{"id":"b0387ae7-ef71-41e1-8db1-2bfb072e735d","arxiv_id":"1908.07929","paper_version":3,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every even N at least 6, the paper produces infinitely many SO_{N+1}-valued residual Galois representations that are SO_{N+1}-irreducible but GL_{N+1}-reducible, each with a geometric lift of Zariski-dense image.","lead":"This number theory paper constructs many Galois representations that are irreducible for a classical group but reducible as ordinary linear representations, and shows they can still be lifted to geometric p-adic representations by a Galois-deformation method. The examples draw a clear boundary where the potential-automorphy toolkit is not known to apply.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Oddness of the constructed SO_{N+1}-representations rests on the real-fibre classification in §5.2/Appendix A; a misidentification of split/non-split types over R would break Lemma 5.3 and the FKP19 lifting hypothesis.","rationale":"The reader's weakest_assumption correctly identifies the real-fibre classification and the resulting trace of complex conjugation as the most load-bearing point. I agree with that assessment. The paper's proof of Lemma 5.3 is a genuine self-contained calculation, but its input from §5.2 is asserted rather than fully demonstrated: the validity of Tate's algorithm split/non-split data over R is a specific, checkable claim, and the sign/Tate-twist comparison in the proof, while plausible, is delicate. I do not see an actual error in the paper, and the construction is otherwise supported by cited external results (Zywina, Hall, Booher, [FKP19]) and internal checks. The proposed test would independently confirm the trace for at least one case; if it passes, the central claim stands. Therefore the reader's ACCEPT verdict should remain unchanged.","tokens_in":19140,"tokens_out":44535,"duration_ms":434090,"concrete_test":"For one representative case, say N ≡ 2 (mod 8) (Proposition A.6 Case (1)), write down the explicit Weierstrass equation from [Zyw19, §6], specialize to a rational w not in the thin set, and use a computer algebra system to classify each singular fibre over R by Kodaira type and real structure (split/non-split, which irreducible components are defined over R), e.g. by real factoring of the cubic near each critical point. Then compute χ(E_x(R)) and the trace of F_∞ on the fibre component spaces, and check that the totals produce Tr(c|H^1(P^1_C, j_*E[ℓ])) = -2 as in Lemma 5.3. If the fibre types match the table in Proposition A.6 and the trace is -2, the concern is settled; if they differ, oddness and hence Theorem 6.1(1) fail for this case.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim depends on the representations in Theorem 6.1 being odd in the sense of Definition 1.1, so that Theorem 2.1 applies. Oddness is verified by Lemma 5.3, whose proof reduces the trace of complex conjugation on H^1(P^1_C, j_*E[ℓ]) to the real fibre data of Proposition A.6. The bridge is the assertion in §5.2 that Zywina's split/non-split classifications of I_n fibres, normally obtained by Tate's algorithm over complete fields with algebraically closed residue fields, remain valid over R (so, e.g., an I_1 fibre at infinity is non-split because −3 is not a square in R), and that the component traces in Lemma A.5 reflect the same real structures. If this bridge is wrong—for instance, if a fibre that is 'split' over a finite field becomes a different real type over R because the relevant square class changes, or if the real structure of an I_n^* fibre has a different number of components defined over R—then the computed traces (5.1) would be off, the fixed subspace of Ad(\\barρ(c)) would have the wrong dimension, and the oddness hypothesis of [FKP19] would fail for the constructed \\barρ. This is the only place where real-geometric input enters, so it is the least secure condition in the chain.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":19417,"tokens_out":29516,"duration_ms":277746,"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].","major_comments":[],"minor_comments":[{"comment":"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.","section":"Theorem 6.1, proof"},{"comment":"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":"Proposition 4.1, proof"},{"comment":"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.","section":"Section 3, paragraph after the matrix display"},{"comment":"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.","section":"Theorem 6.1, proof, Case 6O row of the table"},{"comment":"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.","section":"Lemma 5.3 and §5.2"}],"recommendation":"minor_revision","confidential_remarks":"This is a well-written note whose central construction is sound. The only point that goes beyond a local clarification is the unproved existence of a solvable totally real F with prescribed completion L; I think this is standard and fixable by a citation, but it should not be left to the reader. The real-fibre trace computation, which was the main technical risk, checks out on close reading."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.","headline":"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.","tokens_in":19979,"tokens_out":2169,"would_cite":true,"duration_ms":55631,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F80","11G05","14J27","11R32"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Galois representations","geometric lifts","classical groups","orthogonal groups","elliptic surfaces","oddness","deformation theory","potential automorphy"],"falsifier":"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.","tokens_in":18931,"feed_emoji":"🔢","tokens_out":9853,"duration_ms":90412,"temperature":0.7,"pith_summary":"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.","feed_headline":"Galois reps irreducible in SO yet reducible in GL now lift","feed_subtitle":"Infinitely many such SO_{N+1} representations over a solvable totally real field admit geometric ℓ-adic lifts, for every even N≥6.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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 $\\mathrm{SO}_N$."],"supporting_citations":[{"why":"Theorem 2.1, the relative deformation theorem that produces the global geometric lift from oddness and local lifts; the central lifting engine.","marker":"[FKP19]"},{"why":"Elliptic-surface construction of orthogonal Galois representations with image $\\Omega(V_\\ell)$, the source of the examples in the main theorem.","marker":"[Zyw19]"},{"why":"Earlier version used for Case 6O fibre data and the corrected root-number calculation in Proposition 5.1.","marker":"[Zyw14]"},{"why":"Local lifting theorem at ramified primes away from $\\ell$, used to satisfy the local hypotheses of the lifting theorem.","marker":"[Boo19]"},{"why":"Classification of Euler characteristics of real elliptic fibres used in Lemma A.4 and hence in the trace computation.","marker":"[Sil84]"},{"why":"Decomposition theorem supplying the direct-sum decomposition of $H^2$ of the elliptic surface used to isolate the trace on the relevant cohomology in Appendix A.","marker":"[BBD82]"},{"why":"Hilbert irreducibility theorem used to produce infinitely many specializations with prescribed Galois image and fixed fields.","marker":"[Ser08]"},{"why":"Potential inverse Galois problem with local conditions used in the softer examples of Section 4, showing the phenomenon is not specific to elliptic surfaces.","marker":"[MB90]"}],"fun_headline_variants":["SO-irreducible yet GL-reducible Galois reps now lift via elliptic surfaces","Geometric ℓ-adic lifts for SO-irreducible but GL-reducible reps","Elliptic surfaces yield geometric lifts for odd SO Galois reps","Odd SO reps that are GL-reducible finally lift geometrically"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["SO-irreducible yet GL-reducible Galois reps now lift via elliptic surfaces","Geometric ℓ-adic lifts for SO-irreducible but GL-reducible reps","Elliptic surfaces yield geometric lifts for odd SO Galois reps","Odd SO reps that are GL-reducible finally lift geometrically"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000976,"raw_usage":{"total_tokens":4162,"prompt_tokens":973,"completion_tokens":3189,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":589,"completion_tokens_details":{"reasoning_tokens":3109}},"tokens_in":589,"tokens_out":3189,"duration_ms":21900,"temperature":1.0,"reasoning_tokens":3109,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:54:31.650306+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}