Effect of increasing the ramification on pseudo-deformation rings
Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 1, pp. 189-236.

Given a continuous, odd, semi-simple 2-dimensional representation of G ,Np over a finite field of odd characteristic p and a prime not dividing Np, we study the relation between the universal deformation rings of the corresponding pseudo-representations for the groups G ,Np and G ,Np . As a related problem, we investigate when the universal pseudo-representation arises from an actual representation over the universal deformation ring. Under some hypotheses, we prove analogues of theorems of Boston and Böckle for the reduced pseudo-deformation rings. We improve these results when the pseudo-representation is unobstructed and p does not divide 2 -1. When the pseudo-representation is unobstructed and p divides +1, we prove that the universal deformation rings in characteristic 0 and p of the pseudo-representation for G ,Np are not local complete intersection rings. As an application of our main results, we prove a big R=𝕋 theorem.

Etant donnée une représentation continue, impaire et semi-simple de dimension 2 de G ,Np sur un corps fini de caractéristique impaire p et un nombre premier ne divisant pas Np, nous étudions la relation entre les anneaux de déformation universels des pseudo-représentations correspondantes pour les groupes G ,Np et G ,Np . Nous nous intéressons aussi au problème connexe de savoir si la pseudo-représentation universelle provient d’une véritable représentation sur l’anneau de déformation universel. Sous certaines hypothèses, nous prouvons des analogues des théorèmes de Boston et Böckle pour les anneaux de pseudo-déformation réduits. Nous améliorons ces résultats dans le cas où la pseudo-représentation est non obstruée et p ne divise pas 2 -1. Lorsque la pseudo-représentation est non obstruée et p divise +1, nous prouvons que les anneaux de déformation universels de la pseudo-représentation de G ,Np en caractéristique 0 et p ne sont pas des anneaux locaux d’intersection complète. Comme application de nos résultats principaux, nous prouvons un théorème R=𝕋 pour les algèbres de Hecke élargies et les anneaux de pseudo-représentations.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1198
Classification: 11F80,  11F70,  11F33,  13H10
Keywords: pseudo-representations, deformation of Galois representations, structure of deformation rings
Shaunak V. Deo 1

1 Department of Mathematics Indian Institute of Science Bangalore 560012, India
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{JTNB_2022__34_1_189_0,
     author = {Shaunak V. Deo},
     title = {Effect of increasing the ramification on pseudo-deformation rings},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {189--236},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {34},
     number = {1},
     year = {2022},
     doi = {10.5802/jtnb.1198},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1198/}
}
TY  - JOUR
AU  - Shaunak V. Deo
TI  - Effect of increasing the ramification on pseudo-deformation rings
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2022
DA  - 2022///
SP  - 189
EP  - 236
VL  - 34
IS  - 1
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1198/
UR  - https://doi.org/10.5802/jtnb.1198
DO  - 10.5802/jtnb.1198
LA  - en
ID  - JTNB_2022__34_1_189_0
ER  - 
%0 Journal Article
%A Shaunak V. Deo
%T Effect of increasing the ramification on pseudo-deformation rings
%J Journal de théorie des nombres de Bordeaux
%D 2022
%P 189-236
%V 34
%N 1
%I Société Arithmétique de Bordeaux
%U https://doi.org/10.5802/jtnb.1198
%R 10.5802/jtnb.1198
%G en
%F JTNB_2022__34_1_189_0
Shaunak V. Deo. Effect of increasing the ramification on pseudo-deformation rings. Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 1, pp. 189-236. doi : 10.5802/jtnb.1198. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1198/

[1] Joël Bellaïche Pseudodeformations, Math. Z., Volume 270 (2012) no. 3-4, pp. 1163-1180 | DOI | MR | Zbl

[2] Joël Bellaïche Image of pseudo-representations and coefficients of modular forms modulo p, Adv. Math., Volume 353 (2019), pp. 647-721 | DOI | MR | Zbl

[3] Joël Bellaïche; Gaëtan Chenevier Families of Galois representations and Selmer groups, Astérisque, 324, Société Mathématique de France, 2009, xii+314 pages | Numdam

[4] Joël Bellaïche; Chandrashekhar Khare Level 1 Hecke algebras of modular forms modulo p, Compos. Math., Volume 151 (2015) no. 3, pp. 397-415 | DOI | MR | Zbl

[5] Nicolas Billerey; Ricardo Menares On the modularity of reducible modl Galois representations, Math. Res. Lett., Volume 23 (2016) no. 1, pp. 15-41 | DOI | Zbl

[6] Frauke Bleher; Ted Chinburg Universal deformation rings need not be complete intersections, Math. Ann., Volume 337 (2007) no. 4, pp. 739-767 | DOI | MR | Zbl

[7] Gebhard Böckle The generic fiber of the universal deformation space associated to a tame Galois representation, Manuscr. Math., Volume 96 (1998) no. 2, pp. 231-246 | MR | Zbl

[8] Gebhard Böckle Explicit universal deformations of even Galois representations, Math. Nachr., Volume 206 (1999), pp. 85-110 | DOI | MR | Zbl

[9] Gebhard Böckle A local-to-global principle for deformations of Galois representations, J. Reine Angew. Math., Volume 509 (1999), pp. 199-236 | DOI | MR | Zbl

[10] Gebhard Böckle On the density of modular points in universal deformation spaces, Am. J. Math., Volume 123 (2001) no. 5, pp. 985-1007 | DOI | MR | Zbl

[11] Nigel Boston Families of Galois representations - Increasing the ramification, Duke Math. J., Volume 66 (1992) no. 3, pp. 357-367 | MR | Zbl

[12] Gaëtan Chenevier The p-adic analytic space of pseudocharacters of a profinite group and pseudorepresentations over arbitrary rings, Automorphic forms and Galois representations (London Mathematical Society Lecture Note Series), Volume 414, Cambridge University Press, 2014, pp. 221-285 | DOI | MR | Zbl

[13] Shaunak V. Deo Structure of Hecke algebras of modular forms modulo p, Algebra Number Theory, Volume 11 (2017) no. 1, pp. 1-38 | MR | Zbl

[14] Shaunak V. Deo On density of modular points in pseudo-deformation rings (2021) (https://arxiv.org/abs/2105.05823)

[15] David Eisenbud Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics, 150, Springer, 1995

[16] Matthew Emerton p-adic families of modular forms (after Hida, Coleman and Mazur), Séminaire Bourbaki. Volume 2009/2010 (Astérisque), Volume 339, Société Mathématique de France, 2011, pp. 1012-1026 | Numdam | Zbl

[17] Naomi Jochnowitz Congruences between systems of eigenvalues of modular forms, Trans. Am. Math. Soc., Volume 270 (1982) no. 1, pp. 269-285 | DOI | MR | Zbl

[18] Mark Kisin The Fontaine–Mazur conjecture for GL 2 , J. Am. Math. Soc., Volume 22 (2009) no. 3, pp. 641-690 | DOI | MR | Zbl

[19] Barry Mazur Deforming Galois representations, Galois groups over (Berkeley, CA, 1987) (Mathematical Sciences Research Institute Publications), Volume 16, Springer, 1987, pp. 385-437 | DOI

[20] Ravi Ramakrishna On a Variation of Mazur’s deformation functor, Compos. Math., Volume 87 (1993) no. 3, pp. 269-286 | MR | Zbl

[21] Raphaël Rouquier Caractérisation des caractéres et pseudo-caractéres, J. Algebra, Volume 180 (1996) no. 2, pp. 571-586 | DOI | Zbl

[22] John Tate Relation between K 2 and Galois cohomology, Invent. Math., Volume 36 (1976), pp. 257-274 | DOI | MR | Zbl

[23] Carl Wang-Erickson Presentations of non-commutative deformation rings via A -algebras and applications to deformations of Galois representations and pseudorepresentations (2020) (https://arxiv.org/abs/1809.02484v2)

[24] Lawrence C. Washington Galois Cohomology, Modular forms and Fermat’s last theorem (Boston, 1995), Springer, 1997, pp. 101-120 | DOI | Zbl

[25] Hwajong Yoo Non-optimal levels of a reducible mod modular representation, Trans. Am. Math. Soc., Volume 371 (2019) no. 6, pp. 3805-3830 | MR | Zbl

Cited by Sources: