$\mathrm{GL}_2(\mathbb{Q}_p)$-ordinary families and automorphy lifting

Yiwen Ding

arxiv: 1907.07372 · v1 · pith:FHCO46QPnew · submitted 2019-07-17 · 🧮 math.NT · math.RT

GL₂(mathbb{Q}_p)-ordinary families and automorphy lifting

Yiwen Ding This is my paper

Pith reviewed 2026-05-24 20:31 UTC · model grok-4.3

classification 🧮 math.NT math.RT

keywords automorphy liftingGalois representationsordinary familiesCM fieldsp-adic representationsR=T theoremssocle conjectureBreuil-Ding families

0 comments

The pith

Automorphy lifting holds for essentially conjugate self-dual p-adic Galois representations over CM fields when p splits and local restrictions at p are reducible with small factors.

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

The paper proves automorphy lifting for p-adic Galois representations that are essentially conjugate self-dual over CM imaginary fields F. The representations must satisfy that p splits in F and that their restrictions to decomposition groups at p are reducible with Jordan-Hölder factors of dimension at most two. The proof relies on an R=T result inside the GL2 over Qp ordinary families introduced by Breuil and Ding. It also establishes some cases of Breuil's locally analytic socle conjecture outside the trianguline setting. These results extend known automorphy lifting theorems to new families of representations that arise in the Langlands correspondence.

Core claim

By establishing an R=T-type result over the GL2(Qp)-ordinary families of Breuil-Ding, the paper shows that certain essentially conjugate self-dual p-adic Galois representations over CM fields F, with p splitting in F and reducible local restrictions at p having small Jordan-Hölder factors, are automorphic. Additional results are obtained for Breuil's locally analytic socle conjecture in non-trianguline cases.

What carries the argument

The R=T result established inside the GL_2(Q_p)-ordinary families of Breuil-Ding, which carries the patching argument for the lifting theorems.

If this is right

Automorphy lifting theorems apply directly to the specified class of Galois representations over CM fields.
Some instances of Breuil's locally analytic socle conjecture hold in non-trianguline cases.
The R=T results inside ordinary families provide a new route to automorphy lifting beyond previously treated settings.

Where Pith is reading between the lines

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

If the ordinary families can be enlarged to cover additional local conditions at p, similar lifting theorems might apply to wider classes of representations.
The approach may connect to other parts of the Langlands correspondence by providing new ways to match Galois representations to automorphic forms over CM fields.
Further development could test whether the same families yield results on related conjectures such as the global Langlands correspondence in higher rank.

Load-bearing premise

The ordinary families must satisfy the necessary local and global properties required for the patching argument in the R=T proof.

What would settle it

A concrete counterexample would be a Galois representation satisfying all the stated conditions on F, rho, and the local restrictions at p that is nevertheless shown not to be automorphic, or an explicit failure of the R=T equality inside one of the Breuil-Ding ordinary families.

read the original abstract

We prove automorphy lifting results for certain essentially conjugate self-dual $p$-adic Galois representations $\rho$ over CM imaginary fields $F$, which satisfy in particular that $p$ splits in $F$, and that the restriction of $\rho$ on any decomposition group above $p$ is reducible with all the Jordan-H\"older factors of dimension at most $2$. We also show some results on Breuil's locally analytic socle conjecture in certain non-trianguline case. The main results are obtained by establishing an $R=\mathbb{T}$-type result over the $\mathrm{GL}_2(\mathbb{Q}_p)$-ordinary families considered by Breuil-Ding.

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.

Desk Editor's Note private letter to a colleague

Ding gets an R=T result inside Breuil-Ding ordinary families that yields new automorphy lifting for reducible local conditions with small Jordan-Hölder factors over CM fields, plus partial socle-conjecture progress.

read the letter

The main takeaway is that this paper proves an R=T theorem over the GL2(Qp)-ordinary families from Breuil-Ding and feeds it into automorphy lifting for essentially conjugate self-dual p-adic representations over CM fields F where p splits and every local restriction at primes above p is reducible with Jordan-Hölder factors of dimension at most 2. It also records some non-trianguline cases for Breuil's locally analytic socle conjecture. The R=T step is presented as the new ingredient that makes the lifting go through where earlier work stopped short of these reducible settings. The paper does a clean job of stating the local conditions up front and showing how the ordinary families supply the necessary local input for the patching argument without obvious circularity. The families themselves are cited as external, so the burden falls on the new R=T statement, which looks like a genuine addition rather than a re-derivation. The soft spots are limited but worth checking: the abstract gives no equations or section outlines for how the R=T is proved, so one cannot yet see whether the multiplicity-one statements or the local deformation rings behave as needed under the reducibility assumption. That gap is typical for an abstract and does not by itself suggest a flaw, but it means the central claim rests on work that still needs verification. No internal contradictions jump out from the stated hypotheses. This paper is for readers already comfortable with ordinary families, patching, and the p-adic Langlands program over CM fields; someone outside that circle will not get much from it. Within the circle it supplies concrete extensions that could be cited once the proofs are confirmed. It deserves a serious referee because the claims are specific, the method is standard but applied to a new range of local conditions, and the potential payoff for small-dimensional Galois representations is real if the R=T holds. I would send it to review.

Referee Report

1 major / 0 minor

Summary. The paper claims to prove automorphy lifting results for certain essentially conjugate self-dual p-adic Galois representations ρ over CM imaginary fields F (with p splitting in F and ρ restricted to any decomposition group at p being reducible with all Jordan-Hölder factors of dimension at most 2). It also claims results on Breuil's locally analytic socle conjecture in certain non-trianguline cases. Both are obtained by establishing an R=T-type result over the GL₂(ℚ_p)-ordinary families considered by Breuil-Ding.

Significance. If the R=T result holds inside the Breuil-Ding families and feeds a valid patching argument, the work would extend automorphy lifting to a class of representations with reducible local conditions at p (beyond the usual irreducible or trianguline settings). The explicit use of existing ordinary families and standard patching techniques is a positive feature that keeps the argument within the scope of current methods.

major comments (1)

The central R=T result over the GL₂(ℚ_p)-ordinary families is load-bearing for all stated lifting theorems, yet the provided text supplies no equations, local-condition verifications, or multiplicity-one statements that would allow checking whether the families satisfy the required global and local properties used in the patching argument.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need for greater explicitness around the central R=T result. We address the major comment below and will revise the paper accordingly.

read point-by-point responses

Referee: The central R=T result over the GL₂(ℚ_p)-ordinary families is load-bearing for all stated lifting theorems, yet the provided text supplies no equations, local-condition verifications, or multiplicity-one statements that would allow checking whether the families satisfy the required global and local properties used in the patching argument.

Authors: We agree that the manuscript would benefit from more explicit detail on this point. The R=T isomorphism is established by adapting the Breuil-Ding ordinary families to the essentially conjugate self-dual setting with the given reducible local conditions at p (JH factors of dimension ≤2). The local conditions at primes above p are inherited directly from the Breuil-Ding construction and are verified to be compatible with the global patching setup in the CM field case (p split in F). Multiplicity one follows from the fact that the ordinary Hecke algebras in these families are étale over the weight space after the R=T identification. Nevertheless, the current text does not spell out the equations or the verifications in sufficient detail for independent checking. In the revised version we will add an expanded subsection (likely in Section 3) containing the explicit local-condition statements, the relevant multiplicity-one result, and the precise form of the patched module used in the argument. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation chain rests on establishing a new R=T result inside the GL2(Qp)-ordinary families of Breuil-Ding, then feeding that into standard patching for automorphy lifting. The abstract presents the R=T step as the novel contribution rather than a re-derivation or fit of the input families themselves. No equations or sections are supplied that reduce any claimed prediction to a fitted parameter or to a self-citation chain by construction. Reliance on the prior Breuil-Ding families is external to the new result and does not trigger any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The paper is a proof in algebraic number theory and relies on standard background results in Galois cohomology, deformation theory, and automorphic forms; no free parameters, invented entities, or ad-hoc axioms are visible from the abstract.

pith-pipeline@v0.9.0 · 5643 in / 1326 out tokens · 43393 ms · 2026-05-24T20:31:53.540778+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages · 1 internal anchor

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Companion points and locally analytic socl e for GL 2(L)

Yiwen Ding. Companion points and locally analytic socl e for GL 2(L). Israel Journal of Mathematics, 231:47–122, 2019

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Jacquet modules of locally analytic r epresentations of p-adic reductive groups I

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Matthew Emerton. Jacquet modules of locally analytic r epresentations of p-adic reductive groups II. The relation to parabolic induction. 2007. to app ear in J. Institut Math. Jussieu

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Ordinary parts of admissible represe ntations of p-adic reductive groups I

Matthew Emerton. Ordinary parts of admissible represe ntations of p-adic reductive groups I. Deﬁnition and ﬁrst properties. Ast´ erisque, 331:355–402, 2010

work page 2010
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Local-global compatibility in the p-adic Langlands programme for GL 2/ Q

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A geometric perspective o n the Breuil-M´ ezard conjecture

Matthew Emerton and Toby Gee. A geometric perspective o n the Breuil-M´ ezard conjecture. Journal of the Institute of Mathematics of Jussieu , 13(1):183–223, 2014

work page 2014
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Patching and the completed ho mology of locally symmetric spaces

Toby Gee and James Newton. Patching and the completed ho mology of locally symmetric spaces. arXiv preprint arXiv:1609.06965 , 2016

work page arXiv 2016
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Modularity lifting theorems for ordin ary galois representations

David Geraghty. Modularity lifting theorems for ordin ary galois representations. Mathematis- che Annalen , 373(3):1341–1427, 2019

work page 2019
[26]

The Fontaine-Mazur conjecture for GL 2

Mark Kisin. The Fontaine-Mazur conjecture for GL 2. Journal of the American Mathematical Society, 22:641–690, 2009

work page 2009
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On Jordan-H¨ older s eries of some locally analytic repre- sentations

Sascha Orlik and Matthias Strauch. On Jordan-H¨ older s eries of some locally analytic repre- sentations. Journal of the American Mathematical Society , 28(1):99–157, 2015

work page 2015
[28]

The Fontaine-Mazur conjecture in the residual ly reducible case

Lue Pan. The Fontaine-Mazur conjecture in the residual ly reducible case. arXiv preprint arXiv: 1901.07166 , 2019

work page arXiv 1901
[29]

On some crystalline representations of GL 2(Qp)

Vytautas Paˇ skunas. On some crystalline representations of GL 2(Qp). Algebra Number Theory, 3:411–421, 2009

work page 2009
[30]

The image of Colmez’s Montreal functor

Vytautas Paˇ sk¯ unas. The image of Colmez’s Montreal functor. Publications math´ ematiques de l’IH ´ES, 118(1):1–191, 2013

work page 2013
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Blocks for modp representations of GL 2(Qp)

Vytautas Paˇ sk¯ unas. Blocks for modp representations of GL 2(Qp). London Math. Soc. Lecture Note Ser. , 415:231–247, 2014

work page 2014
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On some consequences of a theorem of J

Vytautas Paˇ sk¯ unas. On some consequences of a theorem of J. Ludwig. ArXiv preprint: arXiv:1804.07567, 2018

work page arXiv 2018
[33]

Banach space re presentations and Iwasawa theory

Peter Schneider and Jeremy Teitelbaum. Banach space re presentations and Iwasawa theory. Israel journal of mathematics , 127(1):359–380, 2002

work page 2002
[34]

On the p-adic cohomology of the Lubin-Ta te tower

Peter Scholze. On the p-adic cohomology of the Lubin-Ta te tower. Ann. Sci. ´Ec. Norm. Sup´ er., 51(4):811–863, 2018. 50

work page 2018
[35]

Automorphy for some ℓ-adic lifts of automorphic mod ℓ Galois representations, II

Richard Taylor. Automorphy for some ℓ-adic lifts of automorphic mod ℓ Galois representations, II. Publications math´ ematiques de l’IH´ES, 108(1):183–239, 2008

work page 2008
[36]

On the automorphy of ℓ-adic Galois representations with small residual image

Jack Thorne. On the automorphy of ℓ-adic Galois representations with small residual image. J. Inst. Math. Jussieu , 11:855–920, 2012. 51

work page 2012

[1] [1]

Potential automorphy and change of weight

Thomas Barnet-Lamb, Toby Gee, David Geraghty, and Richa rd Taylor. Potential automorphy and change of weight. Annals of Mathematics , 179(2):501–609, 2014

work page 2014

[2] [2]

A family of calabi- yau varieties and potential automorphy ii

Tom Barnet-Lamb, David Geraghty, Michael Harris, and Ri chard Taylor. A family of calabi- yau varieties and potential automorphy ii. Publications of the Research Institute for Mathe- matical Sciences Kyoto , 47:29–98, 2011

work page 2011

[3] [3]

Families of Galois representations and Selmer groups

Jo¨ el Bella ¨ ıche and Ga¨ etan Chenevier. Families of Galois representations and Selmer groups. Ast´ erisque, 324:1–314, 2009

work page 2009

[4] [4]

Sur quelques repr ´ esentations potentiellement cristallines de GL 2(Qp)

Laurent Berger and Christophe Breuil. Sur quelques repr ´ esentations potentiellement cristallines de GL 2(Qp). Ast´ erisque, 330:155–211, 2010

work page 2010

[5] [5]

Vers le socle localement analytique pour GLn, II

Christophe Breuil. Vers le socle localement analytique pour GLn, II. Mathematische Annalen , 361:741–785, 2015

work page 2015

[6] [6]

Vers le socle localement analytique pour GL n, I

Christophe Breuil. Vers le socle localement analytique pour GL n, I. Annales de l’Institut Fourier, 66:633–685, 2016

work page 2016

[7] [7]

Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and local-global compatibility

Christophe Breuil and Yiwen Ding. Higher L-invariants for GL 3(Qp) and local-global compat- ibility. arXiv preprint arXiv:1803.10498

work page internal anchor Pith review Pith/arXiv arXiv

[8] [8]

A local model for the trianguline variety and applications

Christophe Breuil, Eugen Hellmann, and Benjamin Schrae n. A local model for the trianguline variety and applications. 2017. to appear in Publications math´ ematiques de l’IH´ES

work page 2017

[9] [9]

Smoothness and classicality on eigenvarieties

Christophe Breuil, Eugen Hellmann, and Benjamin Schrae n. Smoothness and classicality on eigenvarieties. Inventiones mathematicae, 209(1):197–274, 2017

work page 2017

[10] [10]

Towards the ﬁnit e slope part for GL n

Christophe Breuil and Florian Herzig. Towards the ﬁnit e slope part for GL n. 2018. to appear in Int. Math. Res. Not

work page 2018

[11] [11]

Cohen-Macaulay rings

Winferied Bruns and J¨ rgen Herzog. Cohen-Macaulay rings. Number 39. Cambridge university press, 1998

work page 1998

[12] [12]

Monodromy and local-global compatibili ty for ℓ =p

Ana Caraiani. Monodromy and local-global compatibili ty for ℓ =p. Algebra Number Theory , 8:1597–1646, 2014

work page 2014

[13] [13]

Patching and the p-adic local Langlands correspondence

Ana Caraiani, Matthew Emerton, Toby Gee, David Geraght y, Vytautas Paˇ sk¯ unas, and Sug Woo Shin. Patching and the p-adic local Langlands correspondence. Cambridge jour- nal of mathematics , 4(2):197–287, 2016

work page 2016

[14] [14]

Patching and the p-adic langlands program for G L2(Qp)

Ana Caraiani, Matthew Emerton, Toby Gee, David Geraght y, Vytautas Paˇ sk¯ unas, and Sug Woo Shin. Patching and the p-adic langlands program for G L2(Qp). Compositio Mathe- matica, 154(3):503–548, 2018

work page 2018

[15] [15]

Au tomorphy for some l-adic lifts of automorphic mod l galois representations

Laurent Clozel, Michael Harris, and Richard Taylor. Au tomorphy for some l-adic lifts of automorphic mod l galois representations. Publications math´ ematiques de l’IH´ES, 108(1):1– 181, 2008

work page 2008

[16] [16]

Repr´ esentations de GL2(Qp) et (ϕ, Γ)-modules

Pierre Colmez. Repr´ esentations de GL2(Qp) et (ϕ, Γ)-modules. Ast´ erisque, 330:281–509, 2010. 49

work page 2010

[17] [17]

Companion points and locally analytic socl e for GL 2(L)

Yiwen Ding. Companion points and locally analytic socl e for GL 2(L). Israel Journal of Mathematics, 231:47–122, 2019

work page 2019

[18] [18]

Jacquet modules of locally analytic r epresentations of p-adic reductive groups I

Matthew Emerton. Jacquet modules of locally analytic r epresentations of p-adic reductive groups I. Construction and ﬁrst properties. Annales scientiﬁques de l’ ´Ecole normale sup´ erieure, 39(5):775–839, 2006

work page 2006

[19] [19]

On the interpolation of systems of eig envalues attached to automorphic Hecke eigenforms

Matthew Emerton. On the interpolation of systems of eig envalues attached to automorphic Hecke eigenforms. Inventiones mathematicae, 164:1–84, 2006

work page 2006

[20] [20]

Jacquet modules of locally analytic r epresentations of p-adic reductive groups II

Matthew Emerton. Jacquet modules of locally analytic r epresentations of p-adic reductive groups II. The relation to parabolic induction. 2007. to app ear in J. Institut Math. Jussieu

work page 2007

[21] [21]

Ordinary parts of admissible represe ntations of p-adic reductive groups I

Matthew Emerton. Ordinary parts of admissible represe ntations of p-adic reductive groups I. Deﬁnition and ﬁrst properties. Ast´ erisque, 331:355–402, 2010

work page 2010

[22] [22]

Local-global compatibility in the p-adic Langlands programme for GL 2/ Q

Matthew Emerton. Local-global compatibility in the p-adic Langlands programme for GL 2/ Q

work page

[23] [23]

A geometric perspective o n the Breuil-M´ ezard conjecture

Matthew Emerton and Toby Gee. A geometric perspective o n the Breuil-M´ ezard conjecture. Journal of the Institute of Mathematics of Jussieu , 13(1):183–223, 2014

work page 2014

[24] [24]

Patching and the completed ho mology of locally symmetric spaces

Toby Gee and James Newton. Patching and the completed ho mology of locally symmetric spaces. arXiv preprint arXiv:1609.06965 , 2016

work page arXiv 2016

[25] [25]

Modularity lifting theorems for ordin ary galois representations

David Geraghty. Modularity lifting theorems for ordin ary galois representations. Mathematis- che Annalen , 373(3):1341–1427, 2019

work page 2019

[26] [26]

The Fontaine-Mazur conjecture for GL 2

Mark Kisin. The Fontaine-Mazur conjecture for GL 2. Journal of the American Mathematical Society, 22:641–690, 2009

work page 2009

[27] [27]

On Jordan-H¨ older s eries of some locally analytic repre- sentations

Sascha Orlik and Matthias Strauch. On Jordan-H¨ older s eries of some locally analytic repre- sentations. Journal of the American Mathematical Society , 28(1):99–157, 2015

work page 2015

[28] [28]

The Fontaine-Mazur conjecture in the residual ly reducible case

Lue Pan. The Fontaine-Mazur conjecture in the residual ly reducible case. arXiv preprint arXiv: 1901.07166 , 2019

work page arXiv 1901

[29] [29]

On some crystalline representations of GL 2(Qp)

Vytautas Paˇ skunas. On some crystalline representations of GL 2(Qp). Algebra Number Theory, 3:411–421, 2009

work page 2009

[30] [30]

The image of Colmez’s Montreal functor

Vytautas Paˇ sk¯ unas. The image of Colmez’s Montreal functor. Publications math´ ematiques de l’IH ´ES, 118(1):1–191, 2013

work page 2013

[31] [31]

Blocks for modp representations of GL 2(Qp)

Vytautas Paˇ sk¯ unas. Blocks for modp representations of GL 2(Qp). London Math. Soc. Lecture Note Ser. , 415:231–247, 2014

work page 2014

[32] [32]

On some consequences of a theorem of J

Vytautas Paˇ sk¯ unas. On some consequences of a theorem of J. Ludwig. ArXiv preprint: arXiv:1804.07567, 2018

work page arXiv 2018

[33] [33]

Banach space re presentations and Iwasawa theory

Peter Schneider and Jeremy Teitelbaum. Banach space re presentations and Iwasawa theory. Israel journal of mathematics , 127(1):359–380, 2002

work page 2002

[34] [34]

On the p-adic cohomology of the Lubin-Ta te tower

Peter Scholze. On the p-adic cohomology of the Lubin-Ta te tower. Ann. Sci. ´Ec. Norm. Sup´ er., 51(4):811–863, 2018. 50

work page 2018

[35] [35]

Automorphy for some ℓ-adic lifts of automorphic mod ℓ Galois representations, II

Richard Taylor. Automorphy for some ℓ-adic lifts of automorphic mod ℓ Galois representations, II. Publications math´ ematiques de l’IH´ES, 108(1):183–239, 2008

work page 2008

[36] [36]

On the automorphy of ℓ-adic Galois representations with small residual image

Jack Thorne. On the automorphy of ℓ-adic Galois representations with small residual image. J. Inst. Math. Jussieu , 11:855–920, 2012. 51

work page 2012