{"id":"5a01d8e6-a6b6-4526-8e3c-aaa80873ea0a","arxiv_id":"2506.21981","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Hilbert modular forms of even weight and trivial character can be computed as orthogonal modular forms, with all Atkin-Lehner eigenspaces captured by a radical character.","lead":"This paper gives a fast algorithm for computing Hilbert modular forms via ternary quadratic forms, using a new 'radical character' to recover all Atkin-Lehner eigenspaces. The method resolves a gap in Birch's 1991 approach and is implemented to handle many new cases, including large-conductor computations.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 7.5's one-sentence proof does not show that orthogonal degeneracy maps preserve the radical-character eigenspaces, so the computed 'new' space may not be the advertised S_new_k(bΓ0(N))_ε.","rationale":"The central claim of the paper is an algorithm that outputs S_new_k(bΓ0(N))_ε. The algorithm computes on the orthogonal side; the bridge is Theorem 1.4 (Corollary 6.12) for full spaces and Theorem 7.5 for new subspaces. The proof of Theorem 7.5 is a single assertion that the new subspaces correspond 'by definition'. The subtlety is real: the radical-character space is not a subspace of the right-invariant space M_k(SO(bΛ)), and the degeneracy maps in Definition 7.4 are defined only on right-invariant spaces. Without proving that pullback from Λ' carries the appropriate character space into the νM-eigenspace, the old subspaces on the two sides could differ, making the algorithm return the wrong Hecke module. The reader's weakest assumption identifies exactly this step. My proposed computational check on a level with oldforms would settle whether the compatibility holds in a concrete instance; this is feasible because the authors ship code and the expected output is known classically. The running-time fluctuation d vs d^2 noted by the reader is a separate issue and does not affect correctness. Hence the CONDITIONAL verdict remains appropriate.","tokens_in":24855,"tokens_out":18596,"duration_ms":199129,"concrete_test":"Use the authors' C++ or Magma implementation for F=Q, k=2, squarefree N=15 with sign vector ε_3=-1, ε_5=+1. Construct Λ via Algorithm 8.1; compute M(SO(Λ), ν_3); define the old subspace as the span of pullbacks from superorders of levels 3 and 5 carrying the appropriate restricted radical characters; take the orthogonal complement; compute Hecke matrices T_2 and T_7. Compare dimension and Hecke eigenvalues with the known space S_new_2(Γ_0(15))_{ε}. Agreement supports Theorem 7.5; disagreement falsifies it.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 7.5 is the only step connecting the space actually computed (orthogonal modular forms with radical character νM) to the advertised output (new Hilbert/quaternionic forms with Atkin-Lehner signs ε). Its proof reads: 'We restrict Corollary 6.12 to the new subspace on quaternionic modular forms; by definition, this corresponds to the new subspace on orthogonal modular forms.' This is insufficient. Corollary 6.12 is about the full spaces M_k(SO(bΛ), νM) and M_k(bO×)_{εM}. The quaternionic old subspace is generated by pullbacks from superorders O' ⊇ O. The orthogonal old subspace (Definition 7.4) is generated by pullbacks from lattices Λ' with SO(bΛ) ≤ SO(bΛ'). But those pullbacks map M_W(SO(bΛ')) into M_W(SO(bΛ)), the space of SO(bΛ)-invariant functions, whereas M_k(SO(bΛ), νM) consists of functions transforming by the character νM under SO(bΛ) and is not contained in M_W(SO(bΛ)). The paper never defines degeneracy maps on the νM-isotypic components, never identifies which character on Λ' restricts to νM, and never proves the commuting diagram that would make the old subspaces correspond. If the radical character mixes with the Atkin-Lehner signs under degeneracy, the orthogonal complement computed by the algorithm would not be S_new_k(bΓ0(N))_ε.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper presents an algorithm to compute Hilbert modular forms of even weight and trivial central character as orthogonal modular forms on a totally definite ternary quadratic space, extending Birch's method to totally real fields, nonsquare levels, and all Atkin-Lehner eigenspaces. The main theoretical result, Theorem 1.4, is a Hecke-equivariant bijection between cuspidal orthogonal modular forms with a radical character and quaternionic cusp forms with specified Atkin-Lehner eigenvalues; Theorem 7.5 upgrades this to new subspaces. The paper further gives a complexity analysis, an algorithm for constructing suitable lattices and orders, and reports on implementations in Magma and C++. The main advertised computational output is Theorem 1.3: for a totally real field F, even weight k, factored level N with [F:Q] even or N nonsquare, and sign vector ε, the new space S_new_k(bΓ0(N))_ε can be computed as a Hecke module with running time eO(d^2 Nm(p)) for T_p, where d is the dimension of the output space.","tokens_in":25125,"tokens_out":7323,"duration_ms":85269,"significance":"If the main theorems are correct, this is a substantial advance: it gives a uniform, practical method for computing Hilbert modular forms in cases where modular symbols are expensive or infeasible, and it resolves both known limitations of Birch's ternary-form method by introducing the radical character. The even Clifford algebra approach is conceptually transparent and provides a canonical, Hecke-equivariant correspondence without analytic theta-series arguments. The paper also contains an explicit categorical inverse to the even Clifford functor (Appendix A) and concrete computational examples, including a large-level example that outperforms modular symbols. These are real strengths. However, the proof of the newform-theoretic bridge, Theorem 7.5, is currently a one-sentence assertion, and this is load-bearing for the advertised output; the complexity statement also contains an internal inconsistency. The underlying strategy appears defensible, but the manuscript as submitted does not yet establish the central claim.","major_comments":[{"comment":"The proof of Theorem 7.5 is a single sentence: 'We restrict Corollary 6.12 to the new subspace on quaternionic modular forms; by definition, this corresponds to the new subspace on orthogonal modular forms.' This is not sufficient. Corollary 6.12 gives a Hecke-equivariant bijection between the full spaces M_k(SO(bΛ), νM) and M_k(bO)_{εM}. The orthogonal old subspace, by Definition 7.4, is generated by pullbacks α_Λ' from lattices Λ' corresponding to superorders O' ⊇ O, with SO(bΛ) ≤ SO(bΛ'). To restrict Corollary 6.12 to new subspaces, one must prove that these degeneracy maps are compatible with the radical character: for each such Λ' there should be a character ν'_M on SO(bΛ') whose restriction to SO(bΛ) equals νM, and the induced map should land in M_k(SO(bΛ), νM). The manuscript does not define such characters on the superorder lattices, does not prove that νM extends or restricts in the required way, and does not show that the orthogonal old subspace corresponds, under Corollary 6.12, to the quaternionic old subspace generated by pullbacks from O' ⊇ O. Without this compatibility, the orthogonal complement computed by the algorithm is not identified with S_new_k(bΓ0(N))_ε. This is the only place where the space actually computed (orthogonal new space with radical character) is connected to the space advertised in Theorem 1.3 (Hilbert new space with Atkin-Lehner signs), so the gap is load-bearing.","section":"§7, Theorem 7.5"},{"comment":"The complexity statement is internally inconsistent. Theorem 1.3 states that, for fixed F and N, the algorithm takes eO(d^2 Nm(p)) bit operations to compute T_p, where d = dim_C S_new_k(bΓ0(N))_ε. The paragraph immediately after Theorem 1.3 claims: 'After a precomputation (to set up the lattice), the running time becomes eO(d Nm(p))'. These two displayed statements cannot both be true with the same meaning of d. Theorem 8.5 gives yet another expression, eO(Nm(p) H_{3n}(∥Λ∥ d^2)), where d = #Cl(Λ), which is not obviously reconciled with either statement in §1. The authors should state clearly the worst-case cost, the amortized cost after precomputation, and the precise role of the output dimension d in each formula.","section":"§1, Theorem 1.3 and following paragraph"},{"comment":"Related to the gap in Theorem 7.5, the definition of the orthogonal old subspace is not extended to the radical-character twisted setting. Definition 7.4 defines the old subspace inside M_W(SO(bΛ)) using degeneracy maps α_Λ' that pull back untwisted functions from SO(bΛ')-invariant classes. But the spaces in Corollary 6.12 and Theorem 7.5 are the twisted spaces M_k(SO(bΛ), νM). The paper never states what the degeneracy maps are on these twisted spaces, which character on the superorder lattice Λ' is used, or why the images land in the νM-isotypic component rather than mixing different radical characters. This is not just a presentational gap: if the radical character mixes under pullback, the orthogonal complement of the old subspace in M_k(SO(bΛ), νM) need not be the νM-component of the new space, and Theorem 7.5 would fail. A complete proof should include a commutative diagram for the degeneracy maps on both sides.","section":"§7, Definition 7.4 and §6, Corollary 6.12"}],"minor_comments":[{"comment":"The phrase 'isotopic decomposition' should be 'isotypic decomposition'.","section":"§3, Remark 3.9"},{"comment":"The symbol O is overloaded: on the right-hand side of (6.3), O(rad_p(Λ)/2NΛ_p) denotes the orthogonal group, while elsewhere O denotes a quaternion order. Using a different symbol, such as Aut or Orth, would avoid confusion.","section":"§6, equation (6.3)"},{"comment":"The role of d = #Cl(Λ) in the Hermite normal form complexity H_{3n}(∥Λ∥ d^2) is not explained; in particular, it is unclear whether the factor d^2 is a worst-case bound for the size of the lattice entries after reduction, and how this relates to the dimension d of the output space in Theorem 1.3.","section":"§8, Theorem 8.5"},{"comment":"Theorem 1.3 promises to 'return for each ideal n coprime to N a matrix [T_n]', but the algorithms and complexity analysis in §8 only discuss the prime Hecke operator T_p. The authors should state how composite T_n are obtained (e.g., by the standard recurrence from the T_p for p not dividing N), or otherwise clarify the scope of the theorem.","section":"§1, theorem statement"}],"recommendation":"major_revision","confidential_remarks":"I recommend major revision. The central idea—using the radical character to recover all Atkin-Lehner eigenspaces via the even Clifford algebra—is promising and the computational evidence is strong. However, the proof of the newform bridge (Theorem 7.5) is missing a genuinely nontrivial compatibility argument, and this is the step that connects the computed object to the advertised Hilbert new space. The complexity inconsistency in §1 should also be fixed. I would not recommend reject, because the gap appears fixable by proving the missing compatibility lemma; but the paper should not be accepted until Theorem 7.5 has a complete proof. The heavy reliance on [Voi21], an author's own monograph, is acceptable as those results are published and independently checkable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: this is a serious algorithmic paper with a real new idea—the radical character—that plausibly fixes the two deficiencies Birch described. But there is a genuine gap in the newform-theoretic bridge (Theorem 7.5), and until it is closed the main theorem should not be taken as proved.\n\nWhat is actually new: the radical character of Section 6 is a clean way to select each Atkin–Lehner eigenspace separately, and the extension to Hilbert modular forms over arbitrary totally real fields goes beyond the theses of Tornaría and Hein. The even Clifford algebra functor gives a transparent bijection between class sets, and the algorithms are concrete and implemented. The example at level 1,062,347 running in seconds is persuasive evidence that the method works in practice.\n\nThe soft spot: Theorem 7.5 is a single sentence: 'We restrict Corollary 6.12 to the new subspace on quaternionic modular forms; by definition, this corresponds to the new subspace on orthogonal modular forms.' That is not enough. The degeneracy maps αΛ' are defined on full spaces M_W(SO(bΛ')) → M_W(SO(bΛ)) via pullback, and their images land in the SO(bΛ)-invariant functions. The space in the theorem, M_k(SO(bΛ), νM), transforms by a nontrivial character. The paper never defines degeneracy maps on the νM-isotypic components, never says which character on Λ' restricts to νM, and never proves the diagram that would make the old subspaces correspond. Without that, the 'new subspace' on the orthogonal side as defined in Definition 7.4 has no obvious relation to the new subspace on the quaternionic side when νM is nontrivial. The squarefree example in Section 9 avoids the issue because there the full space is already new; the gap only shows up for levels with oldforms, which is exactly the case Birch said was missing. So this is a load-bearing gap, not a cosmetic one.\n\nMinor: the running-time bound in Theorem 1.3 says O~(d^2 Nm(p)) but the discussion and the later Theorem 8.5 suggest O~(d Nm(p)) after precomputation; that should be reconciled.\n\nThe rest of the paper is careful: the Clifford algebra background is standard, the cited results are independent, and the radical character is not fitted—it is defined from the lattice and then shown to match Atkin–Lehner signs. I believe the gap is fixable: one needs to define the correct character twist on the source level and prove the naturality. As written, the paper deserves a serious referee, but the referee should ask for a full proof of Theorem 7.5, not just an assertion.\n\nRecommendation: send it to peer review with a request for major revision on Section 7. I would not desk-reject.","headline":"Strong algorithmic paper with a genuinely new idea (the radical character), but the one-sentence proof of Theorem 7.5 leaves a real gap in the newform theory that needs to be closed before the main theorem is established.","tokens_in":25678,"tokens_out":6375,"would_cite":false,"duration_ms":62990,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F41","11F27","11R52","11E12","11Y16"],"pacs":[],"model":"deepseek-v4-flash","headline":"Even-weight Hilbert modular newforms can be computed through ternary quadratic forms, with each Atkin–Lehner sign sector obtained by a radical character.","keywords":["Hilbert modular forms","orthogonal modular forms","ternary quadratic forms","even Clifford algebra","quaternion orders","Atkin-Lehner operators","p-neighbors","radical character"],"falsifier":"For a composite squarefree level such as $N=33$ over $\\mathbb{Q}$, compute the degeneracy matrix between the radical-character component $\\nu_r$ on $\\mathrm{Cl}(\\Lambda)$ and the component for a super-lattice $\\Lambda'$ via the pullback defined in Section 7; if the matrix is not diagonal with respect to the Atkin–Lehner eigenvalues, the new subspace obtained by restricting Corollary 6.12 is not the Hilbert new space. A simpler check: the multiplicities from Proposition 4.20 predict the dimensions of $S_k^{\\mathrm{new}}(\\Gamma_0(33))_\\varepsilon$, and comparing those dimensions with the output of an orthogonal implementation for each $\\varepsilon$ would settle the one-sentence step in Theorem 7.5.","tokens_in":24638,"feed_emoji":"🧮","tokens_out":6678,"duration_ms":73207,"temperature":0.7,"pith_summary":"This paper claims that Hilbert modular forms of even weight and trivial central character can be computed by converting them into orthogonal modular forms on ternary quadratic forms, and it gives an explicit algorithm that does this for every totally real field except the parity case where the field degree is odd and the level is a square. The core of the method is a Hecke-equivariant bijection between cuspidal orthogonal modular forms for a ternary lattice with a prescribed radical character and quaternionic cusp forms with matching Atkin–Lehner eigenvalues; the Eichler–Shimizu–Jacquet–Langlands correspondence then carries these to Hilbert newforms. If the claims are correct, computing a new space is reduced to enumerating isometry classes of ternary lattices and applying p-neighbor operators, with running time roughly linear in the output dimension after a precomputation.","feed_headline":"Ternary quadratic forms yield all Hilbert modular newforms","feed_subtitle":"Algorithm computes each Atkin–Lehner eigenspace of even-weight Hilbert cusp forms as a Hecke module in near-linear time.","key_machinery":"The even Clifford algebra functor $\\Lambda \\mapsto \\mathrm{Clf}^0(\\Lambda)$ from integral ternary quadratic lattices to quaternion orders, which identifies the class set of $\\Lambda$ with the type set of $\\mathcal{O}$; and the radical character $\\nu_p : SO(\\Lambda_p) \\to \\{\\pm 1\\}$, defined by the action of an isometry on the line $\\mathrm{rad}(\\Lambda_p)/2N\\Lambda_p$ and extended multiplicatively to $\\nu_M$. The functor makes the orthogonal and quaternionic Hecke modules canonically isomorphic, and the radical character selects the Atkin–Lehner eigenspace that the Hilbert newforms occupy. On the computational side, Kneser's $p$-neighbor method and reduction of ternary quadratic forms supply the Hecke operators.","core_discovery":"On the paper's own terms, the central discovery is that the missing half of Birch's method is supplied by a character: for each Atkin–Lehner sign vector $\\varepsilon$ the paper defines a radical character $\\nu_\\varepsilon$ on the special orthogonal group of a ternary lattice, and proves (Theorem 1.4, generalized in Theorem 7.5) a Hecke-equivariant bijection between the cuspidal orthogonal modular forms $S_k(SO(\\Lambda), \\nu_\\varepsilon)$ and the quaternionic forms $S_k(\\widehat{O})_\\varepsilon$ with Atkin–Lehner signs $\\varepsilon$. Combined with the Eichler–Shimizu–Jacquet–Langlands correspondence (Theorem 4.10), this yields all spaces $S_k^{\\mathrm{new}}(\\Gamma_0(N))_\\varepsilon$ whenever a suitable quaternion order exists, i.e., unless $[F:\\mathbb{Q}]$ is odd and $N$ is a square. The bijection is canonical because the even Clifford algebra functor sends class sets of ternary lattices to type sets of quaternion orders, so the two function spaces are literally identified; the radical character files the classes according to the action on the one-dimensional radical of the $p$-adic lattice, which matches the Atkin–Lehner signs.","pith_inferences":["The radical-character construction is likely to adapt to nontrivial central characters or bounded level: the paper's framework already handles residually unramified orders, and its Remark 4.9 points toward residually ramified orders as a further extension.","Because the spinor norm has the explicit formula $\\theta(\\sigma)=1+\\operatorname{tr}(\\sigma)$ in dimension three, the radical character can be evaluated directly from the trace of an isometry, which may simplify implementations and extend the method to lattices whose Clifford order is not residually unramified.","A natural test beyond the paper is to compare the orthogonal-side output with quaternionic or modular-symbol computations at composite levels where the old subspace has several layers; the multiplicity formula in Proposition 4.20 predicts exactly which newforms appear in each component.","The speed demonstrated on $F=\\mathbb{Q}$ suggests that sampling Hecke eigenvalues in a fixed Atkin–Lehner eigenspace could make searches for elliptic curves of moderately large conductor practical, an application the authors hint at but do not develop."],"forward_implications":["For any totally real field $F$, even weight $k$, factored level $N$ with $[F:\\mathbb{Q}]$ even or $N$ nonsquare, and any sign vector $\\varepsilon$, the new space $S_k^{\\mathrm{new}}(\\Gamma_0(N))_\\varepsilon$ is exhibited as a Hecke module by matrices obtained from ternary lattice classes.","The Hecke matrices are sparse, and the running time for $T_p$ is $\\widetilde{O}(d^2 \\operatorname{Nm}(p))$ for fixed $F,N$, reducing to $\\widetilde{O}(d \\operatorname{Nm}(p))$ after lattice setup, so the cost is roughly linear in the dimension $d$ of the output space.","The two deficiencies Birch identified—retrieving only half the forms and losing information when the level is not squarefree—are both removed: every Atkin–Lehner eigenspace is obtained, for arbitrary level outside the square-odd-degree exception.","The dimension of each $\\varepsilon$-eigenspace is asymptotically $2^{-r} \\operatorname{Nm}(N) |k-1|$ by the mass formula, so splitting the new space by sign vectors does not make any individual piece asymptotically negligible."],"supporting_citations":[{"why":"supplies the original ternary-quadratic-form method and documents the two failures the paper addresses.","marker":"[Bir91]"},{"why":"introduces $\\Theta$-equivalence, refining isometry to recover the missing forms in the squarefree case.","marker":"[Tor05]"},{"why":"extends Birch's construction to totally real fields and interprets the resulting forms via the even Clifford algebra.","marker":"[Hei16]"},{"why":"provides the even Clifford functor, type sets, residually unramified orders, and the normalizer computations used throughout.","marker":"[Voi21]"},{"why":"supplies the Eichler–Shimizu–Jacquet–Langlands correspondence that moves quaternionic forms to Hilbert modular forms.","marker":"[JL70]"},{"why":"supplies the Atkin–Lehner newform theory for Hilbert modular forms used to compute multiplicities and signs.","marker":"[RS07]"},{"why":"gives the lattice-method algorithms for orthogonal modular forms that the paper analyzes and extends.","marker":"[GV14]"},{"why":"furnishes the p-neighbor theory that underlies both class-set enumeration and the Hecke operator computation.","marker":"[Kne57]"}],"fun_headline_variants":["Character completes Birch's method for Hilbert forms","Radical character unlocks all Hilbert newforms","Hecke-equivariant bijection yields all newforms","Ternary lattices and quaternions unify Hilbert forms","Missing half of Birch's method supplied by character"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is one sentence in the proof of Theorem 7.5: restricting the bijection of Corollary 6.12 to new subspaces preserves the radical-character labeling, so that the orthogonal degeneracy maps behave exactly like the quaternionic ones; if those maps mix the $\\nu_\\varepsilon$ components, the computed new space would be the wrong Hecke module.","fun_headline_variants_meta":{"raw":{"variants":["Character completes Birch's method for Hilbert forms","Radical character unlocks all Hilbert newforms","Hecke-equivariant bijection yields all newforms","Ternary lattices and quaternions unify Hilbert forms","Missing half of Birch's method supplied by character"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001215,"raw_usage":{"total_tokens":4930,"prompt_tokens":806,"completion_tokens":4124,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":422,"completion_tokens_details":{"reasoning_tokens":4066}},"tokens_in":422,"tokens_out":4124,"duration_ms":28554,"temperature":1.0,"reasoning_tokens":4066,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T22:14:39.696168+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a composite squarefree level such as $N=33$ over $\\mathbb{Q}$, compute the degeneracy matrix between the radical-character component $\\nu_r$ on $\\mathrm{Cl}(\\Lambda)$ and the component for a super-lattice $\\Lambda'$ via the pullback defined in Section 7; if the matrix is not diagonal with respect to the Atkin–Lehner eigenvalues, the new subspace obtained by restricting Corollary 6.12 is not the Hilbert new space. A simpler check: the multiplicities from Proposition 4.20 predict the dimensions of $S_k^{\\mathrm{new}}(\\Gamma_0(33))_\\varepsilon$, and comparing those dimensions with the output of an orthogonal implementation for each $\\varepsilon$ would settle the one-sentence step in Theorem 7.5.","supporting_citations":[],"review_version":1}