pith. sign in

arxiv: 2107.03114 · v2 · submitted 2021-07-07 · 🧮 math.NT · math.RT

Deformation rings and images of Galois representations

Gebhard B\"ockle , Sara Arias-de-Reyna This is my paper

Pith reviewed 2026-05-24 12:26 UTC · model grok-4.3

classification 🧮 math.NT math.RT
keywords deformation ringsGalois representationsreductive groupsclosed subgroupsresidual imageuniversal deformationWitt ringprofinite groups
0
0 comments X

The pith

Every closed subgroup of G(R) with full residual image is a conjugate of G(A) for a closed subring A.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves three results for a connected reductive almost simple group G over the Witt ring of a finite field F of characteristic p. Under a mild condition on p, every closed subgroup of G(R) whose reduction modulo the maximal ideal is all of G(F) must be conjugate to the image of G(A) for some closed local subring A of R with the same residue field. Surjective homomorphisms between two such groups G(R) and G(R') are likewise induced, up to conjugation, by ring homomorphisms from R to R'. The identity map on G(R) is shown to be the universal deformation of the reduction homomorphism to G(F). These statements supply both a classification of subgroups and a universal property for the deformation ring in this setting.

Core claim

Under a mild condition on p relative to the structural constants of G, every closed subgroup H of G(R) with full residual image G(F) is a conjugate of G(A) for A a closed local subring of R with residue field F; every surjective homomorphism G(R) to G(R') is, up to conjugation, induced from a ring homomorphism R to R'; and the identity map on G(R) represents the universal deformation of the reduction map G(R) to G(F).

What carries the argument

The group scheme G(R) together with its reduction map to G(F); the argument shows that closed subgroups with full residual image arise exactly from closed subrings via the functor of points.

If this is right

  • The identity map on G(R) is the universal deformation of the given residual representation.
  • Surjective maps between two groups G(R) and G(R') arise from ring maps R to R' up to conjugation.
  • Closed subgroups with full residual image are classified by closed local subrings with the same residue field.
  • The results supply an abstract classification for images of Galois representations valued in G(R).

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The classification reduces questions about images of Galois representations to questions about subrings of deformation rings.
  • The axiomatic framework sketched in the paper could be checked for other groups such as tori or non-connected reductive groups.
  • Explicit verification for small groups like SL_2 or GL_2 over small Witt rings would test the necessity of the condition on p.
  • The universal property may simplify calculations of tangent spaces or obstructions in deformation theory.

Load-bearing premise

A mild condition relating the prime p to the structural constants of the reductive group G must hold.

What would settle it

Exhibit a closed subgroup H inside G(R) with reduction equal to all of G(F) that is not conjugate inside G(R) to the image of G(A) for any closed local subring A of R, while the condition on p holds.

read the original abstract

Let $\mathcal{G}$ be a connected reductive almost simple group over the Witt ring $W(\mathbb{F})$ for $\mathbb{F}$ a finite field of characteristic $p$. Let $R$ and $R'$ be complete noetherian local $W(\mathbb{F})$ -algebras with residue field $\mathbb{F}$. Under a mild condition on $p$ in relation to structural constants of $\mathcal{G}$, we show the following results: (1) Every closed subgroup $H$ of $\mathcal{G}(R)$ with full residual image $\mathcal{G}(\mathbb{F})$ is a conjugate of a group $\mathcal{G}(A)$ for $A\subset R$ a closed subring that is local and has residue field $\mathbb{F}$ . (2) Every surjective homomorphism $\mathcal{G}(R)\to\mathcal{G}(R')$ is, up to conjugation, induced from a ring homomorphism $R\to R'$. (3) The identity map on $\mathcal{G}(R)$ represents the universal deformation of the representation of the profinite group $\mathcal{G}(R)$ given by the reduction map $\mathcal{G}(R)\to\mathcal{G}(\mathbb{F})$. This generalizes results of Dorobisz and Eardley-Manoharmayum and of Manoharmayum, and in addition provides an abstract classification result for closed subgroups of $\mathcal{G}(R)$ with residually full image. We provide an axiomatic framework to study this type of question, also for slightly more general $\mathcal{G}$, and we study in the case at hand in great detail what conditions on $\mathbb{F}$ or on $p$ in relation to $\mathcal{G}$ are necessary for the above results to hold.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper proves three results for a connected reductive almost simple group 𝒢 over the Witt ring W(𝔽) (𝔽 finite field of char p): under a mild condition on p relative to structural constants of 𝒢, (1) every closed subgroup H ≤ 𝒢(R) with full residual image 𝒢(𝔽) is conjugate to 𝒢(A) for some closed local subring A ⊂ R with residue field 𝔽; (2) every surjective homomorphism 𝒢(R) → 𝒢(R') is, up to conjugation, induced by a ring homomorphism R → R'; (3) the identity map on 𝒢(R) represents the universal deformation of the reduction 𝒢(R) → 𝒢(𝔽). The authors develop an axiomatic framework applicable to slightly more general 𝒢 and analyze in detail the necessary conditions on 𝔽 and p.

Significance. If the results hold, they generalize the theorems of Dorobisz–Eardley-Manoharmayum and Manoharmayum on images of Galois representations and deformation rings, while adding an abstract classification of closed subgroups with residually full image. The axiomatic setup and explicit study of the conditions on p and 𝔽 relative to the root system and structural constants of 𝒢 constitute a clear strengthening; the framework itself may be reusable for other groups.

minor comments (3)
  1. The abstract states the three results under a 'mild condition on p'; the introduction should quote the precise numerical bound (in terms of the rank or the constants appearing in the root datum) already on page 2 so that readers can immediately assess applicability.
  2. Notation for the group scheme 𝒢 and its R-points is introduced in §2; a short table collecting the standing assumptions on R, R' and the residue field 𝔽 would improve readability of the statements of Theorems 3.1, 4.2 and 5.3.
  3. The axiomatic framework of §2 is used for both the main theorems and the necessity analysis; the paper should indicate explicitly which axioms are used only for the necessity statements and which are required for the three positive results.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of the main results, and recommendation for minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives its three main theorems from the algebraic structure of connected reductive groups over Witt rings and an axiomatic framework for studying images of representations, under explicitly stated mild conditions on p. The results generalize independent prior work by Dorobisz, Eardley-Manoharmayum, and Manoharmayum with no author overlap, and the central claims (subgroup classification, homomorphism classification, and universal deformation property) rest on properties of G(R) and residue maps rather than any self-definition, fitted parameters renamed as predictions, or load-bearing self-citations. The derivation is therefore self-contained against external algebraic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the standard setup that G is a connected reductive almost simple group over the Witt ring W(F) together with the mild condition on p; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption G is a connected reductive almost simple group over the Witt ring W(F) for F a finite field of characteristic p.
    This is the explicit setup stated at the beginning of the abstract on which all three results depend.

pith-pipeline@v0.9.0 · 5850 in / 1314 out tokens · 23397 ms · 2026-05-24T12:26:37.471151+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

46 extracted references · 46 canonical work pages

  1. [1]

    A note on the automorphic L anglands group

    James Arthur. A note on the automorphic L anglands group. volume 45, pages 466--482. 2002. Dedicated to Robert V. Moody

    work page 2002

  2. [2]

    Bleher, Ted Chinburg, and Bart de Smit

    Frauke M. Bleher, Ted Chinburg, and Bart de Smit. Inverse problems for deformation rings. Trans. Amer. Math. Soc. , 365(11):6149--6165, 2013

    work page 2013

  3. [3]

    e l Bella\

    Jo\" e l Bella\" che. Image of pseudo-representations and coefficients of modular forms modulo p . Adv. Math. , 353:647--721, 2019

    work page 2019

  4. [4]

    Linear algebraic groups , volume 126 of Graduate Texts in Mathematics

    Armand Borel. Linear algebraic groups , volume 126 of Graduate Texts in Mathematics . Springer-Verlag, New York, second edition, 1991

    work page 1991

  5. [5]

    Explicit deformation of G alois representations

    Nigel Boston. Explicit deformation of G alois representations. Invent. Math. , 103(1):181--196, 1991

    work page 1991

  6. [6]

    Bourbaki

    N. Bourbaki. \' E l \' e ments de math \' e matique. F asc. XXXIV . G roupes et alg \`e bres de L ie. C hapitre IV : G roupes de C oxeter et syst \`e mes de T its. C hapitre V : G roupes engendr \' e s par des r \' e flexions. C hapitre VI : syst \`e mes de racines . Actualit \' e s Scientifiques et Industrielles, No. 1337. Hermann, Paris, 1968

    work page 1968

  7. [7]

    J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson. Atlas of finite groups . Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups, With computational assistance from J. G. Thackray

    work page 1985

  8. [8]

    Conti, J

    A. Conti, J. Lang, and A. Medvedovky. Big image of two-dimensional pseudorepresentations, 2019

    work page 2019

  9. [9]

    Reductive group schemes

    Brian Conrad. Reductive group schemes. In Autour des sch\' e mas en groupes. V ol. I , volume 42/43 of Panor. Synth\`eses , pages 93--444. Soc. Math. France, Paris, 2014

    work page 2014

  10. [10]

    Cohomology of finite groups of L ie type

    Edward Cline, Brian Parshall, and Leonard Scott. Cohomology of finite groups of L ie type. I . Inst. Hautes \'Etudes Sci. Publ. Math. , (45):169--191, 1975

    work page 1975

  11. [11]

    Cohomology of finite groups of L ie type

    Edward Cline, Brian Parshall, and Leonard Scott. Cohomology of finite groups of L ie type. II . J. Algebra , 45(1):182--198, 1977

    work page 1977

  12. [12]

    Curtis and Irving Reiner

    Charles W. Curtis and Irving Reiner. Representation theory of finite groups and associative algebras . Pure and Applied Mathematics, Vol. XI. Interscience Publishers, a division of John Wiley & Sons, New York-London, 1962

    work page 1962

  13. [13]

    Groupes alg\'ebriques

    Michel Demazure and Pierre Gabriel. Groupes alg\'ebriques. T ome I : G \'eom\'etrie alg\'ebrique, g\'en\'eralit\'es, groupes commutatifs . Masson & Cie, \'Editeur, Paris; North-Holland Publishing Co., Amsterdam, 1970. Avec un appendice t Corps de classes local par Michiel Hazewinkel

    work page 1970

  14. [14]

    The inverse problem for universal deformation rings and the special linear group

    Krzysztof Dorobisz. The inverse problem for universal deformation rings and the special linear group. Trans. Amer. Math. Soc. , 368(12):8597--8613, 2016

    work page 2016

  15. [15]

    Adelic openness for D rinfeld modules in special characteristic

    Anna Devic and Richard Pink. Adelic openness for D rinfeld modules in special characteristic. J. Number Theory , 132(7):1583--1625, 2012

    work page 2012

  16. [16]

    The inverse deformation problem

    Timothy Eardley and Jayanta Manoharmayum. The inverse deformation problem. Compos. Math. , 152(8):1725--1739, 2016

    work page 2016

  17. [17]

    Abelian varieties

    Bas Edixhoven, Gerhard van der Geer, and Ben Moonen. Abelian varieties. (preliminary version), http://gerard.vdgeer.net/AV.pdf, 2014

    work page 2014

  18. [18]

    A finiteness theorem for the symmetric square of an elliptic curve

    Matthias Flach. A finiteness theorem for the symmetric square of an elliptic curve. Invent. Math. , 109(2):307--327, 1992

    work page 1992

  19. [19]

    Cohomologie non ab\' e lienne

    Jean Giraud. Cohomologie non ab\' e lienne . Springer-Verlag, Berlin-New York, 1971. Die Grundlehren der mathematischen Wissenschaften, Band 179

    work page 1971

  20. [20]

    Griess, Jr

    Robert L. Griess, Jr. Schur multipliers of the known finite simple groups. II . In The S anta C ruz C onference on F inite G roups ( U niv. C alifornia, S anta C ruz, C alif., 1979) , volume 37 of Proc. Sympos. Pure Math. , pages 279--282. Amer. Math. Soc., Providence, R.I., 1980

    work page 1979

  21. [21]

    \' E l\'ements de g\'eom\'etrie alg\'ebrique

    Alexander Grothendieck. \' E l\'ements de g\'eom\'etrie alg\'ebrique. IV . \' E tude locale des sch\'emas et des morphismes de sch\'emas. III (r \'e dig \'e s avec la collaboration de J ean D ieudonn \'e ). Inst. Hautes \'Etudes Sci. Publ. Math. , (28):255, 1966

    work page 1966

  22. [22]

    Algebraic geometry I

    Ulrich G \"o rtz and Torsten Wedhorn. Algebraic geometry I . Advanced Lectures in Mathematics. Vieweg + Teubner, Wiesbaden, 2010. Schemes with examples and exercises

    work page 2010

  23. [23]

    Big G alois representations and p -adic L -functions

    Haruzo Hida. Big G alois representations and p -adic L -functions. Compos. Math. , 151(4):603--664, 2015

    work page 2015

  24. [24]

    Die adjungierten D arstellungen der C hevalley- G ruppen

    Gerhard Hiss. Die adjungierten D arstellungen der C hevalley- G ruppen. Arch. Math. (Basel) , 42(5):408--416, 1984

    work page 1984

  25. [25]

    G. M. D. Hogeweij. Almost-classical L ie algebras. I , II . Nederl. Akad. Wetensch. Indag. Math. , 44(4):441--452, 453--460, 1982

    work page 1982

  26. [26]

    On the rationality of algebraic monodromy groups of compatible systems, 2018

    Chun Yin Hui. On the rationality of algebraic monodromy groups of compatible systems, 2018

    work page 2018

  27. [27]

    M. Larsen. Maximality of G alois actions for compatible systems. Duke Math. J. , 80(3):601--630, 1995

    work page 1995

  28. [28]

    A structure theorem for subgroups of GL_n over complete local N oetherian rings with large residual image

    Jayanta Manoharmayum. A structure theorem for subgroups of GL_n over complete local N oetherian rings with large residual image. Proc. Amer. Math. Soc. , 143(7):2743--2758, 2015

    work page 2015

  29. [29]

    B. Mazur. Deforming G alois representations. In Galois groups over Q ( B erkeley, CA , 1987) , volume 16 of Math. Sci. Res. Inst. Publ. , pages 385--437. Springer, New York, 1989

    work page 1987

  30. [30]

    James S. Milne. \' E tale cohomology , volume 33 of Princeton Mathematical Series . Princeton University Press, Princeton, N.J., 1980

    work page 1980

  31. [31]

    J. S. Milne. Algebraic groups , volume 170 of Cambridge Studies in Advanced Mathematics . Cambridge University Press, Cambridge, 2017. The theory of group schemes of finite type over a field

    work page 2017

  32. [32]

    Linear algebraic groups and finite groups of L ie type , volume 133 of Cambridge Studies in Advanced Mathematics

    Gunter Malle and Donna Testerman. Linear algebraic groups and finite groups of L ie type , volume 133 of Cambridge Studies in Advanced Mathematics . Cambridge University Press, Cambridge, 2011

    work page 2011

  33. [33]

    Mazur and A

    B. Mazur and A. Wiles. On p -adic analytic families of G alois representations. Compositio Math. , 59(2):231--264, 1986

    work page 1986

  34. [34]

    Compact subgroups of linear algebraic groups

    Richard Pink. Compact subgroups of linear algebraic groups. J. Algebra , 206(2):438--504, 1998

    work page 1998

  35. [35]

    Propri\'et\'es conjecturales des groupes de G alois motiviques et des repr\'esentations l -adiques

    Jean-Pierre Serre. Propri\'et\'es conjecturales des groupes de G alois motiviques et des repr\'esentations l -adiques. In Motives ( S eattle, WA , 1991) , volume 55 of Proc. Sympos. Pure Math. , pages 377--400. Amer. Math. Soc., Providence, RI, 1994

    work page 1991

  36. [36]

    I : P ropri\'et\'es g\'en\'erales des sch\'emas en groupes

    Sch\'emas en groupes. I : P ropri\'et\'es g\'en\'erales des sch\'emas en groupes . S\'eminaire de G\'eom\'etrie Alg\'ebrique du Bois Marie 1962/64 (SGA 3). Dirig\'e par M. Demazure et A. Grothendieck. Lecture Notes in Mathematics, Vol. 151. Springer-Verlag, Berlin, 1970

    work page 1962

  37. [37]

    auser Classics. Birkh\

    T. A. Springer. Linear algebraic groups . Modern Birkh\"auser Classics. Birkh\"auser Boston, Inc., Boston, MA, second edition, 2009

    work page 2009

  38. [38]

    Representations of algebraic groups

    Robert Steinberg. Representations of algebraic groups. Nagoya Math. J. , 22:33--56, 1963

    work page 1963

  39. [39]

    Regular elements of semisimple algebraic groups

    Robert Steinberg. Regular elements of semisimple algebraic groups. Inst. Hautes \' E tudes Sci. Publ. Math. , (25):49--80, 1965

    work page 1965

  40. [40]

    Generators, relations and coverings of algebraic groups

    Robert Steinberg. Generators, relations and coverings of algebraic groups. II . J. Algebra , 71(2):527--543, 1981

    work page 1981

  41. [41]

    A course on finite flat group schemes and p -divisible groups

    Jakob Stix. A course on finite flat group schemes and p -divisible groups. https://www.uni-frankfurt.de/52288632/Stix n finflat n Grpschemes.pdf, 2009

    work page arXiv 2009

  42. [42]

    Deformations of G alois representations and H ecke algebras

    Jacques Tilouine. Deformations of G alois representations and H ecke algebras . Published for The Mehta Research Institute of Mathematics and Mathematical Physics, Allahabad, 1996

    work page 1996

  43. [43]

    On the 1 -cohomology of the general and special linear groups

    Olga Taussky and Hans Zassenhaus. On the 1 -cohomology of the general and special linear groups. Aequationes Math. , 5:129--201, 1970

    work page 1970

  44. [44]

    Surjectivity criteria for p -adic representations

    Adrian Vasiu. Surjectivity criteria for p -adic representations. I . Manuscripta Math. , 112(3):325--355, 2003

    work page 2003

  45. [45]

    1 -cohomology of C hevalley groups

    Helmut V \" o lklein. 1 -cohomology of C hevalley groups. J. Algebra , 127(2):353--372, 1989

    work page 1989

  46. [46]

    The 1 -cohomology of the adjoint module of a C hevalley group

    Helmut V \"o lklein. The 1 -cohomology of the adjoint module of a C hevalley group. Forum Math. , 1(1):1--13, 1989

    work page 1989