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DEFORMATION CONDITIONS FOR PSEUDOREPRESENTATIONS

Part of:Algebraic number theory: local and $p$-adic fieldsDiscontinuous groups and automorphic formsRings with polynomial identity

Published online by Cambridge University Press:  18 July 2019

PRESTON WAKEOpen the ORCID record for PRESTON WAKE[Opens in a new window] and
CARL WANG-ERICKSONOpen the ORCID record for CARL WANG-ERICKSON[Opens in a new window]
PRESTON WAKE
Affiliation:
Institute for Advanced Study, Princeton, NJ 08540, USA; pwake@ias.edu
CARL WANG-ERICKSON
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK; c.wang-erickson@imperial.ac.uk

Abstract

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Given a property of representations satisfying a basic stability condition, Ramakrishna developed a variant of Mazur’s Galois deformation theory for representations with that property. We introduce an axiomatic definition of pseudorepresentations with such a property. Among other things, we show that pseudorepresentations with a property enjoy a good deformation theory, generalizing Ramakrishna’s theory to pseudorepresentations.

MSC classification

Primary:11F70: Representation-theoretic methods; automorphic representations over local and global fields
Secondary:16R99: None of the above, but in this section11S20: Galois theory

Information

Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 7 , 2019, e20
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2019

References

Allen, P. B., Calegari, F., Caraiani, A., Gee, T., Helm, D., Le Hung, B. V., Newton, J., Scholze, P., Taylor, R. and Thorne, J. A., ‘Potential automorphy over CM fields’, Preprint, 2018, arXiv:1812.09999v1 [math.NT].Google Scholar
Amitsur, S. A., ‘On the characteristic polynomial of a sum of matrices’, Linear Multilinear Algebra 8(3) (1979/80), 177–182.Google Scholar
Bellaïche, J. and Chenevier, G., ‘Families of Galois representations and Selmer groups’, Astérisque 324 (2009), xii+314.Google Scholar
Calegari, F. and Specter, J., ‘Pseudo-representations of weight one’. Undated preprint. Accessed January 17, 2019 at http://www.math.jhu.edu/∼jspecter/Pseudo.pdf. First version of [CS19], circa 2016.Google Scholar
Calegari, F. and Specter, J., ‘Pseudo-representations of weight one are unramified’, Algebra Number Theory (2019) Preprint, arXiv:1906.10473v1 [math.NT], to appear.Google Scholar
Chenevier, G., ‘The p-adic analytic space of pseudocharacters of a profinite group, and pseudorepresentations over arbitrary rings’, inAutomorphic Forms and Galois Representations: Vol. I, London Mathematical Society Lecture Note Series, 414 (Cambridge University Press, Cambridge, 2014), 221–285. We follow the numbering of the online version arXiv:0809.0415v2, which differs from the print version.Google Scholar
Grothendieck, A., ‘Éléments de géométrie algébrique. I. Le langage des schémas’, Publ. Math. Inst. Hautes Études Sci. 4 (1960), 5–228.Google Scholar
Liu, T., ‘Torsion p-adic Galois representations and a conjecture of Fontaine’, Ann. Sci. Éc. Norm. Supér. (4) 40(4) (2007), 633–674.Google Scholar
Mazur, B., ‘Modular curves and the Eisenstein ideal’, Publ. Math. Inst. Hautes Études Sci. 47 (1977), 33–186.Google Scholar
Mazur, B., ‘Deforming Galois representations’, inGalois Groups Over Q (Berkeley, CA, 1987), Math. Sci. Res. Inst. Publ., 16 (Springer, New York, 1989), 385–437.Google Scholar
Nyssen, L., ‘Pseudo-représentations’, Math. Ann. 306(2) (1996), 257–283.Google Scholar
Procesi, C., ‘A formal inverse to the Cayley–Hamilton theorem’, J. Algebra 107(1) (1987), 63–74.Google Scholar
Ramakrishna, R., ‘On a variation of Mazur’s deformation functor’, Compos. Math. 87(3) (1993), 269–286.Google Scholar
Rouquier, R., ‘Caractérisation des caractères et pseudo-caractères’, J. Algebra 180(2) (1996), 571–586.Google Scholar
Taylor, R., ‘Galois representations associated to Siegel modular forms of low weight’, Duke Math. J. 63(2) (1991), 281–332.Google Scholar
Wang-Erickson, C., ‘Algebraic families of Galois representations and potentially semi-stable pseudodeformation rings’, Math. Ann. 371(3–4) (2018), 1615–1681.Google Scholar
Wang-Erickson, C., ‘Deformations of residually reducible Galois representations via $A_{\infty }$ -algebra structure on Galois cohomology’, Preprint, 2018, arXiv:1809.02484v1 [math.NT].Google Scholar
Wiles, A., ‘On ordinary 𝜆-adic representations associated to modular forms’, Invent. Math. 94(3) (1988), 529–573.Google Scholar
Wake, P. and Wang-Erickson, C., ‘Ordinary pseudorepresentations and modular forms’, Proc. Amer. Math. Soc. B 4 (2017), 53–71.Google Scholar
Wake, P. and Wang-Erickson, C., ‘The rank of Mazur’s Eisenstein ideal’, Duke Math. J., to appear. Preprint, 2017, arXiv:1707.01894v2 [math.NT].Google Scholar
Wake, P. and Wang-Erickson, C., ‘Pseudo-modularity and Iwasawa theory’, Amer. J. Math. 140(4) (2018), 977–1040.Google Scholar
Wake, P. and Wang-Erickson, C., ‘The Eisenstein ideal with squarefree level’, Preprint, 2018, arXiv:1804.06400v2 [math.NT].Google Scholar