Motivic action conjecture for Doi-Naganuma lifts

Aleksander Horawa; Yingkun Li

REVIEW 2 major objections 4 minor 1 cited by

Motivic action conjecture for Doi-Naganuma lifts

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves the motivic action conjecture for base changes of weight one newforms with solvable projective image, giving an explicit rational basis for the f-isotypic coherent cohomology of Hilbert modular surfaces.

desk verdict A substantial new case of the motivic action conjecture, proved by a genuine theta-lift-back method; the main soft spot is a load-bearing local fact deferred to an unpublished preprint. read the letter →

arxiv 2506.01699 v1 pith:OVLHRQYU submitted 2025-06-02 math.NT

Aleksander Horawa , Yingkun Li This is my paper

classification math.NT MSC 11F4111F6711F2711R42

keywords motivicactionconjectureDoi-NaganumaliftsweightonemodularformsHilbertStarkunitscoherentcohomologythetaArtinrepresentations

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

The paper proves the motivic action conjecture for Hilbert modular forms over real quadratic fields that arise as quadratic base change of weight one newforms with odd, squarefree level and solvable projective image. The conjecture says that the rational structure on the f-isotypic coherent cohomology group $H^{1}$(X,omega)_f is governed by logarithms of Stark units attached to the adjoint Artin representation. The authors show explicitly that the classes ($omega^{1}$_f+$omega^{2}$_f)/ and ($omega^{1}$_f-$omega^{2}$_f)/(/) form an E-basis, where E is the field generated by the Hecke eigenvalues, and that under Stark's conjecture this basis coincides with the logarithmic Stark-unit basis predicted by the motivic action conjecture. A corollary settles the question whether any multiple of $omega^{1}$_f or $omega^{2}$_f can be rational: none is, for these forms.

What carries the argument

The Doi-Naganuma $\theta$ lift from SL2 to the orthogonal group of a signature (2,2) quadratic space realizes the base-change form explicitly as a $\theta$ integral. The rationality step is carried by a functional built from weighted Hirzebruch-Zagier cycles: C_phi(eta) = $\sqrt$(D) sum_s $\varphi$($s^{{-1}}$x0) integral over Z1(x0,s,K1) of iota_s^* eta, with a Schwartz function phi assembled from local data. Unfolding the $\theta$ integral shows that C_phi(eta_-) lies in E^times and C_phi(eta_+) = 0; since eta_- spans the negative eigenspace of the swap involution, one-dimensionality and Serre duality promote this to rationality of both eta_+ and eta_-.

What would settle it

At a prime p dividing D, compute omega_p(kappa)phi_p for kappa in SL2(Zp) using the explicit formula (4.3); if any such vector is not invariant under the full SL2(Zp), then equivariance (4.4) fails and the identification of the theta lift with the base-change newform breaks down.

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Extended reading notes

Core claim

If f0 in S1(N,chi0) is a weight one newform of odd, squarefree level N coprime to 2D and f is its base change to F=Q($\sqrt$(D)), then, provided the projective image of the associated Galois representation is not A5, an E-basis of $H^{1}$(X,omega)_f is given by ($omega^{1}$_f+$omega^{2}$_f)/<f0,f0> and ($omega^{1}$_f-$omega^{2}$_f)/(<f,f>/<f0,f0>). The same statement holds when the projective image is A5 provided Stark's conjecture is assumed for $Ad^{0}$ rho_f0 and $Ad^{0}$ rho_f0 tensor chi_F. This proves the motivic action conjecture for these forms because Stark's conjecture identifies the Petersson norms appearing in the denominators with the logarithms of Stark units, giving exactly the predicted rational classes.

Load-bearing premise

The proof relies on an unpublished assertion that the local Schwartz function at primes dividing D is invariant under SL2(Zp); without that property, the theta-lift identifications and the rationality functional collapse, and additionally the A5 case depends on Stark's conjecture.

Editorial extensions

If this is right

For every such base-change form, the rational structure of H^1(X,omega)_f is explicitly known: the two f-isotypic classes are rational after dividing by Petersson norms, rather than by period invariants.
No nonzero multiple of omega^1_f or omega^2_f is Q-rational, answering in the negative the long-standing rationality question for weight one Hilbert modular forms of this type.
Stark's conjecture supplies the bridge from Petersson norms to logarithms of units, so the motivic action conjecture holds in full for these forms whenever the A5 case satisfies Stark's conjecture.
The proof reveals a new rationality mechanism: a weighted-cycle functional is E-valued because the Galois action on connected components compensates the Galois action on Fourier coefficients, a phenomenon tied to nontrivial nebentypus in odd weight.
The theta-lift-back strategy is expected to generalize to middle-degree coherent cohomology for arbitrary quadratic extensions of totally real fields, with 2^{d0-1} functionals replacing the single functional used here.

Reading between the lines

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

Editorial inference: the emphasis on odd weight and nontrivial nebentypus suggests a broader principle, namely that rationality of coherent classes is governed by twisted Stark units whenever the automorphic form is a quadratic twist of a form with smaller coefficient field.
Editorial inference: the functional C_phi is explicitly computable from Fourier expansions, so the E-rationality of C_phi(eta_-) can be tested numerically for concrete forms beyond the special cases computed in the paper.
Editorial inference: if the local invariance assumption in Lemma 4.1 holds, the odd-squarefree restrictions should be removable with more involved local Fourier computations, as the authors state; the coprime condition between D and N appears more essential because it is used to identify the coefficient fields of f and f0.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The one thing to know: this paper proves Horawa's motivic action conjecture for quadratic base changes of weight one newforms, excluding the A5 case which is handled conditionally under Stark's conjecture. The main theorem gives an explicit E-rational basis of H^1(X,ω)_f, and Corollary 5.14 answers Harris's period question for weight one by showing no multiple of ω_i^f is rational. The proof is real and the result is new; this is not a repackaging of earlier work.

The method is the genuinely original part. The authors lift f back down to SL_2 via a theta integral and show the constant term is a rational multiple of f0, with the scalar computed through a weighted Hirzebruch–Zagier cycle. The Galois compensation in Lemma 5.9 is the key idea, and it is worked out concretely. The local computations in Propositions 3.1 and 4.2 are detailed and convincing, and the series of reductions—theta identification, unfolding, Galois invariance of the cycle, Serre duality—is coherent.

Two soft spots. First, the abstract omits the coprimality condition (D,N)=1, which appears in the theorem statement and is used in the proof, for example in Lemma 4.5 and Proposition 4.8. The abstract overclaims as written. Second, and more substantively, Lemma 4.1 defers the p|D case of the SL2(Z_p)-invariance of φ_p to the unpublished preprint [LZ25]. This invariance is load-bearing: it gives the equivariance (4.4) used in the unfolding to eliminate all but the N'=1 term, and from there the identifications in Proposition 4.8 and Theorem 4.11. I could not verify the invariance directly from the explicit formula (4.3), and the Weil representation at a ramified prime is not trivial enough to call this routine. This is not a fatal flaw—it is likely correct and checkable—but the authors need to either prove it in the paper or give a precise statement with conventions that match theirs. A referee should insist on this before publication.

Who is this for: number theorists working on Stark units, coherent cohomology, and theta lifts. The paper deserves a serious referee. I would send it to a good journal, with the requirement that the [LZ25] dependence be resolved. The conditional A5 case is clearly labeled and handled by known results or Stark's conjecture, so that is not a problem. Overall, if the local fact checks out, the theorem stands.

Referee Report

2 major / 4 minor

Summary. The paper proves the motivic action conjecture for the base change to a real quadratic field F of a weight one newform f0 of odd squarefree level N coprime to 2D, under the assumption that the projective image of the associated Artin representation is not A5; the A5 case is proved conditional on Stark's conjecture. The proof identifies the base change f and f0 with theta lifts for a quadratic space of signature (2,2), computes the Fourier expansion of the theta lift by unfolding (Theorem 4.11), constructs an E-rational functional from Kudla's weighted Hirzebruch-Zagier cycles (Lemma 5.9, Corollary 5.10), and uses Serre duality to show that both η+ and η− are rational. A corollary shows that no multiple of ω1_f or ω2_f is rational.

Significance. This is a substantial result: it is the first proof of the motivic action conjecture for a family of weight one Hilbert modular forms and answers a question of Harris on period invariants. The proof is constructive and combines several deep tools: theta correspondence, explicit local Schwartz functions, Kudla's weighted cycles, and Stark's conjecture. The paper is well structured, with detailed local computations in Section 3 and Proposition 4.2, and a concrete worked example in Section 5.5. The main caveat is the reliance on an unpublished preprint [LZ25] for a key local invariance statement, which must be resolved for the proof to be considered complete.

major comments (2)

[§4.1, Lemma 4.1] The proof of Lemma 4.1 for primes p|D is deferred to the unpublished preprint [LZ25]: the sentence 'When p | D, we have ωp(κ)φp = φp for all κ ∈ SL2(Zp) since φp is an invariant vector [LZ25, §5.1]' is the entire justification for (4.4) at ramified primes. This equivariance is load-bearing: it is used in the unfolding in Section 4.3 to eliminate every N′ > 1 term, leading to the Fourier expansion (4.12), Proposition 4.8, and Theorem 4.11, which in turn give the E-rationality of Cφ (Corollary 5.10) and the rationality of η− (Corollary 5.11). If the asserted invariance failed, the theta lift would not be Hecke-equivariant at the stated level and the identification in Proposition 4.6 would be unsupported. The authors should supply a complete proof of the p|D case within this manuscript, or replace the citation with a published and verifiable reference.
[§4.2, Proposition 4.6] The proof of the second identity in (4.10) is too terse to be fully checked. It says that both sides have the same Hecke eigenvalues for almost all split primes p and that, because (D,N)=1, 'the field F is fixed by the Galois representations associated to the newforms,' so Chebotarev and Lemma 4.4 apply. As written, equality at split primes only implies agreement of the two Galois representations on Gal(Q̄/F); it does not by itself rule out the possibility that the theta lift is the twist of f0 by χF (or another form whose traces agree on split primes but differ at inert primes). Since this identification is the starting point for Theorem 4.11 and the denominators in Theorem 5.5, the authors should expand this into a complete argument using the level and nebentypus of both sides, or explicitly state why twisting is impossible.

minor comments (4)

[§5.3, Corollary 5.10] The phrase 'Lemma 5.9 (applied to φ instead of φ)' contains an apparent typo: the two symbols are identical; should one be the complex conjugate φ̄?
[§5.3, proof of Lemma 5.9, equation (5.16)] The symbol ϖp in equation (5.16) should presumably be ωp (the Weil representation action).
[§4.3, near equation (4.12)] The notation Q0(ν) is used in the unfolding computation without a definition; please define it explicitly.
[References] Reference [LZ25] is listed as '2025' with no publication venue or arXiv identifier; since the current manuscript relies on it for a key local computation, please provide a stable reference or include the argument in the paper.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the E-basis result is derived from an explicit theta-lift computation and Serre duality, with Stark's conjecture only assumed in the A5 case.

full rationale

The derivation of the E-basis in Theorem 5.5(1) is self-contained and does not reduce to the conjecture being proved. The rationality of η− is established through the explicit theta-lift functional Cφ, whose nonvanishing value is computed as Cφ(η−) = C1,1(f;φ) = r0(φ) in Theorem 4.11, with r0 proportional to the ratio ⟨f,f⟩/⟨f0,f0⟩ and with r0 ∈ Q(f0)× obtained from actual Fourier coefficients cp(f0) (Prop. 4.8). This is a genuine computation, not a restatement of the desired basis. The rationality of η+ follows from Serre duality and the explicit pairing ⟨η+, η−⟩ = −2 (Cor. 5.13), again independent of the motivic action conjecture. Stark's conjecture is not used to prove the rational basis; it appears only in Theorem 5.5(2), where the projective image is A5, and there it is clearly stated as an assumption. For the solvable cases, the paper invokes Stark's theorem as external support, not as a premise equivalent to the conclusion. The citations to [Hor23] concern the formulation of the conjecture, the unit-group decomposition, a Serre-duality fact, and the conductor being a square; these are supporting lemmas, not the target theorem. The one caveat worth flagging is Lemma 4.1: the claim that φp is SL2(Zp)-invariant for p | D is deferred to the unpublished, same-second-author preprint [LZ25, §5.1], and the equivariance (4.4) built on it is load-bearing for the unfolding in Section 4.3. However, this is an omitted local computation and a potential correctness/verification risk, not a circular step: the asserted invariance is a concrete Weil-representation calculation, not an equivalent form of the motivic action conjecture or of the rationality being derived. It does not make the paper's prediction equal to its input by construction.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No free parameters or invented entities. The central claim rests on standard background (Chebotarev, strong multiplicity one, Serre duality) plus two domain-specific inputs: an unproved local invariant-vector result cited to an unpublished preprint [LZ25], and for the A5 subcase, Stark's conjecture, which is explicitly assumed. For solvable projective image all external results are published theorems.

assumptions (7)

standard math Strong multiplicity one for cuspidal automorphic representations of GL(2)
Invoked in Proposition 4.6 to identify theta lifts with f0 and f.
standard math Chebotarev density theorem
Used in Lemmas 4.4 and 4.5 to compare newforms and to show Q(f)=Q(f0).
domain assumption Invariant-vector property of phi_p at p|D ([LZ25, Section 5.1])
Used in Lemma 4.1 to prove equivariance (4.4); not proved in this paper and cited to an unpublished preprint.
ad hoc to paper Stark's conjecture for Ad0 rho0 and Ad0 rho0 tensor chi_F in the A5 case
Assumed in Theorem 5.5(2) and Remark 1.1(2); not needed for solvable projective image.
domain assumption Serre duality polarization for H^1(X,omega)_f ([Hor23, Prop 5.14])
Used in Proposition 5.12 and Corollary 5.13 to transfer rationality from eta- to eta+.
domain assumption Harris-Tilouine period relation (5.21)
Used to relate Petersson norms to adjoint L-functions in the proof of Theorem 5.5(2).
standard math Baker's theorem on linear independence of logarithms
Used in Corollary 5.14 to show the two Stark logarithms are not Q-rationally dependent.

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Pith. "Pith review of Motivic action conjecture for Doi-Naganuma lifts." pith.science (2026). https://pith.science/paper/OVLHRQYU

@misc{pith2026250601699,
  author       = {Pith},
  title        = {Pith review of: Motivic action conjecture for Doi-Naganuma lifts},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OVLHRQYU}},
  note         = {Machine review of arXiv:2506.01699}
}

read the original abstract

We prove the motivic action conjecture for the base change to real quadratic fields of weight one newforms with odd, squarefree level and solvable projective image.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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

Works this paper leans on

9 extracted references · 9 canonical work pages · cited by 1 Pith paper

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Deformations of theta integrals and a conjecture of Gross-Zagier

[BLY25] Jan H. Bruinier, Yingkun Li, and Tonghai Yang. “Deformations of theta integrals and a conjecture of Gross-Zagier”. In: Forum Math. Sigma 13 (2025), Paper No. e54, 59 (cit. on pp. 12, 20, 21). [DR80] Pierre Deligne and Kenneth A. Ribet. “Values of abelian L-functions at negative integers over totally real fields”. In: Invent. Math. 59.3 (1980), pp....

work page 2025

[2] [5]

eprint: arXiv:2307.04115 (cit. on pp. 2, 5). [IP21] Atsushi Ichino and Kartik Prasanna. Periods of quaternionic Shimura varieties. I. Vol

work page arXiv

[3] [8]

Artin L-functions with rational char- acters”

II. Artin L-functions with rational char- acters”. In: Advances in Math. 17.1 (1975), pp. 60–92 (cit. on pp. 1, 26, 31, 33). [Sta80] Harold M. Stark. “ L-functions at s =

work page 1975

[4] [9]

Derived Hecke algebra and cohomology of arithmetic groups

IV. First derivatives at s = 0”. In: Adv. in Math. 35.3 (1980), pp. 197–235 (cit. on p. 31). [Ven19] Akshay Venkatesh. “Derived Hecke algebra and cohomology of arithmetic groups”. In: Forum Math. Pi 7 (2019), e7, 119 (cit. on p. 1). Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX2 6GG, UK Email address: horawa@maths.ox.ac.uk Max P...

work page 1980

[5] [19]

Automorphic cohomology, motivic cohomology, and the adjoint L-function

Progress in Mathe- matics. Birkh¨ auser, Boston, MA, 1982, pp. xvi+123 (cit. on p. 3). [PV21] Kartik Prasanna and Akshay Venkatesh. “Automorphic cohomology, motivic cohomology, and the adjoint L-function”. In: Ast´ erisque428 (2021), pp. viii+132 (cit. on pp. 1, 2). [Ral82] Stephen Rallis. “Langlands’ functoriality and the Weil representation”. In: Amer. ...

work page 2021

[6] [63]

Springer-Verlag, Berlin-New York, 1974, pp

Zweite Auflage. Springer-Verlag, Berlin-New York, 1974, pp. xii+222 (cit. on p. 13). [Gar90] Paul B. Garrett. Holomorphic Hilbert modular forms. The Wadsworth & Brooks/Cole Mathematics Series. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1990, pp. xvi+304 (cit. on p. 6). [GLZ23] Giada Grossi, David Loeffler, and Sarah Livia Zerbes...

work page 1974

[7] [1447]

Period invariants of Hilbert modular forms. II

Lecture Notes in Math. Springer, Berlin, 1990, pp. 155–202 (cit. on pp. 2, 24, 27, 31). [Har94] Michael Harris. “Period invariants of Hilbert modular forms. II”. In: Compositio Math. 94.2 (1994), pp. 201–226 (cit. on p. 31). [HK92] Michael Harris and Stephen S. Kudla. “Arithmetic automorphic forms for the nonholomorphic discrete series of GSp(2)”. In:Duke...

work page 1994

[8] [2021]

Theta-functions and Hilbert modular forms

©2021, pp. vi+214 (cit. on p. 30). [Kud78] Stephen S. Kudla. “Theta-functions and Hilbert modular forms”. In: Nagoya Math. J. 69 (1978), pp. 97–106 (cit. on p. 1). [Kud97] Stephen S. Kudla. “Algebraic cycles on Shimura varieties of orthogonal type”. In: Duke Math. J. 86.1 (1997), pp. 39–78 (cit. on pp. 3, 9–11). [Kud03] Stephen S. Kudla. “From modular for...

work page 1978