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Products of prime ideals in ray class groups

T0 review · 2 major / 2 minor · reviewed 2026-06-30 · grok-4.3

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Pith's one-line read Every class in the narrow ray class group modulo q is a product of three prime ideals of norm at most (Nq)^{103/64 + κ} for any κ > 0.

desk verdict This extends the Matomäki–Teräväinen dense-model method to narrow ray class groups and plugs in Wu's subconvexity to improve the three-prime bound from cubic to (Nq)^{103/64+κ}, plus a positive-proportion two-prime result. read the letter →

arxiv 2606.30567 v1 pith:IKZS6CKO submitted 2026-06-29 math.NT

Likun Xie This is my paper · ORCID
classification math.NT
keywords narrowrayclassgroupsprimeidealproductssubconvexitycharactersumsHeckeL-functionsnumberfieldsdensemodeltransference
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 that for any fixed number field, every element of the narrow ray class group modulo an integral ideal q arises as a product of three prime ideals whose norms are bounded by a power of Nq. The exponent obtained is 103/64 plus an arbitrarily small positive amount, coming from Wu's subconvexity bound applied inside an extended version of the Matomäki–Teräväinen multiplicative dense-model and transference method. This replaces an earlier uniform bound of size roughly (Nq)^3 that held for the same groups. The same argument also yields that a positive proportion of the classes can be written as products of only two such primes.

What carries the argument

The multiplicative dense-model and transference framework extended to narrow ray class groups of a fixed number field.

What would settle it

Exhibit a sequence of ideals q in a fixed number field together with a ray class modulo q that cannot be written as a product of three prime ideals each of norm at most (Nq)^{103/64 + 0.01}.

Watch

Extended reading notes

Core claim

Every class in the narrow ray class group modulo an integral ideal q of a fixed number field is represented by a product of three prime ideals of norm at most (N q)^{max(1,3α,4α_0)+κ} for any κ>0, where α is the exponent in short character sum bounds for general non-principal ray class characters and α_0 comes from a bounded-order subconvexity input for Hecke L-functions. Wu's subconvexity bound gives the admissible choice α=α_0=103/256, hence the explicit bound (N q)^{103/64+κ}. A positive proportion of ray classes are represented by products of two prime ideals.

Load-bearing premise

The multiplicative dense-model and transference framework of Matomäki–Teräväinen extends without essential change to narrow ray class groups of a fixed number field.

Editorial extensions

If this is right

  • The earlier O_K((N q)^3) bound of Deshouillers, Gun, Ramaré and Sivaraman is replaced by the sharper exponent 103/64 + κ.
  • A positive proportion of the classes in the narrow ray class group are represented by products of two prime ideals of the same norm bound.
  • The result holds uniformly for the narrow ray class groups of any fixed number field.

Reading between the lines

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

  • If a stronger subconvexity bound for the relevant Hecke L-functions becomes available, the exponent on Nq would improve immediately by substituting the new α and α_0.
  • The same transference method might apply to other arithmetic progressions inside class groups once the dense-model step is verified for those settings.
  • The two-prime representation result suggests that the set of products of two small-norm primes is already dense in a positive-density subset of the ray class group.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The manuscript proves that every class in the narrow ray class group Cl_q^+ modulo an integral ideal q of a fixed number field K is represented by a product of three prime ideals each of norm at most (Nq)^{max(1,3α,4α0)+κ} for any κ>0, where α=α0=103/256 comes from Wu's subconvexity bound for Hecke L-functions; this yields the explicit exponent 103/64 + κ and improves the prior O_K((Nq)^3) result. It further shows that a positive proportion of ray classes admit a representation by a product of two such primes. The argument extends the multiplicative dense-model and transference framework of Matomäki–Teräväinen to the narrow ray class group setting.

Significance. If the claimed extension of the dense-model/transference method holds without degrading the exponent, the result supplies a concrete improvement in the scale of prime-ideal generators for narrow ray classes, with an explicit subconvexity-derived bound. The positive-proportion statement for two-prime products and the direct importation of Wu's bound are additional strengths. The work ships an explicit, parameter-free exponent (modulo the external subconvexity input) rather than an existence result.

major comments (2)
  1. [section containing the extension of the dense-model and transference framework to narrow ray class groups] The central claim rests on the assertion that the Matomäki–Teräväinen multiplicative dense-model construction and transference principle extend to narrow ray class groups Cl_q^+ without altering the admissible exponent max(1,3α,4α0). The narrow condition introduces an extra 2-torsion factor in the character group together with a sign condition at the real places; the manuscript must supply a self-contained verification (in the section containing the proof of the main theorem) that these features do not force an extra factor into the short-sum estimates or the transference step that would worsen the bound.
  2. [section on short-sum bounds for ray-class characters] The short character sum bounds for non-principal ray-class characters are invoked uniformly in the conductor q. The manuscript should confirm, with an explicit reference to the relevant lemma or proposition, that the narrow positivity constraint does not invalidate the application of these bounds inside the dense-model construction at the scale required for the three-prime representation.
minor comments (2)
  1. [Introduction and Theorem 1.1] The dependence of implied constants on the fixed field K should be stated explicitly in the introduction and in the statement of the main theorem.
  2. [preliminaries] The notation Cl_q^+ for the narrow ray class group and the precise definition of the narrow positivity condition should appear before the statement of the main results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for explicit verification of the narrow-ray-class extension. We address both major comments below and will revise the manuscript accordingly to improve clarity without changing the main results or the exponent.

read point-by-point responses

  1. Referee: [section containing the extension of the dense-model and transference framework to narrow ray class groups] The central claim rests on the assertion that the Matomäki–Teräväinen multiplicative dense-model construction and transference principle extend to narrow ray class groups Cl_q^+ without altering the admissible exponent max(1,3α,4α0). The narrow condition introduces an extra 2-torsion factor in the character group together with a sign condition at the real places; the manuscript must supply a self-contained verification (in the section containing the proof of the main theorem) that these features do not force an extra factor into the short-sum estimates or the transference step that would worsen the bound.

    Authors: The proof of the main theorem (Section 4) already carries out the extension by working directly with the group law and character group of Cl_q^+, incorporating the 2-torsion and the sign conditions at real places into the definition of the narrow ray-class characters. The short-sum estimates and transference step are identical to those in Matomäki–Teräväinen because they depend only on the conductor and the subconvexity input, not on the narrow versus ordinary distinction; the 2-torsion is absorbed into the existing character-sum bounds of bounded order. To address the request for self-contained verification, we will insert a dedicated paragraph (or short subsection) in Section 4 that explicitly checks the effect of the narrow positivity constraint and 2-torsion on each step of the dense-model construction and transference, confirming that no extra factor appears in the admissible exponent. revision: yes

  2. Referee: [section on short-sum bounds for ray-class characters] The short character sum bounds for non-principal ray-class characters are invoked uniformly in the conductor q. The manuscript should confirm, with an explicit reference to the relevant lemma or proposition, that the narrow positivity constraint does not invalidate the application of these bounds inside the dense-model construction at the scale required for the three-prime representation.

    Authors: Lemma 3.2 states the short-sum bounds for all non-principal ray-class characters of conductor dividing q, with uniformity in q; the narrow positivity constraint is a condition only at the infinite places and does not alter the analytic properties or the length of the finite sums used in the dense-model construction. The same bounds therefore apply verbatim inside the three-prime representation argument. We will add an explicit sentence in the proof of the main theorem (Section 4) that references Lemma 3.2 and notes that the narrow condition is compatible with the scale at which the bounds are applied. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity; derivation extends external Matomäki–Teräväinen framework and Wu subconvexity without self-referential reduction.

full rationale

The paper's main claim adapts the multiplicative dense-model and transference framework of Matomäki–Teräväinen (external) to narrow ray class groups and inserts Wu's subconvexity bound (α=α0=103/256) for the explicit exponent. No load-bearing self-citations, self-definitional steps, fitted inputs renamed as predictions, or ansatzes smuggled via author citations appear in the abstract or derivation outline. The improvement over the prior O_K((Nq)^3) bound is also external. The derivation chain remains self-contained against independent inputs.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The result rests on Wu's subconvexity bound and on the applicability of the Matomäki–Teräväinen framework to ray class groups; no free parameters or new entities are introduced in the abstract.

assumptions (2)
  • domain assumption Wu's subconvexity bound supplies α=α0=103/256
    Directly supplies the numerical exponent in the final bound.
  • domain assumption The Matomäki–Teräväinen dense-model framework extends to narrow ray class groups
    The proof strategy is described as an extension of this framework.

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Cite this review

Pith. "Pith review of Products of prime ideals in ray class groups." pith.science (2026). https://pith.science/paper/IKZS6CKO

@misc{pith2026260630567,
  author       = {Pith},
  title        = {Pith review of: Products of prime ideals in ray class groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IKZS6CKO}},
  note         = {Machine review of arXiv:2606.30567}
}
abstract

We prove that every class in the narrow ray class group modulo an integral ideal $\mathfrak q$ of a fixed number field is represented by a product of three prime ideals of norm at most $ ( N\mathfrak q)^{\max(1,3\alpha,4\alpha_0)+\kappa} $ for any $\kappa>0$, where $\alpha$ is the exponent in short character sum bounds for general non-principal ray class characters and $\alpha_0$ comes from a bounded-order subconvexity input for Hecke $L$-functions. Wu's subconvexity bound gives the admissible choice $\alpha=\alpha_0=103/256$, hence the explicit bound $(N\mathfrak q)^{103/64+\kappa}$. This improves the previous $O_K((N\mathfrak q)^3)$-scale bound of Deshouillers, Gun, Ramar\'e, and Sivaraman. We also prove that a positive proportion of ray classes are represented by products of two prime ideals. The proof extends the multiplicative dense-model and transference framework of Matom\"aki--Ter\"av\"ainen to narrow ray class groups.

Discussion (0). Continue with ORCID to comment.

Reference graph

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