{"id":"b819b9de-b0e2-40a0-8734-b31ea0784be6","arxiv_id":"2506.01688","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Incoherent Hilbert Eisenstein series over real quadratic fields are realized as Doi-Naganuma theta lifts, yielding explicit Rankin-Selberg L-functions and a non-unit theorem for Borcherds products at CM points.","lead":"The paper proves that certain Hilbert Eisenstein series over real quadratic fields can be written as Doi-Naganuma lifts of ordinary Eisenstein series over the rationals. It then uses this identity to show that special values of Borcherds products at Heegner points are not integral units once the underlying discriminants are large.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Local surjectivity Proposition 4.10 rests on an unverified ramified-prime nonvanishing computation; the related p=2 case is deferred in Lemma 5.1, and the O(s^n) matching step inherits this fragility.","rationale":"I read the manuscript in good faith and found the main architecture coherent: the global theta integral is unfolded to a product of local factors, the archimedean and unramified non-archimedean matching are standard, and the arithmetic application follows from the explicit L-function formula plus standard coefficient bounds. The most load-bearing step is the local surjectivity statement Proposition 4.10, exactly as the Reader identified. Its proof is a single display whose nonvanishing assertion depends on the structure of the local Fourier transform and on the invariance of the displayed Schwartz function under an Iwahori-type subgroup. These are precisely the kinds of calculations that can fail at ramified primes, especially p=2, and the paper itself defers the p=2 case of the closely related Lemma 5.1. I do not see an actual contradiction in the text, and the argument may well be correct; the issue is that the key computation is not fully written out. I also checked the final exponential decay claim in Section 6.3: using the displayed Theorem 6.4, Duke-type coefficient bounds for the Shintani lift, and the denominator |D1 D2|^{(k-1)/2} / Lambda(1,chi_1) Lambda(1,chi_2), the resulting exponent is indeed negative, so that part of the application does not raise a separate objection. Therefore the Reader's conditional verdict is appropriate; my concern does not change it, but it does identify the precise place where the proof needs strengthening.","tokens_in":37208,"tokens_out":46335,"duration_ms":443173,"concrete_test":"Work out the ramified p=2 case of Proposition 4.10 explicitly, e.g. take F_2 = Q_2(sqrt(2)) and E_2 a ramified quadratic extension. (1) Compute the Fourier transform of the displayed phi and verify directly that phi is invariant under omega(k) for all k in J_2 cap N_2^-; (2) compute integral_{d_2^{-1}} psi(-Tr(mb)) db with the measure used in Section 2 and confirm it is nonzero for every m in O_2; (3) complete the omitted p=2 step of Lemma 5.1 for d = 8 by enumerating the finite quadratic module L_0^vee/L_0 and checking H_mu subset ker(chi) for all nonzero isotropic mu. If either the invariance or the nonvanishing fails at a ramified p=2 prime, Theorem 4.1 and Theorem 1.1 need a modified statement or an additional hypothesis.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central matching theorem (Theorem 4.1, hence Theorem 1.1) is proved by reducing to the local statement Proposition 4.10: the map (Xi_p, phi) -> F_p(.,0; Xi_p, phi) from I(0,chi_{2,p}) x S(V(Q_p)) to I(0,chi_p) is surjective. The proof of Proposition 4.10 supplies, for each local class alpha, a single pair (Xi_p, phi) and asserts the Whittaker coefficient W_m(F) = vol(1+p^n Z_p) integral_{d_p^{-1}} psi(-Tr(mb)) db is nonzero for every m in O_p. This display is the only justification that the projection to every irreducible component is nonzero, so a failure at any ramified prime would invalidate Theorem 4.1. The calculation is genuinely compressed: the invariance of phi under J_p cap N_p^- is established only by asserting invariance of the Fourier transform under a conjugate subgroup, and the normalization of the additive measure and inverse different d_p^{-1} is not stated. Moreover, the paper explicitly defers the p=2 version of the analogous local statement in Lemma 5.1 with 'the argument can be suitably adapted', so the same computation in Proposition 4.10 has no independent check in the ramified p=2 setting. Because Remark 4.11 converts this s=0 surjectivity into the full O(s^n) local matching statement, any gap or counterexample in this nonvanishing computation is load-bearing for the whole paper.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that incoherent Hilbert Eisenstein series for a real quadratic field F, associated with a CM extension E/F with E/Q Galois, can be expressed as the Doi-Naganuma theta lift of an incoherent Eisenstein series over Q (Theorem 1.1). The proof proceeds by unfolding the theta integral to a local matching problem and solving it locally at archimedean, unramified, and ramified places, with the ramified case reduced to a local surjectivity statement (Proposition 4.10). An explicit version is given for biquadratic fields (Theorem 1.2), yielding a formula for a generalized Rankin-Selberg L-function (Theorems 1.4 and 6.4). This is applied to prove that Borcherds products on X_0(N)^2 with effective divisors are not integral units at Heegner points of sufficiently large discriminants (Theorem 1.6), generalizing earlier work of the first author. The paper also contains a new construction of fundamental invariant vectors in the Weil representation of finite quadratic modules (Section 5.1).","tokens_in":37493,"tokens_out":22390,"duration_ms":233603,"significance":"If correct, the main theorem is a significant structural result: it realizes level-one incoherent Hilbert Eisenstein series as Doi-Naganuma lifts, filling a gap for small and non-parallel weight. The explicit local matching and the resulting Rankin-Selberg formula are valuable and give a concrete handle on L-functions that appeared in the work of Bruinier-Kudla-Yang. The non-integral-unit application generalizes a known result for singular moduli to higher level and is a natural target of the method. The paper is largely explicit and builds on standard machinery; the main risk is the compressed local nonvanishing computation and a deferred p=2 case, as detailed below.","major_comments":[{"comment":"The nonvanishing computation is load-bearing for Theorem 4.1 and hence Theorem 1.1, but it is not fully proved. The invariance of φ under J_p∩N_p^- is asserted after a one-sentence Fourier-transform computation, and the normalization of the additive measure and of the volume of J_p∩N_p^- in the definition of C_1(φ) is not stated. In particular, the displayed equality C_1(φ)=φ in the proof should be C_1(φ)=vol(J_p∩N_p^-)φ up to a Weil-index scalar; this does not affect the nonvanishing conclusion, but it must be written correctly. Please expand the computation, including the split case and the action of the conjugate subgroup, so that the surjectivity claim is checkable by the reader.","section":"4.3 (Proposition 4.10)"},{"comment":"Lemma 5.1 defers the p=2 case with 'the argument can be suitably adapted' and footnote 3 'Alternatively, one can check all cases by hand.' Theorem 6.4 is stated for D_1,D_2 not both even, and the construction of φ_{D_1,N_1,N_2} in (6.2) invokes u_{Q_p(√D_1)} for p|D_1; when D_1 is even, this requires exactly the p=2 case of Lemma 5.1. Since Theorem 1.6 relies on Theorem 6.4, the p=2 gap is load-bearing for the full statement of Theorem 1.6. Either prove the p=2 case, or restrict Theorem 6.4 to odd D_1 and explain, using the symmetry of L(s,G;D_1,D_2), why Theorem 1.6 follows for all allowed D_1,D_2.","section":"5.1 (Lemma 5.1) and 6.2"},{"comment":"The proof says 'we denote the set of the other places by S, which is non-empty.' For a Siegel section that is spherical at every finite place, S is empty and the argument does not cover arbitrary C(s). The statement of Theorem 4.1 does not restrict to incoherent sections. Please either restrict the statement to the incoherent setting, where the finite ramified places are naturally present, or modify the proof so that when S is empty one applies Proposition 4.10 at an arbitrary finite prime.","section":"4.4 (Proof of Theorem 4.1)"}],"minor_comments":[{"comment":"In Proposition 4.10, the phrase 'When F_p=Q_p^2, let them be Z_p^2' for O_p and d_p^{-1} is confusing; the split case should be written explicitly in terms of the product structure F_p=Q_p×Q_p.","section":"4.3"},{"comment":"The measure on J_p∩N_p^- in (4.7) is not normalized; since C_1(φ) is defined without a normalization factor, the equality C_1(φ)=φ in the proof of Proposition 4.10 is dimensionally incorrect and should be corrected.","section":"4.3"},{"comment":"Reference [14] and [15] are the same Duke article and should be merged.","section":"References"},{"comment":"The estimate |L'(0,ξ(f);D_1,D_2)| ≪_N |D_1D_2|^{-δ} is asserted from [15]; please state the precise Fourier-coefficient bound used and the resulting δ, since the standard convexity bound would not suffice.","section":"6.3 (Proof of Theorem 1.6)"},{"comment":"In (6.17), the notation Im(R_{H_0}) should be defined; presumably it denotes the image of the raising operator on functions on H_0(R).","section":"6.2 (Lemma 6.3)"}],"recommendation":"major_revision","confidential_remarks":"The paper is substantial and likely correct after revision, but the two load-bearing points—the local nonvanishing computation in Proposition 4.10 and the p=2 case in Lemma 5.1 as used in Theorem 6.4—must be fully resolved before publication. The third major comment on the empty set S is a smaller but real gap in the written proof of Theorem 4.1."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Let me give you my read on 2506.01688. The headline: this is a substantial, likely correct paper that fills a genuine gap. Doi-Naganuma lifts were previously for holomorphic modular forms of parallel weight k large; this paper treats incoherent Hilbert Eisenstein series, which are weight (1,1) and vanish at s=0. The matching theorem (Theorems 1.1 and 4.1) is new, and the applications are real: an explicit Rankin-Selberg L-function left open in Bruinier-Kudla-Yang, and a higher-level generalization of the singular-moduli non-unit theorem.\n\nWhat the paper does well: the local-to-global structure is coherent. Archimedean and unramified matching are done explicitly, and the use of the Kudla-Rallis decomposition plus Whittaker coefficients to prove surjectivity is a sound strategy. Proposition 5.2's construction of chi-isotypic invariant vectors is elegant and independently useful. The explicit matching example in Section 5 is a substantial calculation, and the resulting formula for L(s, G; D1, D2) is concrete and checkable. No circularity; the constants are canonical, and the self-citations are antecedents being generalized.\n\nThe soft spots are real but narrow. Proposition 4.10 is the load-bearing local surjectivity statement, and its proof compresses the nonvanishing of W_m(F(Xi_p, phi)) for all m in O_p. The invariance of phi under the relevant subgroup is asserted through a conjugate subgroup without the measure normalization spelled out. If this fails at a ramified prime, Theorem 4.1 and everything downstream goes with it. The p=2 case is explicitly deferred in Lemma 5.1 with the argument can be suitably adapted, so the same computation has no independent check. I do not see an actual contradiction; the computation looks plausible and likely fixable, but a referee should ask for the details. The subconvexity input in Section 6.3 is used without stating the exponent; that is minor.\n\nWho this is for: anyone working on theta lifts, incoherent Eisenstein series, or CM values of Borcherds products. It deserves a serious referee. I would send it to review and ask for an expanded proof of Proposition 4.10 and the p=2 cases.","headline":"Substantial, likely correct, and fills a real gap; the only serious worry is the compressed local surjectivity proof in Prop 4.10.","tokens_in":38082,"tokens_out":2528,"would_cite":true,"duration_ms":26612,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F41","11F27","11F37","11F67","11G15"],"pacs":[],"model":"deepseek-v4-flash","headline":"For real quadratic fields with Galois CM, every incoherent Hilbert Eisenstein series is a Doi–Naganuma lift of a rational Eisenstein series.","keywords":["incoherent Hilbert Eisenstein series","Doi–Naganuma lift","theta correspondence","Weil representation","Borcherds products","Heegner points","Rankin–Selberg L-function","integral units"],"falsifier":"At a ramified prime $p$ (especially $p=2$), compute the explicit Whittaker coefficient $W_m(F_p(\\cdot,0;\\Xi_p,\\varphi))$ for every class $m\\in\\mathcal{O}_p$ required by Proposition 4.10; a single zero where the proof needs a nonzero value would break the local surjectivity and with it Theorems 1.1 and 1.6.","tokens_in":36961,"feed_emoji":"🧮","tokens_out":17860,"duration_ms":156757,"temperature":0.7,"pith_summary":"This paper shows that incoherent Hilbert Eisenstein series attached to a real quadratic field $F$ are Doi–Naganuma $\\theta$ lifts of incoherent Eisenstein series over $\\mathbb{Q}$, provided the CM extension $E/F$ is Galois over $\\mathbb{Q}$. The proof converts the global identity into a local matching problem: at each place one chooses a rational Eisenstein section $\\Xi$ and a Schwartz function $\\varphi$ so that the $\\theta$ integral reproduces the given Siegel section $\\Phi$ up to $O(s^n)$. The payoff is an explicit description of the generalized Rankin–Selberg $L$-function appearing in the theory of big CM cycles, and a new construction of fundamental invariant vectors for Weil representations of finite quadratic modules. As an arithmetic application, the paper proves that for odd square-free $N$, values of Borcherds products on $X_0(N)^2$ with effective divisors are not integral units when the involved discriminants are sufficiently large.","feed_headline":"Incoherent Hilbert Eisenstein series become Doi–Naganuma lifts","feed_subtitle":"With Galois CM, every such series is a rational theta lift, and Borcherds values fall short of being units.","key_machinery":"The machinery is the Doi–Naganuma $\\theta$ integral $I(h,\\Xi,\\varphi,s)=\\int_{G(\\mathbb{Q})\\backslash G(\\mathbb{A})} E(g,\\Xi,s)\\,\\theta_V(g,h,\\varphi)\\,dg$, which unfolds to a sum over $B(F)\\backslash G(F)$ of local sections $F(h,s;\\Xi,\\varphi)$ as in (2.32). The global matching theorem is assembled from a standard archimedean identity, the spherical matching at unramified primes, and a surjectivity statement (Proposition 4.10) for the map $(\\Xi,\\varphi)\\mapsto F(\\cdot,0;\\Xi,\\varphi)$ from $I(0,\\chi_{2,p})\\times S(V(\\mathbb{Q}_p))$ into $I(0,\\chi_p)$. Surjectivity is detected through Whittaker coefficients: a suitable coefficient is shown to be nonzero for every class $\\alpha\\in F_p^\\times/\\mathrm{Nm}(E_p^\\times)$, and a lemma on simple modules then forces the image to be the whole space. The explicit examples use new fundamental invariant vectors $u_K$ in the finite Weil representation, obtained as the one-dimensional $\\chi$-isotypic part of the group ring.","core_discovery":"The central claim is Theorem 1.1: for any incoherent Siegel section $\\Phi \\in I(s,\\chi)$ attached to a CM extension $E/F$ with $E/\\mathbb{Q}$ Galois, there are a standard rational Eisenstein section $\\Xi \\in I(s,\\chi_2)$ and a Schwartz function $\\varphi\\in S(V(\\mathbb{A}))$ such that $I(h,\\Xi,\\varphi,s)=C(s)E(h,\\Phi,s)+O(s^n)$ for a real-analytic $C(s)$ and any $n\\ge 1$. In the simplest level-one case the identity is exact, with $C(s)$ supplied by the completed $L$-function $\\Lambda(s+1,\\chi_1)$. The paper proves this by showing the local matching map is surjective at every finite prime, and it records explicit matching data in terms of new invariant vectors in the Weil representation. It then substitutes the $\\theta$-integral expression into the CM-value formula for higher Green functions to compute the generalized Rankin–Selberg $L$-function and to derive the non-unit result for Borcherds products.","pith_inferences":["A natural next step is to complete the deferred $p=2$ cases of Lemma 5.1; doing so should remove the odd-square-free technical condition on $N$ in Theorem 1.6, as the paper itself anticipates.","The one-dimensionality of the $\\chi$-isotypic part of the finite Weil representation suggests that the same invariant-vector construction works for other ramified quadratic extensions and finite quadratic modules, yielding explicit local matching data beyond the quartic-Galois setting.","The explicit formula for $L(s,G;D_1,D_2)$ can be tested numerically for small $D_1,D_2$ and $N$ by computing both sides from Fourier expansions, which would also probe the conjectural arithmetic intersection interpretation attached to this $L$-function in [11]."],"forward_implications":["For real quadratic fields with quartic Galois CM, every incoherent Hilbert Eisenstein series is, up to $O(s^n)$, a Doi–Naganuma lift, so its diagonal restriction inherits a spectral expansion from a rational Eisenstein series.","The generalized Rankin–Selberg $L$-function $L(s,G;D_1,D_2)$ studied in [11] is computed explicitly in terms of $L(G,s+k)$, partial zeta values, and the Fourier coefficients $c_{\\tilde G}(|D_1|)c_{\\tilde G}(|D_2|)$ of the Shintani lift; its vanishing is controlled by the Fricke eigenvalue $\\epsilon(G)$.","For odd square-free $N$, the value $\\Psi(\\tau_1,\\tau_2)$ of a Borcherds product on $X_0(N)^2$ with effective divisor is not an integral unit when $\\max(|D_1|,|D_2|)$ is sufficiently large, extending the level-one singular-moduli result.","The new invariant vectors $u_K$ give explicit Schwartz functions at ramified primes, making the matching identity exact in the level-one case and computable in genus-zero examples such as differences of hauptmoduls."],"supporting_citations":[{"why":"It decomposes the local induced representation $I(0,\\chi_p)$ into irreducible summands, the basis of the local surjectivity criterion.","marker":"[31]"},{"why":"It introduces incoherent Eisenstein series, the object whose first derivative is being studied and lifted.","marker":"[27]"},{"why":"It defines the normalized Hilbert Eisenstein series and the generalized Rankin–Selberg $L$-function that the paper computes explicitly.","marker":"[11]"},{"why":"It provides the CM-value formula for higher Green functions that converts the theta-lift identity into the Borcherds-product application.","marker":"[10]"},{"why":"It supplies the higher Green functions, Heegner points, and the Shintani-lift Fourier coefficients used in the explicit $L$-function formula.","marker":"[18]"},{"why":"It establishes the level-one singular-moduli non-unit result and the overall strategy that Theorem 1.6 generalizes.","marker":"[34]"},{"why":"It supplies the unfolding-and-matching method for theta integrals and the regularized theta-lift framework for Green functions.","marker":"[9]"},{"why":"It provides the subconvexity and Fourier-coefficient bounds for half-integral weight forms behind the size estimate in Theorem 1.6.","marker":"[15]"}],"fun_headline_variants":["Incoherent Hilbert Eisenstein series are Doi–Naganuma lifts","Doi–Naganuma lifts realize all incoherent Hilbert Eisenstein series","Borcherds values fail to be integral units in higher levels","Hilbert Eisenstein series: a Doi–Naganuma lift with new invariants"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof stands on the local assertion that for every ramified prime the map $(\\Xi,\\varphi)\\mapsto F(\\cdot,0;\\Xi,\\varphi)$ is surjective onto $I(0,\\chi_p)$, which hinges on a nonvanishing calculation of Whittaker coefficients whose $p=2$ cases are deferred.","fun_headline_variants_meta":{"raw":{"variants":["Incoherent Hilbert Eisenstein series are Doi–Naganuma lifts","Doi–Naganuma lifts realize all incoherent Hilbert Eisenstein series","Borcherds values fail to be integral units in higher levels","Hilbert Eisenstein series: a Doi–Naganuma lift with new invariants"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000931,"raw_usage":{"total_tokens":3971,"prompt_tokens":914,"completion_tokens":3057,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":530,"completion_tokens_details":{"reasoning_tokens":2973}},"tokens_in":530,"tokens_out":3057,"duration_ms":23984,"temperature":1.0,"reasoning_tokens":2973,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T11:36:38.264757+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"At a ramified prime $p$ (especially $p=2$), compute the explicit Whittaker coefficient $W_m(F_p(\\cdot,0;\\Xi_p,\\varphi))$ for every class $m\\in\\mathcal{O}_p$ required by Proposition 4.10; a single zero where the proof needs a nonzero value would break the local surjectivity and with it Theorems 1.1 and 1.6.","supporting_citations":[{"cited_title":"Kudla and Stephen Rallis,Ramified degenerate principal series representations forSp(n), Israel J","cited_arxiv_id":null,"evidence_quote":"It decomposes the local induced representation $I(0,\\chi_p)$ into irreducible summands, the basis of the local surjectivity criterion."},{"cited_title":"J.86(1997), no","cited_arxiv_id":null,"evidence_quote":"It introduces incoherent Eisenstein series, the object whose first derivative is being studied and lifted."},{"cited_title":"Kudla, and Tonghai Yang,Special values of Green functions at big CM points, Int","cited_arxiv_id":null,"evidence_quote":"It defines the normalized Hilbert Eisenstein series and the generalized Rankin–Selberg $L$-function that the paper computes explicitly."},{"cited_title":"Math.225(2021), no","cited_arxiv_id":null,"evidence_quote":"It provides the CM-value formula for higher Green functions that converts the theta-lift identity into the Borcherds-product application."},{"cited_title":"Gross, W","cited_arxiv_id":null,"evidence_quote":"It supplies the higher Green functions, Heegner points, and the Shintani-lift Fourier coefficients used in the explicit $L$-function formula."},{"cited_title":"Math.157(2021), no","cited_arxiv_id":null,"evidence_quote":"It establishes the level-one singular-moduli non-unit result and the overall strategy that Theorem 1.6 generalizes."},{"cited_title":"Deformations of Theta Integrals and A Conjecture of Gross-Zagier","cited_arxiv_id":"2204.10604","evidence_quote":"It supplies the unfolding-and-matching method for theta integrals and the regularized theta-lift framework for Green functions."},{"cited_title":"Math.92 (1988), no","cited_arxiv_id":null,"evidence_quote":"It provides the subconvexity and Fourier-coefficient bounds for half-integral weight forms behind the size estimate in Theorem 1.6."}],"review_version":1}