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Hilbert Eisenstein series as Doi-Naganuma lift

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Pith's one-line read For real quadratic fields with Galois CM, every incoherent Hilbert Eisenstein series is a Doi–Naganuma lift of a rational Eisenstein series.

desk verdict Substantial, likely correct, and fills a real gap; the only serious worry is the compressed local surjectivity proof in Prop 4.10. read the letter →

arxiv 2506.01688 v1 pith:EL7CZTUK submitted 2025-06-02 math.NT

Yingkun Li , Mingkuan Zhang This is my paper
classification math.NT MSC 11F4111F2711F3711F6711G15
keywords incoherentHilbertEisensteinseriesDoi–NaganumaliftthetacorrespondenceWeilrepresentationBorcherdsproductsHeegnerpointsRankin–SelbergL-functionintegralunits
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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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].
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

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).

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 (3)
  1. [4.3 (Proposition 4.10)] 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.
  2. [5.1 (Lemma 5.1) and 6.2] 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.
  3. [4.4 (Proof of Theorem 4.1)] 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.
minor comments (5)
  1. [4.3] 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.
  2. [4.3] 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.
  3. [References] Reference [14] and [15] are the same Duke article and should be merged.
  4. [6.3 (Proof of Theorem 1.6)] 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.
  5. [6.2 (Lemma 6.3)] 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).

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the local-to-global matching proof is self-contained; self-citations are antecedents, and the deferred p=2 computation is a completeness gap, not circularity.

full rationale

The claimed derivation chain is a local-to-global matching theorem. Theorem 1.1 is obtained by unfolding the theta integral (Lemma 2.4) and proving the local matching statement Theorem 4.1. No parameter is fitted to the quantity being 'predicted': the constants C(s), Gamma_R, and L(s, chi_1) are canonical, and the matching data (Xi, phi) are constructed for every Siegel section Phi rather than calibrated to a subset of its coefficients. The central surjectivity Proposition 4.10 is proved from the Kudla-Rallis decomposition of I(0, chi_p) (external reference [31]) plus an explicit Whittaker-coefficient computation; the nonvanishing is not imported from a self-citation. Self-citations [33] and [34] are antecedents being generalized and enter as technique, not as load-bearing premises; [9] supplies a theta-lift identification and Fock-model notation but does not posit Theorem 1.1. The review-rule flags in the manuscript are completeness gaps rather than circular steps: Proposition 4.10's ramified-prime Whittaker computation is compressed, and Lemma 5.1 explicitly defers p=2 ('the argument can be suitably adapted for p=2'); if those fail, the derived theorems would not follow, but that is a soundness risk, not a reduction of the conclusion to the hypotheses. I therefore find no significant circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard external machinery rather than fitted parameters. No constants are tuned to data. The main axioms are the Kudla-Rallis decomposition of degenerate principal series, the Siegel-Weil embedding, the CM-value formula from [10], equidistribution of CM points, and subconvexity bounds for GL1 twists. None of these are introduced ad hoc for this paper.

assumptions (5)
  • standard math Kudla-Rallis decomposition of I(0,chi_p) into irreducible summands R(W_alpha) [31]
    Used in Propositions 4.8 and 4.10 to reduce surjectivity of the local theta matching to nonvanishing of one Whittaker coefficient per component.
  • standard math Siegel-Weil embedding and local theta correspondence for quadratic chi2
    Used to realize sections Xi from Schwartz functions on E2 plus V in Remark 2.5 and Section 5.
  • domain assumption CM-value formula for higher Green functions at big CM cycles [10, Theorem 5.10]
    Imported as Theorem 3.1; it connects the regularized theta lift to the constant term and the L-function term in the arithmetic formula.
  • domain assumption Equidistribution of CM points on X0(N)^2 (Duke [14]) and Brauer-Siegel lower bound for class numbers
    Used in Section 6.3 to get a uniform lower bound for the Green function value at big CM cycles.
  • domain assumption Subconvexity bounds for GL1-twists of GL2 forms and non-trivial bounds for half-integral weight Fourier coefficients
    Used in Section 6.3 to show the L-derivative term decays like |D1D2|^{-delta}; the precise statement is not given.

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Pith. "Pith review of Hilbert Eisenstein series as Doi-Naganuma lift." pith.science (2026). https://pith.science/paper/EL7CZTUK

@misc{pith2026250601688,
  author       = {Pith},
  title        = {Pith review of: Hilbert Eisenstein series as Doi-Naganuma lift},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EL7CZTUK}},
  note         = {Machine review of arXiv:2506.01688}
}
abstract

In this paper, we show that incoherent Hilbert Eisenstein series for a real quadratic fields can be expressed as the Doi-Naganums lift of an incoherent Eisenstein series over $\mathbb{Q}$. As an application, we show when $N$ is odd and square-free, the values at Heegner points of Borcherds product on $X_0(N)^2$ with effective divisors are not integral units when the discriminants are sufficiently large. This generalizes a result of the first author to higher levels. In the process, we explicitly describe the Rankin-Selberg type L-function that appeared in the work of Bruinier-Kudla-Yang when the quadratic space has signature (2, 2), and give a new construction of fundamental invariant vectors appearing in Weil representations of finite quadratic modules.

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Forward citations

Cited by 1 Pith paper

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  1. Twisted Siegel-Weil formulas for $\mathrm{GL}_2$ over non-Galois quartic CM fields

    math.NT 2026-07 conditional novelty 7.0 of 10

    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.

Reference graph

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