Skip to main content Accessibility help
×
Home

DEFORMATION CONDITIONS FOR PSEUDOREPRESENTATIONS

  • PRESTON WAKE (a1) and CARL WANG-ERICKSON (a2)

Abstract

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.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      DEFORMATION CONDITIONS FOR PSEUDOREPRESENTATIONS
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      DEFORMATION CONDITIONS FOR PSEUDOREPRESENTATIONS
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      DEFORMATION CONDITIONS FOR PSEUDOREPRESENTATIONS
      Available formats
      ×

Copyright

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.

References

Hide All
[ACC+18] 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].
[Ami80] Amitsur, S. A., ‘On the characteristic polynomial of a sum of matrices’, Linear Multilinear Algebra 8(3) (1979/80), 177182.
[BC09] Bellaïche, J. and Chenevier, G., ‘Families of Galois representations and Selmer groups’, Astérisque 324 (2009), xii+314.
[CS16] 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.
[CS19] Calegari, F. and Specter, J., ‘Pseudo-representations of weight one are unramified’, Algebra Number Theory (2019) Preprint, arXiv:1906.10473v1 [math.NT], to appear.
[Che14] 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), 221285. We follow the numbering of the online version arXiv:0809.0415v2, which differs from the print version.
[Gro60] Grothendieck, A., ‘Éléments de géométrie algébrique. I. Le langage des schémas’, Publ. Math. Inst. Hautes Études Sci. 4 (1960), 5228.
[Liu07] Liu, T., ‘Torsion p-adic Galois representations and a conjecture of Fontaine’, Ann. Sci. Éc. Norm. Supér. (4) 40(4) (2007), 633674.
[Maz77] Mazur, B., ‘Modular curves and the Eisenstein ideal’, Publ. Math. Inst. Hautes Études Sci. 47 (1977), 33186.
[Maz89] Mazur, B., ‘Deforming Galois representations’, inGalois Groups Over Q (Berkeley, CA, 1987), Math. Sci. Res. Inst. Publ., 16 (Springer, New York, 1989), 385437.
[Nys96] Nyssen, L., ‘Pseudo-représentations’, Math. Ann. 306(2) (1996), 257283.
[Pro87] Procesi, C., ‘A formal inverse to the Cayley–Hamilton theorem’, J. Algebra 107(1) (1987), 6374.
[Ram93] Ramakrishna, R., ‘On a variation of Mazur’s deformation functor’, Compos. Math. 87(3) (1993), 269286.
[Rou96] Rouquier, R., ‘Caractérisation des caractères et pseudo-caractères’, J. Algebra 180(2) (1996), 571586.
[Tay91] Taylor, R., ‘Galois representations associated to Siegel modular forms of low weight’, Duke Math. J. 63(2) (1991), 281332.
[WE18a] Wang-Erickson, C., ‘Algebraic families of Galois representations and potentially semi-stable pseudodeformation rings’, Math. Ann. 371(3–4) (2018), 16151681.
[WE18b] Wang-Erickson, C., ‘Deformations of residually reducible Galois representations via $A_{\infty }$ -algebra structure on Galois cohomology’, Preprint, 2018, arXiv:1809.02484v1 [math.NT].
[Wil88] Wiles, A., ‘On ordinary 𝜆-adic representations associated to modular forms’, Invent. Math. 94(3) (1988), 529573.
[WWE17a] Wake, P. and Wang-Erickson, C., ‘Ordinary pseudorepresentations and modular forms’, Proc. Amer. Math. Soc. B 4 (2017), 5371.
[WWE17b] 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].
[WWE18a] Wake, P. and Wang-Erickson, C., ‘Pseudo-modularity and Iwasawa theory’, Amer. J. Math. 140(4) (2018), 9771040.
[WWE18b] Wake, P. and Wang-Erickson, C., ‘The Eisenstein ideal with squarefree level’, Preprint, 2018, arXiv:1804.06400v2 [math.NT].
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

MSC classification

DEFORMATION CONDITIONS FOR PSEUDOREPRESENTATIONS

  • PRESTON WAKE (a1) and CARL WANG-ERICKSON (a2)

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed