Addendum to "Measured foliations and Hilbert 12th problem"

Igor V. Nikolaev

REVIEW 4 major objections 4 minor 7 references

Addendum to "Measured foliations and Hilbert 12th problem"

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

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Pith's one-line read The paper claims that Hecke eigenforms of level $m^2p$ generate the ring class fields of $\mathbb{Q}(\sqrt{p})$ for every prime $p \equiv 3 \bmod 4$.

desk verdict The addendum's central containment is false: the p=7 example itself shows a totally real ring class field cannot sit inside the CM coefficient field. read the letter →

arxiv 2505.22272 v1 pith:EDCY3A6U submitted 2025-05-28 math.NT math.OA

Igor V. Nikolaev This is my paper

classification math.NTmath.OA MSC 11G4546L85

keywords classfieldtheoryrealquadraticfieldsHeckeeigenformsHilbert12thproblemmeasuredfoliationsringmodularformsexplicitabelianextensions

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 addendum proposes an explicit class field theory for real quadratic fields of prime discriminant $p \equiv 3 \bmod 4$. It claims that for each such prime there exists a conductor $f_1$ and a Hecke eigenform of level $m^2p$ whose Fourier coefficients generate a field containing the ring class field of $\mathbb{Q}(\sqrt{p})$ modulo $f_1$. The intended mechanism is to transfer an established modular construction for imaginary quadratic fields to the real side through an isomorphism of class groups. Numerical tables for many primes are presented as evidence, listing the relevant conductors, levels, and minimal polynomials of the resulting ring class fields.

What carries the argument

The load-bearing identity is the class group isomorphism $\mathrm{Cl}(\mathbb{Q}(\sqrt{p}) \bmod f_1) \cong \mathrm{Cl}(\mathbb{Q}(\sqrt{-p}) \bmod f_2)$, combined with the inclusion $H(\mathbb{Q}(\sqrt{-p})) \subseteq K_f$ for eigenforms of level $p$. The paper's new step is to assert that this inclusion lifts to the composite levels $m^2p$ and that the class group action transfers to the real side, with passage to the measured foliation $F = \mathrm{Re}\,f$ turning imaginary algebraic numbers such as $i\sqrt{4 \pm \sqrt{7}}$ into real ones $\sqrt{4 \pm \sqrt{7}}$.

What would settle it

For $p=7$ the table gives the ring class field minimal polynomial $x^4 - 8x^2 + 9$ and the coefficient field minimal polynomial $x^4 + 8x^2 + 9$; since neither polynomial divides the other, $H(\mathbb{Q}(\sqrt{7}) \bmod 3)$ is not a subfield of that $K_f$. A decisive test for the theorem is to find one prime $p \equiv 3 \bmod 4$ where the same polynomial-divisibility check fails, or to prove that a reformulation such as taking real parts is the intended meaning.

Watch

Extended reading notes

Core claim

The central assertion is Theorem 4: for every prime $p \in \{4n+3 \mid n \geq 1\}$ there is an integer $f_1 \geq 1$ such that $H(\mathbb{Q}(\sqrt{p})) \bmod f_1 \subseteq K_f$, where $K_f$ is the field generated by the Fourier coefficients of a Hecke eigenform $f \in S_2(\Gamma(m^2p))$ for some $m \geq 1$. If correct, this would give an explicit modular construction of the abelian extensions of $\mathbb{Q}(\sqrt{p})$ for all such primes. The proof rests on the class group isomorphism $\mathrm{Cl}(\mathbb{Q}(\sqrt{p}) \bmod f_1) \cong \mathrm{Cl}(\mathbb{Q}(\sqrt{-p}) \bmod f_2)$ together with the known inclusion of the imaginary ring class field into $K_f$; the numerical examples, such as the $p=7$ case with minimal polynomial $x^4 - 8x^2 + 9$, illustrate how the real field is obtained from the coefficient field.

Load-bearing premise

Everything rests on assuming that the known modular construction for imaginary quadratic fields survives on the real side under the class group isomorphism, and that the resulting real extension is literally contained in the field generated by the modular form's coefficients; the $p=7$ example in the paper shows that literal containment already fails.

Editorial extensions

If this is right

For every prime $p \equiv 3 \bmod 4$, the abelian extensions of $\mathbb{Q}(\sqrt{p})$ would be generated by algebraic numbers extracted from the Fourier coefficients of a single modular form of level $m^2p$.
The level $m^2p$ and conductor $f_1$ would be computable from class group data, making the construction explicit rather than existential.
The real-imaginary class group isomorphism would become the bridge transferring known imaginary quadratic explicit class field theory to real quadratic fields.
The numerical tables would provide a family of minimal polynomials for the ring class fields, with degrees matching the corresponding class group orders in every computed case.

Reading between the lines

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

The $p=7$ computation suggests the theorem as literally stated is not what the numerical data show: the ring class field has minimal polynomial $x^4 - 8x^2 + 9$ while the coefficient field has $x^4 + 8x^2 + 9$, so the real field is obtained by replacing $i$ with $1$ rather than by literal subfield containment.
If the intended statement is a real-part correspondence rather than an inclusion, the paper would need a reformulated theorem; the numerical evidence supports that reformulation better than the stated one.
The unproved lifting of the inclusion to levels $m^2p$ could be tested numerically for the primes left unresolved in the table, such as $p=71,79,131,151$, once computational tools can handle the required levels and conductors.
A structural proof of the class group isomorphism for all relevant primes would be the natural next step, since the table already exhibits the isomorphism for all primes it computes.

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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,

The addendum's Theorem 4 is not established. The proof's central step—from a class group isomorphism to a field containment—is a non sequitur, and the paper's own Example 5 contradicts the statement. If correct, the result would be a big step on Hilbert's 12th problem for real quadratic fields; as it stands, the claimed containment fails for p=7.

What is genuinely new is the specific level formula N=m²p and the numerical table for primes p≡3 mod 4. The examples are computed carefully, sources are cited, and the data may be useful as computational data points for someone exploring explicit class field theory. The paper is short and honest about the limits of the database searches.

The soft spots are not minor. The proof of Theorem 4 is a few lines: formula (2) gives an isomorphism of finite abelian groups Cl(Q(√p) mod f1) ≅ Cl(Q(√−p) mod f2). From an action of the second group on K_f, the paper concludes that the first ring class field is contained in K_f. That does not follow; a group isomorphism does not produce a field embedding. The p=7 example makes the failure explicit: K_f is the CM field with minimal polynomial x⁴+8x²+9, while the claimed ring class field is the totally real field x⁴−8x²+9. A totally real quartic cannot be a subfield of a CM quartic. The phrase 'passage to the measured foliation is equivalent to taking imaginary component' replaces i by 1, which is not a field embedding and changes the field. The proof also leans on unproved assertions from the author's earlier papers [5], [6]; no independent verification is offered. The extension of Hecke's inclusion to levels m²p is asserted with 'It remains to notice', without argument.

This is a load-bearing flaw, not a typo. The central theorem is false as stated, and the example is its counterexample. What remains is a set of computational observations which, stripped of the invalid interpretation, might be worth a short note.

Who is this for? Someone interested in the numerical side of explicit class field theory might browse the table, but the paper as a research contribution does not hold. I'd still send it to a referee if I were the editor, mainly because the claim is important and the error is instructive; a referee could quickly document the failure and maybe salvage the data. As a reader, I wouldn't cite it.

Recommendation: reject, but with a report that points to Example 5 and the non-embedding argument. If the author can replace the field containment by a different statement (e.g., a congruence or a Galois action), the data might find a home.

Referee Report

4 major / 4 minor

Summary. The paper is an addendum to the author's earlier work on measured foliations and the Hilbert 12th problem. Its main theorem (Theorem 4) asserts that for every prime p ≡ 3 mod 4 there exist an integer f1 ≥ 1 and a Hecke eigenform f of level m²p such that the ring class field H(Q(√p) mod f1) is contained in the coefficient field K_f. The proof combines a class group isomorphism from the author's earlier paper [5] with Hecke's theorem on imaginary quadratic fields, and the paper presents a table of numerical examples constructed with computer algebra systems.

Significance. If Theorem 4 were correct, it would constitute a striking step toward an explicit class field theory for real quadratic fields. The paper does provide a useful set of computational examples and it cites the relevant classical literature. However, the central claim is contradicted by the paper's own Example 5, where the alleged ring class field is totally real while the coefficient field K_f is a CM field; a totally real field cannot be a subfield of a CM field. Because the main theorem fails on the first worked example and the proof contains an unsupported logical leap, the significance of the paper as a research contribution is very limited.

major comments (4)

[§1, proof of Theorem 4] The step 'In other words, the ring class field H(Q(√p)) mod f1 belongs to K_f' is not a logical consequence of the preceding statements. Formula (2) gives only an isomorphism of finite abelian class groups. Even if the action of Cl(Q(√p) mod f1) extends to K_f, that action does not produce a Q-embedding of the totally real field H(Q(√p) mod f1) into the CM field K_f. This is the load-bearing step of the proof, and it is unjustified.
[§2, Example 5] Example 5 explicitly contradicts the containment asserted in Theorem 4. For p = 7, the coefficient field K_f is given by x^4 + 8x^2 + 9, whose roots are ±i√(4±√7), so K_f has no real embeddings. The claimed ring class field H(Q(√7) mod 3) is given by Q(±√(4±√7)), whose minimal polynomial x^4 − 8x^2 + 9 has four real roots. A totally real quartic field cannot be a subfield of a CM quartic field, so H(Q(√7) mod 3) ⊄ K_f. The 'passage to the measured foliation' described in the example replaces i by 1, which is not a field embedding and changes the field.
[§1, proof of Theorem 4, final sentence] The proof ends by asserting that the inclusion (1) is 'also true for the levels {m²p | m ≥ 1}', but no proof or reference for this extension is supplied. Remark 2 states that a rigorous proof of even the composite-level inclusions is 'unknown to the author'. Since Theorem 4 relies on this extension for every m ≥ 1, the proof is incomplete at this point.
[§1, Theorem 4 statement] The theorem is stated for all primes p ≡ 3 mod 4, but the proof depends on the isomorphism (2) taken from [5, formula (2)] and on Theorem 3 from the author's previous paper [6] without independent verification. This is not by itself an error, but it makes the theorem a direct concatenation of unproved prior results rather than an independent derivation; combined with the contradiction in Example 5, it leaves the central claim unsupported.

minor comments (4)

[§2, Example 5] The notation 'N = 63 = 3 27' is ambiguous; it should be written as 3² · 7 or 3^2 * 7.
[Abstract and title] There are OCR-like typographical errors, including 'folia tions' in the title and 'noncommutaive geometry' in the key words, which should be corrected.
[References] Reference [2] contains the typo 'Gesselschaft' for 'Gesellschaft', and the page-span notation 'Kl.6 (1910), 619-623' is nonstandard.
[Data availability] The statement 'All data are available as part of the manuscript' is inconsistent with the later 'Data availability' section saying no datasets were generated or analyzed.

Circularity Check

3 steps flagged · score 8.0 of 10

Theorem 4 is not derived: the key containment is asserted by the phrase 'In other words' and the table's ring class fields are obtained by replacing i with 1; the proof loads on the author's own [5] and [6].

self definitional [Section 1, Theorem 4 proof]
"Since Gal (H (Q(√−p)) mod f2) ∼= Cl (Q(√−p) mod f2), the action of the group Cl (Q(√p) mod f1) extends to the field Kf (Theorem 1). In other words, the ring class field H (Q(√p)) mod f1 belongs to Kf ."

The sentence 'In other words' equates the existence of an action of Cl(Q(√p) mod f1) on Kf with the containment H(Q(√p)) mod f1 ⊆ Kf, which is exactly the theorem's conclusion. An action supplies only some subfield of Kf with that Galois group; identifying it with the actual ring class field is precisely what needs proof. The paper's own p=7 example shows the identification is false: Kf has polynomial x^4+8x^2+9 and no real embeddings, while H(Q(√7) mod 3) is totally real. Thus the claimed containment is put in by definition rather than derived.
self citation load bearing [Section 1, Introduction]
"This was settled in [5, Theorem 1] by a general formula N = f′D, where f′ is an integer number depending on the conductor f ≥ 1. The aim of this Addendum is a precise formula based on Hecke’s Theorem 1."

Theorem 4 is advertised as a 'precise formula' for the converse of Theorem 3, but the paper states that the problem was already 'settled' in the author's own paper [5]. The proof of Theorem 4 then cites [5, formula (2)] for the class group isomorphism and [6] for the 'passage to measured foliation'. Both sources are by the same author and are not independently verified (no machine-checked proof or external confirmation is supplied). The new theorem therefore reduces to a chain of self-citations: the prior work already contained the claimed converse, and the addendum adds no independent derivation.

1 more flagged steps

renaming known result [Section 2, Example 5, equations (4)-(5)]
"A passage to the measured foliation F = Re f [6, p. 273] is equivalent to taking imaginary component of the complex numbers. Therefore the roots (4) become ±√(4±√7) having the minimal polynomial x^4 − 8x^2 + 9. We conclude that: H(Q(√7) mod 3) ∼= Q(±√(4±√7))."

The 'ring class field' in the numerical example is manufactured by the map i ↦ 1 applied to the roots of Kf. This is a renaming of the already-known CM coefficient field, not a derivation of the real ring class field: Kf = Q(i√(4±√7)) has signature (0,2), while Q(±√(4±√7)) has signature (4,0) and is a different field. The containment H(Q(√7) mod 3) ⊆ Kf fails for this very example, so the table's conclusion (5) is an artifact of the renaming operation, making the numerical 'prediction' equivalent to its input by construction.

full rationale

The central claim of Theorem 4 is not established by independent mathematics. The proof's only substantive step is the sentence 'In other words, the ring class field H(Q(√p)) mod f1 belongs to Kf', which is the theorem itself; the preceding group action merely provides a subfield of Kf with an isomorphic Galois group. The p=7 worked example confirms that the actual ring class field is totally real and non-isomorphic to Kf, so the identification is false, not just unproved. The remaining inputs are the author's earlier papers [5] and [6]: [5] is said to have already settled the converse by N=f′D, and [6] supplies the measured-foliation 'equivalence' used to replace i by 1. Since neither is independently verified and both are load-bearing, the derivation reduces to a self-citation chain plus a definitional identification. This warrants a circularity score of 8: the result is forced by the author's own prior claims and by the 'in other words' definition, rather than derived from the cited classical theorems (Hecke, Shimura).

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The proof of Theorem 4 rests on the class group isomorphism from [5], an unproved extension of Hecke's theorem to levels m²p, and a non-rigorous correspondence from measured foliations. These are assumed without independent verification in this addendum.

assumptions (4)

domain assumption For each prime p ≡ 3 mod 4 there exist integers f1,f2 with Cl(Q(√p) mod f1) ≅ Cl(Q(√-p) mod f2).
Cited from [5, formula (2)] by the same author; no proof or independent verification is given in this addendum.
standard math Hecke's theorem: for each prime p ≡ 3 mod 4, there exists a Hecke eigenform f ∈ S2(Γ(p)) with H(Q(√-p)) ⊆ K_f.
Classical result by Hecke, but the paper's statement is used for levels m²p as well.
ad hoc to paper The inclusion (1) is also true for levels m²p for all m ≥ 1.
Stated without proof in the final line of the proof of Theorem 4.
domain assumption The measured foliation correspondence from [6] maps coefficient fields to real subfields by taking imaginary parts.
Used in Example 5 to replace roots ±i√(4±√7) by ±√(4±√7).

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Cite this review

Pith. "Pith review of Addendum to "Measured foliations and Hilbert 12th problem"." pith.science (2026). https://pith.science/paper/EDCY3A6U

@misc{pith2026250522272,
  author       = {Pith},
  title        = {Pith review of: Addendum to "Measured foliations and Hilbert 12th problem"},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EDCY3A6U}},
  note         = {Machine review of arXiv:2505.22272}
}

read the original abstract

We study numerical examples of the abelian extensions of the real quadratic number fields based on the results in Acta Mathematica Vietnamica 48 (2023), 271-281 (arXiv:0804.0057)

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Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages

[5]

I. V. Nikolaev, Real multiplication revisited, J. Generalized Lie Theory Appl. 10 (2016), S2- 007; arXiv:1404.4999

work page Pith review arXiv 2016
[6]

I. V. Nikolaev, Measured foliations and Hilbert 12th problem, Acta Math. Vietnam. 48 (2023), 271-281; arXiv:0804.0057

work page Pith review arXiv 2023
[1]

Diamond and J

F. Diamond and J. Shurman, A First Course in Modular Forms , GTM 228, Springer, 2005

work page 2005
[2]

E. Hecke, Ueber die Konstruktion der Klassenk¨ orper reeller quadratischer K¨ orper mit Hilfe von automorphen Funktionen , Nachrichten von der Gesselschaft der Wissenschaften zu G¨ ottingen, Mathematisch-Physikalische Kl.6 (1910), 619-623

work page 1910
[3]

Hecke, Bestimmung der Perioden gewisser Integrale durch die Theorie der Klassenk¨ orper, Math

E. Hecke, Bestimmung der Perioden gewisser Integrale durch die Theorie der Klassenk¨ orper, Math. Z. 28 (1928), 708-727

work page 1928
[4]

The legacy of Niels Hendrik Abel

Yu. I. Manin, Real multiplication and noncommutative geometry , in “The legacy of Niels Hendrik Abel”, 685-727, Springer, Berlin, 2004

work page 2004
[7]

Shimura, Class fields over real quadratic fields and Hecke operators , Annals of Math

G. Shimura, Class fields over real quadratic fields and Hecke operators , Annals of Math. 95 (1972), 130-190. 1 Department of Mathematics and Computer Science, St. John’s University, 8000 Utopia Parkway, New York, NY 11439, United States. Email address : igor.v.nikolaev@gmail.com

work page 1972

Pith tools

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

[1] [5]

I. V. Nikolaev, Real multiplication revisited, J. Generalized Lie Theory Appl. 10 (2016), S2- 007; arXiv:1404.4999

work page Pith review arXiv 2016

[2] [6]

I. V. Nikolaev, Measured foliations and Hilbert 12th problem, Acta Math. Vietnam. 48 (2023), 271-281; arXiv:0804.0057

work page Pith review arXiv 2023

[3] [1]

Diamond and J

F. Diamond and J. Shurman, A First Course in Modular Forms , GTM 228, Springer, 2005

work page 2005

[4] [2]

E. Hecke, Ueber die Konstruktion der Klassenk¨ orper reeller quadratischer K¨ orper mit Hilfe von automorphen Funktionen , Nachrichten von der Gesselschaft der Wissenschaften zu G¨ ottingen, Mathematisch-Physikalische Kl.6 (1910), 619-623

work page 1910

[5] [3]

Hecke, Bestimmung der Perioden gewisser Integrale durch die Theorie der Klassenk¨ orper, Math

E. Hecke, Bestimmung der Perioden gewisser Integrale durch die Theorie der Klassenk¨ orper, Math. Z. 28 (1928), 708-727

work page 1928

[6] [4]

The legacy of Niels Hendrik Abel

Yu. I. Manin, Real multiplication and noncommutative geometry , in “The legacy of Niels Hendrik Abel”, 685-727, Springer, Berlin, 2004

work page 2004

[7] [7]

Shimura, Class fields over real quadratic fields and Hecke operators , Annals of Math

G. Shimura, Class fields over real quadratic fields and Hecke operators , Annals of Math. 95 (1972), 130-190. 1 Department of Mathematics and Computer Science, St. John’s University, 8000 Utopia Parkway, New York, NY 11439, United States. Email address : igor.v.nikolaev@gmail.com

work page 1972