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

Mingkuan Zhang

REVIEW 3 major objections 4 minor 52 references

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

T0 review · 3 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2607.16893 v1 pith:WVSC74TU submitted 2026-07-18 math.NT

Mingkuan Zhang This is my paper

classification math.NT MSC 11F2711F4111G1511F67

keywords twistedSiegel-WeilformulaDoi-Naganumaliftnon-GaloisquarticCMfieldJacquet-LanglandscorrespondenceBorcherdsformsvaluesthetaintegralWeilrepresentation

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 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.

What carries the argument

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

What would settle it

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.

Watch

Extended reading notes

Core claim

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

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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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

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.

What 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.

Now 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.

Second, 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.

The 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.

Overall: 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.

Referee Report

3 major / 4 minor

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).

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 (3)

[§2.1, Eq. (2.1); Theorems 1.1–1.3 and 5.3] 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.
[§5.3 / proof of Theorem 5.3] 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.
[Theorem 1.3 and §3.5/Remark 3.2] 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.

minor comments (4)

[§2.2 and §4.1] 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.
[References] 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.
[Throughout] 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.
[§2.1 diagram] 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.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the central theorem is proved by independent local Whittaker calculations and strong multiplicity one, with only minor non-load-bearing self-citation.

full rationale

The derivation chain does not reduce to its inputs by construction. The characters χ and ρ are defined explicitly via Artin maps and the Frobenius table of M32; Lemma 2.2 verifies a central-character compatibility, but this is only a necessary condition and does not by itself imply the twisted theta integral equals the Doi-Naganuma lift. The main equality π_χ ≅ Θ(π_ρ) is established by comparing the local Whittaker newform values computed in Propositions 4.2 and 4.5 at almost all primes (Lemma 5.5), then invoking strong multiplicity one. These Whittaker computations are independent evaluations of the Weil representation against the chosen characters, not restatements of Theorem 5.3. The 'for every ϕ there exist (φ,Ξ)' consequence follows from global theta correspondence and the representation isomorphism, so it is not self-definitional. Theorem 1.3 relates a genuinely different quantity—regularized theta lifts of Borcherds forms at twisted CM cycles—to Whittaker coefficients Wf of the twisted theta integral; Wf appears as the output of a Fourier-coefficient computation, not as an input disguised as a prediction. The only self-citation, [32] by the author with Y. Li, supplies explicit SL2-invariant Schwartz functions and motivation; it is not load-bearing for the central theorem, and the paper itself notes its invariant vectors do not match at ramified places. The auxiliary degree-32 field M32 is assumed and cited to [31]; this is an external existence/conditional assumption and a correctness risk, not a circular step. No equation in the paper is equivalent by construction to a fitted parameter or to a prior conclusion of the same authors.

Assumptions & free parameters 3 free parameters · 5 assumptions · 3 invented entities

The central claim rests on a network of external theorems (Harris–Kudla, Jacquet–Langlands, strong multiplicity one) and on the auxiliary M32 construction. The characters χ and ρ are custom-built to satisfy the central-character identity; this is a construction, not a fit, but it limits the theorem's scope to those characters.

free parameters (3)

local quadratic form constants a1,p, a2,p = p^{-1} or 1; p^n with 1/p^{n-1} a norm of β0 (Section 5.2)
Chosen by hand to make the quadratic spaces isometric and the local Whittaker functions match; they are not measured from data.
values of ρ on Gal(M32/F) generators = η(1,0,0)=-1, η(0,1,0)=i (Lemma 2.2)
Chosen so that the central-character identity holds; the main theorem is only for this special ρ.
ε and β0 = ε ∈ ɵF2^+ with -ε locally a norm; β0 ∈ A_{ɵK4,f} with -ε = Nm β0
Used to construct W− and the isometry W+ ≅ W−; the paper notes the twisted CM values depend on this choice.

assumptions (5)

domain assumption Existence of M32 with Gal(M32/Q) ≅ (Z/4Z)^2 ⋊ Z/2Z and prescribed Frobenius/splitting data
Section 2.1 'suppose M32 is a degree 4 Galois extension...' based on [31, Prop 2.1]; not proved in this paper.
domain assumption Harris–Kudla local-global compatibility π_χ ≅ Θ(χ) and π_ρ ≅ Θ(ρ)
Invoked verbatim as Proposition 5.1 via [17, Section 13].
standard math Strong multiplicity one for cuspidal automorphic representations of GL2 over number fields
Used in the proof of Theorem 5.3 citing [38, Theorem 4.10].
standard math Jacquet–Langlands local description of theta lifts (Proposition 1.5, Theorem 4.6 of [23])
Used to define local representations and central characters in Section 5.1.
domain assumption Existence of totally positive ε with -ε locally a norm at all finite places
Used to define W− in Section 3.3; existence is asserted without proof.

invented entities (3)

M32, degree-32 Galois extension of Q with group (Z/4Z)^2 ⋊ Z/2Z
purpose: Constructs the quadratic character χ via the Artin map and controls splitting behavior for local matching.
Auxiliary field introduced for the proof; its existence is cited to [31, Prop 2.1], not independently evidenced here.
The quadratic character χ (from ψ on Gal(M32/ɵK4))
purpose: Defines the twisted theta integral and twisted CM cycles.
Defined via the auxiliary field; no existence outside this construction.
Hecke character ρ on A_F^×
purpose: Target character whose Hecke integral is lifted; chosen to satisfy central-character matching.
Defined via η with values -1, i on generators; chosen ad hoc to make Lemma 2.2 true.

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Pith. "Pith review of Twisted Siegel-Weil formulas for $\mathrm{GL}_2$ over non-Galois quartic CM fields." pith.science (2026). https://pith.science/paper/WVSC74TU

@misc{pith2026260716893,
  author       = {Pith},
  title        = {Pith review of: Twisted Siegel-Weil formulas for $\mathrmGL_2$ over non-Galois quartic CM fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WVSC74TU}},
  note         = {Machine review of arXiv:2607.16893}
}

abstract

We establish twisted Siegel-Weil formulas for non-Galois quartic CM fields, identifying the twisted theta integral against a quadratic character with the Doi-Naganuma lift of Hecke's integral. This implies the base change of Jacquet-Langlands correspondence for certain Hecke characters from $\mathbb{Q}$ to real quadratic fields via Doi-Naganuma lift is an isomorphism. As an application, we prove that the twisted CM values of Borcherds forms are algebraic multiple of logarithm of units, explicitly described by the Fourier coefficients of twisted theta integrals.

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Works this paper leans on

52 extracted references · 4 linked inside Pith

[1]

J. H. Bruinier, Y. Li, and T. Yang.Deformations of Theta Integrals and a Conjecture of Gross-Zagier. arxiv:2204.10604, submitted. 2022 (cit. on pp. 6, 7, 16, 26, 34–36, 38, 42)

arXiv 2022
[32]

Li and M

Y. Li and M. Zhang.Hilbert Eisenstein series as Doi-Naganuma lift. 2025. arXiv: 2506.01688 [math.NT](cit. on pp. 3, 4, 8, 15, 16, 33)

arXiv 2025
[2]

CM values of higher automorphic Green functions for orthogonal groups

J. H. Bruinier, S. Ehlen, and T. Yang. “CM values of higher automorphic Green functions for orthogonal groups”. In:Invent. Math.225.3 (2021), pp. 693–785 (cit. on pp. 19, 22, 40, 41)

2021
[3]

On two geometric theta lifts

J. H. Bruinier and J. Funke. “On two geometric theta lifts”. In:Duke Math. J.125.1 (2004), pp. 45–90 (cit. on pp. 5, 21)

2004
[4]

Special values of Green functions at big CM points

J. H. Bruinier, S. S. Kudla, and T. Yang. “Special values of Green functions at big CM points”. In:Int. Math. Res. Not. IMRN9 (2012), pp. 1917–1967 (cit. on pp. 4, 6–8, 18–23, 26)

2012
[5]

CM-values of Hilbert modular functions

J. H. Bruinier and T. Yang. “CM-values of Hilbert modular functions”. In:Invent. Math.163.2 (2006), pp. 229–288 (cit. on pp. 4, 6, 19, 20)

2006
[6]

CM-values of Hilbert modular functions

J. H. Bruinier and T. Yang. “CM-values of Hilbert modular functions”. In:Invent. Math.163.2 (2006), pp. 229–288 (cit. on p. 8)

2006
[7]

Twisted Borcherds products on Hilbert modular surfaces and their CM values

J. H. Bruinier and T. Yang. “Twisted Borcherds products on Hilbert modular surfaces and their CM values”. In:Amer. J. Math.129.3 (2007), pp. 807–841 (cit. on p. 20)

2007

Show all 52 references

[8]

Harmonic Maass forms associated to real quadratic fields

P. Charollois and Y. Li. “Harmonic Maass forms associated to real quadratic fields”. In:J. Eur. Math. Soc. (JEMS)22.4 (2020), pp. 1115–1148 (cit. on pp. 26, 35, 36, 38)

2020
[9]

On the functional equation of certain Dirichlet series

K. Doi and H. Naganuma. “On the functional equation of certain Dirichlet series”. In:Invent. Math.9 (1969/70), pp. 1–14 (cit. on p. 2)

1969
[10]

Twisted arithmetic Siegel Weil formula onX 0(N)

T. Du and T. Yang. “Twisted arithmetic Siegel Weil formula onX 0(N)”. In:J. Number Theory203 (2019), pp. 95–117 (cit. on p. 2)

2019
[11]

Freitag.Hilbert modular forms

E. Freitag.Hilbert modular forms. Springer-Verlag, Berlin, 1990, pp. viii+250 (cit. on p. 17)

1990
[12]

A Siegel-Weil formula for automorphic characters: cubic variation of a theme of Snitz

W. T. Gan. “A Siegel-Weil formula for automorphic characters: cubic variation of a theme of Snitz”. In:J. Reine Angew. Math.625 (2008), pp. 155–185 (cit. on p. 2)

2008
[13]

The regularized Siegel-Weil formula (the second term identity) and the Rallis inner product formula

W. T. Gan, Y. Qiu, and S. Takeda. “The regularized Siegel-Weil formula (the second term identity) and the Rallis inner product formula”. In:Invent. Math.198.3 (2014), pp. 739–831 (cit. on p. 2)

2014
[14]

A proof of the Howe duality conjecture

W. T. Gan and S. Takeda. “A proof of the Howe duality conjecture”. In:J. Amer. Math. Soc.29.2 (2016), pp. 473–493 (cit. on p. 2)

2016
[15]

P. 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. 16)

1990
[16]

Arithmetic automorphic forms for the nonholomorphic discrete series ofGSp(2)

M. Harris and S. S. Kudla. “Arithmetic automorphic forms for the nonholomorphic discrete series ofGSp(2)”. In:Duke Mathematical Journal66.1 (1992), pp. 59–121 (cit. on p. 13)

1992
[17]

The central critical value of a triple productL-function

M. Harris and S. S. Kudla. “The central critical value of a triple productL-function”. In:Ann. of Math. (2)133.3 (1991), pp. 605–672 (cit. on pp. 4, 31). 44 REFERENCES

1991
[18]

Special values of Green’s functions on hyperbolic 3-space

S. Herrero et al. “Special values of Green’s functions on hyperbolic 3-space”. In:Duke Math. J.174.15 (2025), pp. 3245–3316 (cit. on p. 4)

2025
[19]

Horawa and Y

A. Horawa and Y. Li.Motivic action conjecture for Doi-Naganuma lifts. 2025. arXiv: 2506.01699 [math.NT](cit. on pp. 16, 27)

2025 arXiv
[20]

Intersections of Hirzebruch–Zagier Divisors and CM Cycles

B. J. Howard and T. Yang. “Intersections of Hirzebruch–Zagier Divisors and CM Cycles”. In: 2012 (cit. on pp. 4, 21)

2012
[21]

θ-series and invariant theory

R. Howe. “θ-series and invariant theory”. In:Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1. Vol. XXXIII. Proc. Sympos. Pure Math. Amer. Math. Soc., Providence, RI, 1979, pp. 275–285 (cit. on p. 2)

1977
[22]

A regularized Siegel-Weil formula for unitary groups

A. Ichino. “A regularized Siegel-Weil formula for unitary groups”. In:Math. Z.247.2 (2004), pp. 241–277 (cit. on p. 2)

2004
[23]

Jacquet and R

H. Jacquet and R. P. Langlands.Automorphic forms onGL(2). Vol. Vol. 114. Lecture Notes in Mathematics. Springer-Verlag, Berlin-New York, 1970, pp. vii+548 (cit. on pp. 4, 14, 31)

1970
[24]

Another product for a Borcherds form

S. Kudla. “Another product for a Borcherds form”. In:Advances in the theory of automorphic forms and theirL-functions. Vol. 664. Contemp. Math. Amer. Math. Soc., Providence, RI, 2016, pp. 261–294 (cit. on p. 14)

2016
[25]

Algebraic cycles on Shimura varieties of orthogonal type

S. S. Kudla. “Algebraic cycles on Shimura varieties of orthogonal type”. In:Duke Math. J.86.1 (1997), pp. 39–78 (cit. on pp. 18, 19)

1997
[26]

Integrals of Borcherds forms

S. S. Kudla. “Integrals of Borcherds forms”. In:Compositio Math.137.3 (2003), pp. 293–349 (cit. on pp. 4, 5, 7, 18)

2003
[27]

Theta-functions and Hilbert modular forms

S. S. Kudla. “Theta-functions and Hilbert modular forms”. In:Nagoya Math. J.69 (1978), pp. 97–106 (cit. on p. 3)

1978
[28]

A regularized Siegel-Weil formula: the first term identity

S. S. Kudla and S. Rallis. “A regularized Siegel-Weil formula: the first term identity”. In:Ann. of Math. (2)140.1 (1994), pp. 1–80 (cit. on p. 2)

1994
[29]

R. P. Langlands.Base change forGL(2). Vol. 96. Annals of Mathematics Studies. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1980, pp. vii+237 (cit. on p. 2)

1980
[30]

From sum of two squares to arithmetic Siegel-Weil formulas

C. Li. “From sum of two squares to arithmetic Siegel-Weil formulas”. In:Bull. Amer. Math. Soc. (N.S.)60.3 (2023), pp. 327–370 (cit. on p. 2)

2023
[31]

Real-dihedral harmonic Maass forms and CM-values of Hilbert modular func- tions

Y. Li. “Real-dihedral harmonic Maass forms and CM-values of Hilbert modular func- tions”. In:Compos. Math.152.6 (2016), pp. 1159–1197 (cit. on pp. 4, 5, 7–9, 11)

2016
[33]

Lion and M

G. Lion and M. Vergne.The Weil representation, Maslov index and theta series. Vol. 6. Progress in Mathematics. Birkh¨ auser, Boston, MA, 1980, pp. vi+337 (cit. on p. 14)

1980
[34]

Lion and M

G. Lion and M. Vergne.The Weil representation, Maslov index and theta series. Vol. 6. Progress in Mathematics. Birkh¨ auser, Boston, MA, 1980, pp. vi+337 (cit. on p. 15)

1980
[35]

The rationality of vector valued modular forms associated with the Weil representation

W. J. McGraw. “The rationality of vector valued modular forms associated with the Weil representation”. In:Math. Ann.326.1 (2003), pp. 105–122 (cit. on p. 16). REFERENCES 45

2003
[36]

The fundamental theorem of complex multiplication

J. S. Milne. “The fundamental theorem of complex multiplication”. In:arXiv preprint arXiv:0705.3446(2007) (cit. on p. 21)

2007 arXiv
[37]

Central values of RankinL-series over real quadratic fields

A. A. Popa. “Central values of RankinL-series over real quadratic fields”. In:Compos. Math.142.4 (2006), pp. 811–866 (cit. on pp. 14, 26, 31)

2006
[38]

Notes on the arithmetic of Hilbert modular forms

A. Raghuram and N. Tanabe. “Notes on the arithmetic of Hilbert modular forms”. In:J. Ramanujan Math. Soc.26.3 (2011), pp. 261–319 (cit. on p. 32)

2011
[39]

Saito.Automorphic forms and algebraic extensions of number fields

H. Saito.Automorphic forms and algebraic extensions of number fields. Department of Mathematics, Kyoto University, Lectures in Mathematics, No. 8. Kinokuniya Book Store Co., Ltd., Tokyo, 1975, pp. iv+183 (cit. on p. 2)

1975
[40]

Borcherds forms and generalizations of singular moduli

J. Schofer. “Borcherds forms and generalizations of singular moduli”. In:J. Reine Angew. Math.629 (2009), pp. 1–36 (cit. on p. 4)

2009
[41]

Thetareihen positiv definiter quadratischer Formen

R. Schulze-Pillot. “Thetareihen positiv definiter quadratischer Formen”. In:Invent. Math.75.2 (1984), pp. 283–299 (cit. on p. 2)

1984
[42]

Darstellungsmaße von Spinorgeschlechtern tern¨ arer quadratischer Formen

R. Schulze-Pillot. “Darstellungsmaße von Spinorgeschlechtern tern¨ arer quadratischer Formen”. In:J. Reine Angew. Math.352 (1984), pp. 114–132 (cit. on p. 2)

1984
[43]

Theta series and automorphic forms on GL 2

H. Shimizu. “Theta series and automorphic forms on GL 2”. In:J. Math. Soc. Japan 24 (1972), pp. 638–683 (cit. on p. 13)

1972
[44]

The special values of the zeta functions associated with Hilbert modular forms

G. Shimura. “The special values of the zeta functions associated with Hilbert modular forms”. In:Duke Math. J.45.3 (1978), pp. 637–679 (cit. on pp. 16, 17)

1978
[45]

On liftings of holomorphic cusp forms

T. Shintani. “On liftings of holomorphic cusp forms”. In:Automorphic forms, rep- resentations andL-functions (Proc. Sympos. Pure Math., Oregon State Univ., Cor- vallis, Ore., 1977), Part 2. Proc. Sympos. Pure Math., XXXIII. Amer. Math. Soc., Providence, RI, 1979, pp. 97–110 (c...

1977
[46]

¨Uber die analytische Theorie der quadratischen Formen

C. L. Siegel. “ ¨Uber die analytische Theorie der quadratischen Formen”. In:Ann. of Math. (2)36.3 (1935), pp. 527–606 (cit. on p. 2)

1935
[47]

Theta correspondence of automorphic characters

K. Snitz. “Theta correspondence of automorphic characters”. In:J. Number Theory 122.2 (2007), pp. 429–465 (cit. on p. 2)

2007
[48]

Correspondances de Shimura

J.-L. Waldspurger. “Correspondances de Shimura”. In:Proceedings of the Interna- tional Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983). PWN, Warsaw, 1984, pp. 525–531 (cit. on p. 2)

1983
[49]

Correspondances de Shimura et quaternions

J.-L. Waldspurger. “Correspondances de Shimura et quaternions”. In:Forum Math. 3.3 (1991), pp. 219–307 (cit. on p. 2)

1991
[50]

Sur la formule de Siegel dans la th´ eorie des groupes classiques

A. Weil. “Sur la formule de Siegel dans la th´ eorie des groupes classiques”. In:Acta Math.113 (1965), pp. 1–87 (cit. on p. 2)

1965
[51]

The lambda invariants at CM points

T. Yang, H. Yin, and P. Yu. “The lambda invariants at CM points”. In:Int. Math. Res. Not. IMRN7 (2021), pp. 5542–5603 (cit. on p. 4)

2021
[52]

Modular forms associated to real quadratic fields

D. Zagier. “Modular forms associated to real quadratic fields”. In:Invent. Math.30.1 (1975), pp. 1–46 (cit. on p. 3). F achbereich Mathematik, Technische Universit¨at Darmstadt, Schlossgartenstrasse 7, D–64289 Darmstadt, Germany Email address:mzhang@mathematik.tu-darmstadt.de

1975

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [1]

J. H. Bruinier, Y. Li, and T. Yang.Deformations of Theta Integrals and a Conjecture of Gross-Zagier. arxiv:2204.10604, submitted. 2022 (cit. on pp. 6, 7, 16, 26, 34–36, 38, 42)

arXiv 2022

[2] [32]

Li and M

Y. Li and M. Zhang.Hilbert Eisenstein series as Doi-Naganuma lift. 2025. arXiv: 2506.01688 [math.NT](cit. on pp. 3, 4, 8, 15, 16, 33)

arXiv 2025

[3] [2]

CM values of higher automorphic Green functions for orthogonal groups

J. H. Bruinier, S. Ehlen, and T. Yang. “CM values of higher automorphic Green functions for orthogonal groups”. In:Invent. Math.225.3 (2021), pp. 693–785 (cit. on pp. 19, 22, 40, 41)

2021

[4] [3]

On two geometric theta lifts

J. H. Bruinier and J. Funke. “On two geometric theta lifts”. In:Duke Math. J.125.1 (2004), pp. 45–90 (cit. on pp. 5, 21)

2004

[5] [4]

Special values of Green functions at big CM points

J. H. Bruinier, S. S. Kudla, and T. Yang. “Special values of Green functions at big CM points”. In:Int. Math. Res. Not. IMRN9 (2012), pp. 1917–1967 (cit. on pp. 4, 6–8, 18–23, 26)

2012

[6] [5]

CM-values of Hilbert modular functions

J. H. Bruinier and T. Yang. “CM-values of Hilbert modular functions”. In:Invent. Math.163.2 (2006), pp. 229–288 (cit. on pp. 4, 6, 19, 20)

2006

[7] [6]

CM-values of Hilbert modular functions

J. H. Bruinier and T. Yang. “CM-values of Hilbert modular functions”. In:Invent. Math.163.2 (2006), pp. 229–288 (cit. on p. 8)

2006

[8] [7]

Twisted Borcherds products on Hilbert modular surfaces and their CM values

J. H. Bruinier and T. Yang. “Twisted Borcherds products on Hilbert modular surfaces and their CM values”. In:Amer. J. Math.129.3 (2007), pp. 807–841 (cit. on p. 20)

2007