Lifting $G$-irreducible but $\mathrm{GL}_n$-reducible Galois representations

Chandrashekhar Khare; Najmuddin Fakhruddin; Stefan Patrikis

Lifting $G$-irreducible but $\mathrm{GL}_n$-reducible Galois representations

T0 review · 0 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For every even N≥6 and all large primes ℓ, the paper constructs infinitely many SO_{N+1}-valued Galois representations that are irreducible as SO_{N+1} representations yet reducible inside GL_{N+1}, and proves each has a geometric lift.

desk verdict A compact, honest paper that supplies the first systematic examples where FKP19-style lifting works and potential automorphy does not; the oddness computation is the part to check, and it looks right. read the letter →

arxiv 1908.07929 v3 pith:FMY6SA6W submitted 2019-08-21 math.NT

Najmuddin Fakhruddin , Chandrashekhar Khare , Stefan Patrikis This is my paper · ORCID

classification math.NT MSC 11F8011G0514J2711R32

keywords Galoisrepresentationsgeometricliftsclassicalgroupsorthogonalellipticsurfacesoddnessdeformationtheorypotentialautomorphy

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper constructs many Galois representations that can be lifted to geometric ℓ-adic representations by relative Galois deformation theory, even though the standard potential-automorphy route cannot reach them. The signature feature of the examples is that they are G-irreducible but GL-reducible: a representation valued in the classical group SO_{N+1} that is irreducible there, yet visibly preserves a subspace after embedding in GL_{N+1}. The main theorem states that for every even N≥6 and every sufficiently large prime ℓ, there are infinitely many non-isomorphic such representations over a solvable totally real field F, each admitting a geometric lift to SO_{N+1}(Z_ℓ) with Zariski-dense image; under a plausible local lifting hypothesis the same holds over F=Q. The paper also shows that in some cases the underlying SO_N representation is not odd and cannot be lifted by any existing method, so the passage to SO_{N+1} is essential.

What carries the argument

The load-bearing identity is $\bar\rho=\bar\theta\oplus 1$: adjoining a trivial line to an orthogonal representation turns a representation that is irreducible in the special orthogonal group into one that is also irreducible in the larger split group $\mathrm{SO}_{N+1}$, because the only invariant subspaces are non-isotropic and hence not stabilized by any parabolic. The oddness check reduces to a trace computation: oddness is equivalent to $\operatorname{Tr}(\operatorname{Ad}\bar\rho(c))=-\operatorname{rk}(G)$ for complex conjugation $c$, and the paper computes this trace from the action of complex conjugation on $H^1(\mathbb{P}^1_{\mathbb{C}}, j_*E[\ell])$ of a real elliptic surface, using a Lefschetz fixed-point lemma and the decomposition theorem to relate the trace to the Euler characteristic of the real fibres. The third ingredient is the authors' relative deformation theorem, which converts the verified oddness and local liftability into a global geometric lift.

What would settle it

Locate a real point $w$ of the parameter space in one of the cases of Lemma 5.3 where the real singular fibres differ from the configurations listed in Proposition A.6, for example the $N=6$ Case 6$\Omega$ family; then by Lemmas A.4 and A.5 the trace of $c$ on the cohomology group $H^1(\mathbb{P}^1_{\mathbb{C}}, j_*E[\ell])$ would differ from the value $2$ recorded in (5.1), invalidating the oddness hypothesis used to apply the lifting theorem.

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Extended reading notes

Core claim

The central discovery is a supply of residual representations $\bar\rho:\Gamma_F\to \mathrm{SO}_{N+1}(\mathbb{F}_\ell)$ that are $\mathrm{SO}_{N+1}$-irreducible but $\mathrm{GL}_{N+1}$-reducible and that satisfy the oddness hypothesis needed for the lifting theorem of the authors' earlier work. The construction takes a Galois representation $\bar\theta:\Gamma_{\mathbb{Q}}\to \mathrm{O}(V_\ell)$ with large image $\Omega(V_\ell)$ obtained from an elliptic-surface monodromy family, and sets $\bar\rho=\bar\theta\oplus 1$. The only proper invariant subspaces of $\bar\rho$ are the $N$-dimensional space and the complementary line; because these are non-isotropic, the image lies in no proper parabolic subgroup of $\mathrm{SO}_{N+1}$, giving SO-irreducibility, while the invariant line makes it GL-reducible. The main technical work is to establish oddness: the trace of complex conjugation on the relevant cohomology of a real elliptic surface is computed by a topological calculation, and this forces the adjoint fixed-space dimension to equal $\dim\mathrm{Flag}(\mathrm{SO}_{N+1})$. The authors then arrange local behaviour at primes above $\ell$ by passing to a solvable totally real extension (or by assuming local Hodge–Tate regular de Rham lifts when $F=\mathbb{Q}$), and apply their relative lifting theorem to obtain geometric lifts with Zariski-dense image.

Load-bearing premise

The whole argument depends on the claim that complex conjugation acts on the cohomology of the real elliptic surfaces with the traces listed in (5.1); if that trace calculation were off, the oddness condition required by the lifting theorem would not be met.

Editorial extensions

If this is right

For each even $N\ge 6$ and all sufficiently large $\ell$, the construction yields infinitely many non-isomorphic liftable $\mathrm{SO}_{N+1}$-valued residual representations, so the phenomenon is not rare.
These examples are outside the reach of potential automorphy lifting theorems, which require $\mathrm{GL}$-irreducibility; the paper's lifting method works precisely in the gap between $G$-irreducibility and $\mathrm{GL}$-irreducibility.
In the cases $N\equiv 2\pmod 8$ and $N\equiv 6\pmod 8$ (Case 6$\Omega$), the underlying $\mathrm{SO}_N$ representation $\bar\theta$ is not odd and cannot be lifted to a Hodge–Tate regular $\mathrm{SO}_N$-valued representation by existing methods; only after embedding into $\mathrm{SO}_{N+1}$ does a lift exist.
The same local-condition argument shows that if local Hodge–Tate regular de Rham lifts for $\mathrm{SO}_{N+1}$ are known over $\mathbb{Q}_\ell$, then the examples can be taken over $F=\mathbb{Q}$.
The authors observe that these examples suggest congruences between non-endoscopic cusp forms on $\mathrm{Sp}_N$ and endoscopic forms on the split endoscopic group $\mathrm{SO}_N$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The parabolic-stabilizer argument is insensitive to the choice of the invariant subspace, so the same construction should work with a character twist or a different embedded representation in place of the trivial line, producing analogous $G$-irreducible/$\mathrm{GL}$-reducible examples for other classical groups.
The trace-of-complex-conjugation technique gives a general recipe: for any monodromy representation coming from an elliptic surface over $\mathbb{Q}$, the archimedean oddness condition is determined by the real Kodaira fibre configuration, so one can systematically search such families for new liftable examples by tabulating real fibre types.
If current work on local crystalline lifting (extending the $\mathrm{GL}_N$ results to other groups) becomes unconditional, the paper's conditional $F=\mathbb{Q}$ statement would follow immediately, placing infinitely many non-$\mathrm{GL}$-irreducible geometric $\mathrm{SO}_{N+1}$-representations over $\mathbb{Q}$.
The twist by a quadratic character used in Case 6$\Omega$ suggests a general 'oddness restoration' step: a non-odd representation valued in one orthogonal group can become odd after a quadratic twist and embedding in the next odd orthogonal group.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper is a clean, useful note. It does not build a new lifting machine; it takes the authors' earlier FKP19 theorem and produces a large family of residual representations that satisfy its hypotheses but sit outside the reach of current potential automorphy methods. The main theorem says exactly that, and the proof is organized around the two genuinely hard things: verifying oddness and verifying the local lifting conditions. The oddness verification is the real mathematical content, and it is done carefully via a topological trace computation on a real elliptic surface. That computation, Lemma 5.3 with Appendix A, is the part of the paper a referee should scrutinize. I did not find an error in it. The authors explicitly address the split-versus-non-split question over R, cite Silhol's classification for real fibres, and compute the relevant square classes. The stress-test worry about a wrong real fibre type is fair as a place to check, but I do not think it lands as an actual flaw. The worst case is that a footnote or a sentence could make the bridge to the real fibre table more explicit. The reliance on the unpublished Zyw14 version for Case 6O is a real but minor weakness; the main theorem does not depend on that case, and the published Zyw19 handles the rest. The notational slip in Proposition 4.1, writing Γ_Q(ζℓ) where Γ_F(ζℓ) is meant, is cosmetic. The self-citation to FKP19 is appropriate: that is a separate theorem with its own proof, and the present paper's contribution is checking its hypotheses in new examples, not restating the theorem. The elementary Section 3 and the Moret-Bailly construction in Section 4 are pleasant illustrations rather than the main event. The paper is also candid about its limitations, especially the conditional F=Q case and the local lifting input it would need. Who should read this? Anyone working on deformation rings, lifting theorems, or the boundary between Galois-theoretic and automorphic methods. It sharpens that boundary and gives reusable examples. It deserves a serious referee and, after the usual checks, acceptance. If I were the editor, I would ask the referee to spend time on the real-fibre trace calculation and then accept.

Referee Report

0 major / 5 minor

Summary. The paper proves Theorem 1.2: for every even N ≥ 6 and every sufficiently large prime ℓ, there are infinitely many non-isomorphic mod-ℓ Galois representations Γ_F → SO_{N+1}(F_ℓ), with F a totally real solvable extension of Q, that are irreducible as SO_{N+1}-valued representations yet reducible as GL_{N+1}-valued representations, and each admits a geometric ℓ-adic lift with Zariski-dense image. Under an additional local lifting hypothesis one may take F = Q. The proof combines the authors' earlier lifting theorem [FKP19] with three families of examples: a simple sum of odd GL_2 representations in §3, a Moret-Bailly potential-inverse-Galois-problem construction in §4, and Zywina's orthogonal Galois representations arising from elliptic surfaces in §5. The main technical work is the verification of the oddness hypothesis in Zywina's examples, which is done by computing the trace of complex conjugation on the relevant ℓ-adic cohomology of a real elliptic surface in Appendix A.

Significance. If the result stands, it provides a genuinely new range of examples where the Galois-deformation lifting method of [FKP19] applies but current potential automorphy methods do not; this sharpens the contrast between G-irreducibility and GL_n-irreducibility and has suggestive consequences for congruences between endoscopic and non-endoscopic automorphic forms. The paper is careful and largely self-contained in its central technical step: the oddness check is reduced to a detailed topological computation on real elliptic surfaces, with explicit trace tables and honest discussion of the conditional F = Q statement. The main risk identified in the stress-testing, namely the passage from Tate-algorithm fibre classifications over finite fields to real fibre types in §5.2, is addressed sufficiently: the only split/non-split information that affects the final traces is explicitly controlled, and the I_n^*-fibre contributions cancel in the trace computation. The paper also gives credit to, and builds on, prior work of Zywina, Booher, and the authors' own [FKP19].

minor comments (5)

[Theorem 6.1, proof] The existence of a solvable totally real extension F/Q with F_v/Q_ℓ isomorphic to L/Q_ℓ for all v|ℓ is asserted without proof or citation. This is not entirely formal, since it is the step that guarantees the final field F is solvable over Q; please add a proof or a precise reference, for example to a Shafarevich-type embedding theorem with local conditions, so that the reader can verify this local-global step.
[Proposition 4.1, proof] In the proof of Proposition 4.1, the notation Γ_{Q(ζ_ℓ)} should be Γ_{F(ζ_ℓ)}: the representation ¯θ is a priori a representation of Γ_F, and the linear-disjointness hypothesis gives the needed image statement after restriction to F(ζ_ℓ), not to Q(ζ_ℓ). This typo appears in at least two places in that proof.
[Section 3, paragraph after the matrix display] The word 'non-isotropic' in the sentence describing the ¯ρ_i should read 'non-isomorphic'; the intended hypothesis is that the restrictions to Γ_{Q(ζ_ℓ)} are pairwise non-isomorphic, not that they are non-isotropic.
[Theorem 6.1, proof, Case 6O row of the table] Please clarify the phrase 'projection to the SO_{N+1}-component' in Case 6O. Since O_{N+1} is identified with SO_{N+1} × {±1}, the projection replaces ϑ_{ℓ,w_i} by (det ϑ_{ℓ,w_i}) ⊗ ϑ_{ℓ,w_i}; as written, the reader has to infer this twist, which is important for the oddness calculation.
[Lemma 5.3 and §5.2] The assertion that the two I_2 fibres in Case 6O comprise one split and one non-split fibre over R is stated rather than shown. A short sign computation using f(−1), f(1), and 1 ± h(w) would make the real-fibre classification transparent. This is worth adding because this is the only delicate point in transferring Zywina's finite-field fibre data to the real trace computation; the I_4^* contributions cancel, so the trace is robust to the unspecified real structure of those fibres.

Circularity Check

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No significant circularity: the paper constructs new examples and verifies the hypotheses of a prior lifting theorem; the trace computation is an independent topological input.

full rationale

The derivation chain is not circular. The main lifting conclusion is delegated to [FKP19, Theorem A], which is a prior theorem by the same authors whose stated hypotheses (oddness, absolute irreducibility after adjoining ζ_ℓ, and local lifts) do not include the target conclusion of this paper. The new content of the paper is the construction of residual representations and the verification of those hypotheses. In particular, the oddness verification in Section 5.2 and Lemma 5.3 is a new topological calculation: the trace of complex conjugation on H^1(P^1_C, j_*E[ℓ]) is reduced, via Appendix A, to real fibre data of elliptic surfaces, and the computation is performed rather than assumed. The twists introduced in the table of Theorem 6.1 are chosen to satisfy the independently stated oddness definition of Definition 1.1, not to reproduce a pre-existing output. No parameter fitting, normalization, or definitional identification forces the constructed representations to have the claimed properties. The conditional F=Q statement is explicitly flagged as depending on an open local lifting question, and the local lifting at places above ℓ is handled by a standard base-extension trivialization, not by importing the desired conclusion. The use of the authors' own [FKP19] is a legitimate citation of an independent theorem with its own proof and assumptions that do not include the present paper's results.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The construction has no fitted parameters. It rests on a stack of external theorems, with one significant self-citation ([FKP19]) that is a separate prior lifting result rather than a restatement of the current conclusions. The paper's own contribution is to verify the hypotheses of that theorem in a new class of examples.

assumptions (7)

domain assumption [FKP19, Theorem A] lifting theorem: odd, G-irreducible residual representations with local lifts admit geometric G-lifts with Zariski-dense derived image.
Used as the main black box in every construction (Theorem 2.1 of the paper). Proved in the authors' prior paper; self-citation but a separate result.
domain assumption Moret-Bailly potential inverse Galois problem with local conditions ([MB90]).
Used in Section 4 to produce extensions F and K with prescribed local behavior and Galois group SO_N(Fℓ).
domain assumption Zywina big orthogonal monodromy construction and specialization theorem ([Zyw19], [Zyw14]).
Supplies residual representations with large image in Ω(Vℓ) and the real fibre configurations used to compute traces.
domain assumption Booher local lifting theorem ([Boo19, Theorem 1.1]).
Provides lifts at ramified primes not above ℓ for the representations constructed in the paper.
domain assumption Müller's crystalline lifting theorem ([Mul13, Theorems 2.5.3 and 2.5.4]).
Provides potentially crystalline local lifts with prescribed Hodge-Tate weights for the GL2 inputs in Section 3.
domain assumption Potential automorphy theorems of [BLGGT14] and Taïbi's Theorem A ([Taï16]).
Used only in Proposition 4.4 to rule out many regular lifts, not used in the main lifting construction.
standard math Standard topological tools: Lefschetz fixed point theorem, decomposition theorem, Betti-étale comparison, Kodaira classification, and Silhol's Euler characteristics of real fibres.
Used in Appendix A to compute the trace of complex conjugation on the cohomology of real elliptic surfaces.

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Cite this review

Pith. "Pith review of Lifting $G$-irreducible but $\mathrm{GL}_n$-reducible Galois representations." pith.science (2026). https://pith.science/paper/FMY6SA6W

@misc{pith2026190807929,
  author       = {Pith},
  title        = {Pith review of: Lifting $G$-irreducible but $\mathrmGL_n$-reducible Galois representations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FMY6SA6W}},
  note         = {Machine review of arXiv:1908.07929}
}

abstract

In recent work, the authors proved a general result on lifting $G$-irreducible odd Galois representations $\mathrm{Gal}(\overline{F}/F) \to G(\overline{\mathbb{F}}_{\ell})$, with $F$ a totally real number field and $G$ a reductive group, to geometric $\ell$-adic representations. In this note we take $G$ to be a classical group and construct many examples of $G$-irreducible representations to which these new lifting methods apply, but to which the lifting methods provided by potential automorphy theorems do not.

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

A. A. Be linson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I ( L uminy, 1981), Ast\'erisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5--171. 751966 (86g:32015)

work page 1981

[2] [2]

Thomas Barnet-Lamb, Toby Gee, David Geraghty, and Richard Taylor, Potential automorphy and change of weight, Ann. of Math. (2) 179 (2014), no. 2, 501--609. 3152941

2014

[3] [3]

Jeremy Booher, Minimally ramified deformations when p , Compos. Math. 155 (2019), no. 1, 1--37. 3875451

work page 2019

[4] [4]

Frank Calegari, Even G alois representations and the F ontaine-- M azur conjecture. II , J. Amer. Math. Soc. 25 (2012), no. 2, 533--554. 2869026

work page 2012

[5] [5]

Frank Calegari , Matthew Emerton , and Toby Gee , Globally realizable components of local deformation rings , arXiv e-prints (2018), arXiv:1807.03529

work page Pith review arXiv 2018

[6] [6]

Matthew Emerton and Toby Gee , Moduli stacks of \'e tale ( , ) -modules and the existence of crystalline lifts , arXiv e-prints (2019), arXiv:1908.07185

work page Pith review arXiv 2019

[7] [7]

Najmuddin Fakhruddin , Chandrashekhar Khare , and Stefan Patrikis , Relative deformation theory and lifting irreducible Galois representations , arXiv e-prints (2019), arXiv:1904.02374

work page Pith review arXiv 2019

[8] [8]

Wushi Goldring and Jean-Stefan Koskivirta, Strata H asse invariants, H ecke algebras and G alois representations , Invent. Math. 217 (2019), no. 3, 887--984. 3989256

work page 2019