{"id":"4b38171c-644c-4526-a3df-7b426a2c67da","arxiv_id":"2607.16893","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Twisted theta integrals over non-Galois quartic CM fields equal Doi–Naganuma lifts of Hecke's integral, making twisted CM values of Borcherds forms algebraic multiples of logarithms of units.","lead":"The paper proves twisted Siegel–Weil formulas for non-Galois quartic CM fields, identifying a twisted theta integral against a quadratic character with the Doi–Naganuma lift of Hecke's integral. This yields algebraicity results: twisted CM values of Borcherds forms are algebraic multiples of logarithms of units, with coefficients given by Fourier coefficients of twisted theta integrals.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Main theorems depend on an auxiliary degree-32 field M32 whose unconditional existence is only cited, not proved; if [31, Prop. 2.1] does not cover every non-Galois quartic CM field, the central formulas collapse.","rationale":"The reader's weakest_assumption identifies exactly the M32 existence issue, and the paper's own text supports this: §2.1 says 'suppose M32 is a degree 4 Galois extension...' and cites rather than proves the existence result. This assumption is load-bearing because Lemma 2.1, Lemma 2.2, and Lemma 5.5 all rely on the concrete Galois group and Frobenius/splitting behavior of M32, without which the character χ and the unramified Whittaker matching are undefined. The central Theorem 5.3 is therefore conditional on this external existence statement, so the reader's CONDITIONAL verdict is appropriate. I do not see an internal inconsistency that would force REJECT; the local matching at unramified primes is detailed, and the acknowledged ramified-place and isometry-dependence gaps are limitations but not by themselves fatal to the representation-theoretic isomorphism claimed in Theorem 5.3. Thus no verdict change is needed.","tokens_in":40234,"tokens_out":18454,"duration_ms":186994,"concrete_test":"Check the cited construction computationally. For a concrete non-Galois quartic CM field, e.g. K4=Q(sqrt(-3+sqrt(2))), form M8=K4(sqrt(-3-sqrt(2))), F=Q(sqrt(7)), and K8 as the fixed field of ⟨(1,0,0)⟩ in the group (Z/4Z)^2⋊Z/2Z. Write K8=F[X]/(f(X)) and construct L=F[X,Y]/(f(X), τ(f)(Y)). Verify with Magma/PARI that L/Q is Galois of degree 32 with Gal(L/Q) ≅ (Z/4Z)^2⋊Z/2Z and that the splitting behavior of unramified p matches table I.(1)–II.(3). Repeat for several K4s, varying D2 and α. If any K4 yields a smaller Galois group or a different Frobenius pattern, the main theorems require an extra existence hypothesis. Alternatively, verify directly that [31, Prop. 2.1] proves existence for all non-Galois quartic CM fields.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is the unconditional existence of M32. Section 2.1 introduces it with 'suppose M32 is a degree 4 Galois extension of M8 such that ...' and the only support is a citation to the proof of [31, Proposition 2.1]; the displayed construction M32 ≅ F[X,Y]/(f(X), τ(f)(Y)) assumes K8 has the form F[X]/(f(X)). This extension is not removable: Lemma 2.1 defines χ through the Artin map to Gal(M32/eK4), Lemma 2.2 derives the central-character identity using the precise Frobenius action in Gal(M32/Q), and Lemma 5.5 matches local Whittaker functions at unramified primes using the same Frobenius table. The statements of Theorems 1.1, 1.2, 5.3, and 1.3 impose no hypothesis that such M32 exists. If the embedding problem behind [31, Prop. 2.1] is obstructed for even one non-Galois quartic CM field, χ is undefined and the proof of Theorem 5.3 cannot be started. The paper itself records related unresolved points (ramified-place invariant vectors fail, Prop. 5.6; twisted CM values depend on a W+≅W− isometry), but those affect refinements; the M32 existence is the load-bearing gap for the central claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims twisted Siegel-Weil formulas for non-Galois quartic CM fields. It constructs a quadratic character χ on norm-one ideles of the reflex field eK4 via an auxiliary degree-32 Galois extension M32, and a Hecke character ρ on A_F^× satisfying a central-character identity (Lemmas 2.1–2.2). It computes Whittaker functions of the twisted theta integral and of the Doi-Naganuma lift of Hecke's integral (Propositions 4.2, 4.5, 5.7), matches them at almost all finite primes (Lemma 5.5), and invokes strong multiplicity one to conclude πχ ≅ Θ(πρ) (Theorem 5.3). As an application, it proves an explicit formula for twisted CM values of Borcherds forms (Theorem 1.3).","tokens_in":40628,"tokens_out":14917,"duration_ms":148232,"significance":"If the proof is completed, this is a substantial contribution: it would give a twisted Siegel-Weil identity in a non-Galois quartic CM setting, an isomorphism interpretation of the base change of Jacquet-Langlands for the relevant characters, and an explicit algebraicity/log-unit formula for twisted CM values. The local Whittaker computations are concrete and checkable, and the strategy of using strong multiplicity one rather than full local matching is attractive. However, the central argument depends on an auxiliary field whose existence is only cited, and some representation-theoretic prerequisites for the strong multiplicity one step are not stated. These gaps are fixable but are load-bearing.","major_comments":[{"comment":"The characters χ and ρ, and the Frobenius table used in Lemma 5.5, are defined through an auxiliary degree-32 extension M32 with Gal(M32/Q) ≅ (Z/4Z)^2 ⋊ Z/2Z. The text says only 'suppose M32 is a degree 4 Galois extension of M8 such that...' and cites the proof of [31, Proposition 2.1]. The main theorems do not state the existence of M32 as a hypothesis. Since χ is undefined without M32, this is load-bearing. Please include a complete proof or precise quotation of the existence result for every non-Galois quartic CM field; if the embedding problem is not always solvable, restrict the main theorems to the fields for which M32 exists.","section":"§2.1, Eq. (2.1); Theorems 1.1–1.3 and 5.3"},{"comment":"The theorem is proved by matching local Whittaker functions at almost all finite primes and then applying strong multiplicity one ([38, Thm 4.10]). Strong multiplicity one applies to irreducible cuspidal automorphic representations. The paper proves (Prop. 5.1) that Θ(χ) is such a representation, but it does not prove that the Doi-Naganuma lift space Θ(πρ) is an irreducible cuspidal automorphic representation of GL2(A_{eF2}), nor that the Whittaker functions computed in Prop. 4.5 are those of local newforms. Without this, the strong multiplicity one step is incomplete. The admitted failure of ramified matching (Prop. 5.6) is not by itself fatal, but it makes this step essential. Please add the required cuspidality/irreducibility statement and a precise derivation of (5.2) from the representation isomorphism.","section":"§5.3 / proof of Theorem 5.3"},{"comment":"The left-hand side of (1.7) involves twisted CM cycles Zχ(W±), which depend on the choice of isometry W+(A_{eF2,f}) ≅ W−(A_{eF2,f}) (as acknowledged in the Outlook). The theorem is stated unconditionally and does not specify this choice. The proof of Theorem 1.3 uses the equality (φ+_μ,Ξ+_μ) = (φ−_μ,Ξ−_μ) from (6.12), which relies on that isometry. Either prove independence of the natural choices (analogous to [4, Lemma 4.2]) or state the result with the choice made explicit and the dependence described.","section":"Theorem 1.3 and §3.5/Remark 3.2"}],"minor_comments":[{"comment":"Lemma 2.1 defines χ only on norm-one ideles, while §4.1 uses χ as a Hecke character on A_{eK4}^×. Please specify the extension of χ to all of A_{eK4}^× and explain how the central character identity in Lemma 2.2 is affected.","section":"§2.2 and §4.1"},{"comment":"References [5] and [6] appear to be duplicates (both Bruinier–Yang, 'CM-values of Hilbert modular functions', Invent. Math. 163.2 (2006), pp. 229–288). Also [33] and [34] are the same Lion–Vergne book. Please consolidate.","section":"References"},{"comment":"There are several typos and notational inconsistencies: 'twiste' in the abstract, 'Therem' in the Outlook, and the mixed notation SL2(bZ) versus SL_2(\\widehat{Z}). Please proofread.","section":"Throughout"},{"comment":"The field diagram is dense and some labels (e.g., K16, M16, K) are not clearly defined in the text. A short explanation of the diagram would improve readability.","section":"§2.1 diagram"}],"recommendation":"major_revision","confidential_remarks":"The manuscript relies heavily on the author's related works [31] and [32]; in particular, the central auxiliary field M32 is only cited to [31]. I recommend that the editor ensure the existence claim in [31] indeed covers the present generality, and that the author be asked to include the statement or proof rather than a citation. The paper is promising but should not be accepted before this gap is closed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague, read this if you work on Siegel-Weil formulas or CM values. The paper proves a twisted Siegel-Weil formula for non-Galois quartic CM fields: the twisted theta integral against a quadratic character χ equals the Doi-Naganuma lift of Hecke's integral for a matching Hecke character ρ. That gives an isomorphism of Jacquet-Langlands base change for these characters, and as an application, twisted CM values of Borcherds forms are expressed as rational multiples of logarithms of units. This is genuinely new; previous work by Li and the author treated the trivial character over biquadratic/cyclic quartic CM fields.\n\nWhat is good: the proof strategy is transparent. They match local Whittaker functions at almost all primes (Lemma 5.5) using explicit calculations in the mixed model, then invoke strong multiplicity one. The archimedean matching is also done (Prop 5.7). The application to Borcherds forms is worked out in detail, following Bruinier-Li-Yang. The paper is honest about its limitations.\n\nNow the soft spots. The most load-bearing gap is the auxiliary degree-32 field M32. Section 2.1 says \"suppose M32 is a degree 4 Galois extension of M8 such that...\" with a specific Galois group and Frobenius behavior, and the only support is a citation to [31, Prop. 2.1]. The character χ is defined via the Artin map to Gal(M32/F4-tilde), and the central-character identity in Lemma 2.2 uses the Frobenius table. If such an extension does not exist for every non-Galois quartic CM field, the main theorems only hold for fields admitting it, but the theorem statements impose no such hypothesis. This is not necessarily fatal if [31, Prop 2.1] indeed proves existence, but the paper should state the result and prove it or put \"exists M32\" as an explicit assumption.\n\nSecond, the paper explicitly says ramified-place matching of invariant vectors fails (Prop 5.6) and \"needs some new ideas.\" I do not think this undercuts Theorem 5.3, because strong multiplicity one only needs almost-all-place matching and the \"for any ϕ\" part follows from the two spaces being the same automorphic representation. But the isomorphism is shown abstractly, not with an explicit local matching at ramified primes. Third, Theorem 1.3 depends on the choice of isometry between W+ and W−, as the paper admits. That is a real caveat for the arithmetic application.\n\nThe paper also leans on two preprints, [1] and [32], one of which is the author's own companion work. That is normal but means the referee must check both.\n\nOverall: the central argument is plausible, the calculations are detailed, and the gaps are explicit and addressable. This deserves a serious referee, not a desk reject. I would recommend conditional acceptance, with the M32 existence question either proved or made into a hypothesis, and the isometry-dependence stated clearly in Theorem 1.3.","headline":"Genuinely new twisted Siegel-Weil formulas for non-Galois quartic CM fields; the proof is coherent and honest, but the main theorem's dependence on an auxiliary degree-32 field M32 needs to be either proved or explicitly assumed.","tokens_in":41134,"tokens_out":4356,"would_cite":true,"duration_ms":46829,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F27","11F41","11G15","11F67"],"pacs":[],"model":"deepseek-v4-flash","headline":"Twisted theta integrals against a quadratic character equal Doi-Naganuma lifts of Hecke integrals, making the Doi-Naganuma base change of Jacquet-Langlands an isomorphism.","keywords":["twisted Siegel-Weil formula","Doi-Naganuma lift","non-Galois quartic CM field","Jacquet-Langlands correspondence","Borcherds forms","CM values","theta integral","Weil representation"],"falsifier":"Fix a non-Galois quartic CM field, say with discriminant d_{K4}=p²q, and compute for a single unramified prime p the local Whittaker values W(t(α)) in Proposition 4.2 and the corresponding Doi-Naganuma-lift values in Proposition 4.5 for all α in a small set; agreement is expected if the theorem is correct, and a mismatch would falsify the local matching lemma on which the theorem rests. A more direct test is to check whether the degree-32 extension M₃₂ with the required Frobenius data exists for that field; if it does not, the main theorems simply do not apply to it.","tokens_in":40086,"feed_emoji":"🧮","tokens_out":9703,"duration_ms":86892,"temperature":0.7,"pith_summary":"The paper establishes twisted Siegel-Weil formulas for GL₂ over non-Galois quartic CM fields: a theta integral of a Schwartz function against a specially constructed quadratic character on the norm-one ideles of the CM field is shown to coincide with a Doi-Naganuma lift of Hecke's integral (a theta lift of a Hecke character from a real quadratic field). The proof matches local Whittaker functions of both sides at almost all places using the Weil representation in the mixed model and the explicit splitting behaviour of primes in an auxiliary degree-32 extension, then invokes strong multiplicity one for Hilbert modular forms. A corollary is that the Doi-Naganuma lift realizes the base change of the Jacquet-Langlands correspondence for these characters. As an application, the paper derives a formula expressing twisted CM values of Borcherds forms on a Hilbert modular surface as algebraic multiples of logarithms of units, with the coefficients coming from Fourier coefficients of the twisted theta integral.","feed_headline":"Twisted theta integral equals Doi-Naganuma lift","feed_subtitle":"For non-Galois quartic CM fields, the identity gives explicit algebraic CM values of Borcherds forms.","key_machinery":"The central mechanism is the Weil representation extended to the subgroup GL₂ × GO(V) via the similitude relation, used to define the twisted theta integral (1.1) and the Doi-Naganuma lift (1.3). The load-bearing identities are the explicit local Whittaker-function computations (Propositions 4.2 and 4.5): for the Schwartz functions ϕ₀ and (φ₀, Ξ₀) attached to maximal integral lattices, the values at t(α) of the local Whittaker functions of θ_{eK4,χ} and I(·, φ₀, Ξ₀, ρ) are matched case-by-case according to the Frobenius element in Gal(M₃₂/Q). The auxiliary degree-32 field M₃₂ with Galois group (Z/4Z)² ⋊ Z/2Z supplies the quadratic character χ (Lemma 2.1) and the character ρ (Lemma 2.2), and","core_discovery":"The central discovery is Theorem 5.3: for the quadratic character χ on the norm-one ideles of the non-Galois quartic CM field eK4 (built from an auxiliary degree-32 extension) and the Hecke character ρ on the real quadratic field F fixed by Lemma 2.2, the automorphic representation π_χ generated by the twisted theta integral θ_{eK4,χ} is isomorphic to the Doi-Naganuma lift Θ(π_ρ) of the representation generated by Hecke's integral θ_ρ. Equivalently, every twisted theta integral against χ equals, as an automorphic form, a Doi-Naganuma lift of Hecke's integral for suitable Schwartz functions. The proof matches local Whittaker functions at unramified primes for the specific Schwartz functions a","pith_inferences":["Beyond the paper: the proof matches local Whittaker functions only at almost all places; the paper notes that at ramified primes the standard invariant vectors do not work. An explicit matching at those places would turn the existence statement in (1.4) into a fully explicit, uniform identity, and would likely illuminate the ramified behaviour of the theta correspondence.","Beyond the paper: since the Fourier coefficients W_f(α, φ_μ) are rational, the twisted CM value formula suggests p-adic or congruence-theoretic analogues of the stated algebraic multiples of log units; such congruences could be tested computationally for small discriminants, where the paper says numerical examples already agree.","Beyond the paper: the auxiliary field M₃₂ is introduced by a 'suppose' statement; determining exactly which non-Galois quartic CM fields admit it with the required Frobenius behaviour is a concrete open problem, and the main theorems apply only to those fields.","Beyond the paper: the explicit Whittaker expansion opens the way to computing L-functions and epsilon factors of the base-changed representation in terms of the original characters, as the author suggests; this could give new evidence for the arithmetic of abelian varieties of GL₂-type over real quadratic fields."],"forward_implications":["The isomorphism π_χ ≅ Θ(π_ρ) makes the Doi-Naganuma lift a base change of the Jacquet-Langlands correspondence: a certain GL₂(Q) representation goes to a GL₂(A_{eF2}) representation under the lift.","For every Schwartz function ϕ on the CM field, there exist finitely many Schwartz functions (φ_i, Ξ_i) such that θ_{eK4,χ}(g, ϕ) = I(h, φ, Ξ, ρ) for g = h ∈ GL₂(A_{eF2}); the equality holds as automorphic forms, not merely as abstract representations.","The twisted CM values of Borcherds forms are algebraic multiples of logarithms of units, and the formula (1.7) gives them explicitly from rational Fourier coefficients c(−m, μ), Legendre polynomials P_r, rational Whittaker values W_f(α, φ_μ), and log|λ_α/λ′_α|.","The method replaces the classical Eisenstein-series route to Siegel-Weil formulas with a Whittaker-matching proof, showing that for these characters the theta correspondence is an isomorphism of automorphic representations."],"fun_headline_variants":["Twisted theta integral matches Doi-Naganuma lift","Explicit algebraic CM values via twisted theta","Quartic CM: theta equals Doi-Naganuma lift","Borcherds CM values from theta integral identity","Non-Galois quartic fields: theta is lift"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"Everything rests on the unconditional existence of the auxiliary degree-32 Galois extension M₃₂ over Q with Galois group (Z/4Z)² ⋊ Z/2Z and the exact Frobenius and splitting behaviour tabulated in Section 2.1; the characters χ and ρ, and hence the local Whittaker matching, are defined through it.","fun_headline_variants_meta":{"raw":{"variants":["Twisted theta integral matches Doi-Naganuma lift","Explicit algebraic CM values via twisted theta","Quartic CM: theta equals Doi-Naganuma lift","Borcherds CM values from theta integral identity","Non-Galois quartic fields: theta is lift"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00032,"raw_usage":{"total_tokens":1591,"prompt_tokens":647,"completion_tokens":944,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":391,"completion_tokens_details":{"reasoning_tokens":867}},"tokens_in":391,"tokens_out":944,"duration_ms":6778,"temperature":1.0,"reasoning_tokens":867,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T19:37:13.818889+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Fix a non-Galois quartic CM field, say with discriminant d_{K4}=p²q, and compute for a single unramified prime p the local Whittaker values W(t(α)) in Proposition 4.2 and the corresponding Doi-Naganuma-lift values in Proposition 4.5 for all α in a small set; agreement is expected if the theorem is correct, and a mismatch would falsify the local matching lemma on which the theorem rests. A more direct test is to check whether the degree-32 extension M₃₂ with the required Frobenius data exists for that field; if it does not, the main theorems simply do not apply to it.","supporting_citations":[],"review_version":1}