galois representations, Arithmetic algebraic geometry (Park City, UT, 1999) IAS/Park City Math. Ser., vol. 9, Amer. Math. Soc., Providence, RI, 2001, pp. 233-406. Appendix 1 by Mark Dickinson, Appendix 2 by Tom Weston and Appendix 3 by Matthew Emerton. MR 1860043 (2003a:11061)
- [Hi] Haruzo Hida, Modular forms and Galois cohomology, Cambridge Studies in Advanced Mathematics, vol. 69, Cambridge University Press, Cambridge, 2000. MR 1779182 (2002b:11071)
- [Hi2] Haruzo Hida, Geometric modular forms and elliptic curves, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. MR 1794402 (2001j:11022)
- [Mat] Hideyuki Matsumura, Commutative ring theory, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1989. Translated from the Japanese by M. Reid. MR 1011461 (90i:13001)
- [Maz1] Barry Mazur, An introduction to the deformation theory of Galois representations, Modular forms and Fermat's last theorem (Boston, MA, 1995) Springer, New York, 1997, pp. 243-311. MR 1638481
- [Maz2]
B. Mazur, Deforming Galois representations, Galois groups over
(Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 16, Springer, New York, 1989, pp. 385-437. MR 1012172 (90k:11057), https://doi.org/10.1007/978-1-4613-9649-9_7
- [Ra] R. Rainone, On the inverse problem for deformation rings of representations, Master's thesis, Universiteit Leiden, June 2010.
- [Se] Jean-Pierre Serre, Linear representations of finite groups, Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott; Graduate Texts in Mathematics, Vol. 42. MR 0450380 (56 #8675)
- [Su]
J. G. Sunday, Presentations of the groups
and
, Canad. J. Math. 24 (1972), 1129-1131. MR 0311782 (47 #344)
Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 11G99, 11F80, 20C20
Retrieve articles in all journals with MSC (2010): 11G99, 11F80, 20C20
Additional Information
Krzysztof Dorobisz
Affiliation:
Mathematisch Instituut, Universiteit Leiden, P.O. Box 9512, 2300 RA Leiden, The Netherlands
Email:
kdorobisz@gmail.com
DOI:
https://doi.org/10.1090/tran6644
Keywords:
Deformations of group representations,
universal deformation rings,
inverse problem,
special linear group
Received by editor(s):
December 5, 2013
Received by editor(s) in revised form:
July 18, 2014, and October 24, 2014
Published electronically:
March 1, 2016
Article copyright:
© Copyright 2016
American Mathematical Society