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An upper bound for counting algebraic tori over $\mathbb{Q}$ by Artin conductor

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Pith's one-line read The paper proves an all-n upper bound, of strength X^{exp(C(log n)^2)}, for the number of n-dimensional algebraic tori over Q with Artin conductor at most X, without classifying finite subgroups of GL_n(Z).

desk verdict A genuinely new uniform upper bound for counting tori by Artin conductor, with a coherent counting argument whose subexponential exponent rests entirely on one unrefereed imported input (Weisfeiler's primitive linear group theorem). read the letter →

arxiv 2608.03651 v1 pith:A4ZUYK3H submitted 2026-08-04 math.NT

Jungin Lee This is my paper · ORCID
classification math.NT MSC 11R3211R4220C15
keywords algebraictorioverQArtinconductortoruscountingtuningsubmodulefinitelineargroupsirreduciblerationalrepresentationscovectororbitsizearithmeticstatistics
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 attacks a natural analogue of counting number fields by discriminant: count n-dimensional algebraic tori over Q by their Artin conductor, which equals the discriminant for the tori obtained from number fields via Weil restriction. The main theorem states that for every n ≥ 2 the number N_n^tor(X) of such tori with conductor at most X is bounded by a constant times X raised to the power exp(C(log n)^2), with an absolute constant C independent of n. The exponent is subexponential in n, and the proof works for all n at once, bypassing the explicit classifications of finite subgroups of GL_2(Z) and GL_3(Z) used in earlier work. The heart is a representation-theoretic statement: every faithful irreducible rational representation of dimension d of a finite group has a nonzero covector whose orbit has size at most exp(C(log 2d)^2). If true, this turns a lattice-counting argument into a uniform torus-counting bound and brings the torus problem into the same qualitative regime as the best number-field upper bounds, though with a larger exponent.

What carries the argument

The central mechanism is the tuning submodule Ξ_ρ(L), the fixed-point module of O_L ⊗ M under the semilinear Galois action, together with the minimal covector orbit μ(H,U). The tuning submodule gives a short nonzero vector in every H-Galois extension, and its determinant is compared with the Artin conductor of the associated representation. A finite H-invariant polynomial map of degree at most μ(H,U), built from the elementary symmetric functions of a minimal covector orbit, converts that short vector into a count with exponent μ(H,U)/2. The dyadic summation over irreducible constituents W_i in Theorem 1.3 assembles these contributions into the exponent Θ(H,V) = max_i μ_i/(2a_i), with a (log

What would settle it

Exhibit a sequence of finite subgroups H_d of GL_d(Z) carrying faithful irreducible rational representations W_d of dimension d for which the minimal nonzero covector orbit |H_d·ℓ| grows faster than exp(C(log 2d)^2) for every fixed C; Theorem 1.2 is exactly the assertion that no such sequence exists, and Theorem 1.1 uses Theorem 1.2 as its only source of the subexponential exponent.

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Extended reading notes

Core claim

The central claim is that counting tori by Artin conductor can be bounded uniformly in n without classifying finite subgroups of GL_n(Z). The paper proves Theorem 1.1: for an absolute constant C and every n ≥ 2, N_n^tor(X) ≪_n X^{exp(C(log n)^2)}. The proof rests on Theorem 1.2, which asserts that for every finite group G and every faithful irreducible rational representation U of dimension d, there is a nonzero covector ℓ in U^∨ whose G-orbit has size at most exp(C(log 2d)^2). That small orbit produces a low-degree invariant polynomial map, while the tuning submodule supplies a short vector whose coordinates determine the splitting field. Combining these two ingredients, each irreducible co

Load-bearing premise

The load-bearing premise is the structure theorem for finite primitive linear groups over the complex numbers imported as Lemma 3.2: every such group has a normal subgroup that is a direct product of alternating groups A_m with m ≥ 10, and its index is at most n^{2 log_2 n + 5}, with no extra Lie-type factor in characteristic zero; if this structure theorem fails, the subexponential orbit bound and the main theorem collapse to a much weaker power bound.

Editorial extensions

If this is right

  • For every fixed n, the number of n-dimensional tori over Q with Artin conductor at most X grows at most like a power of X whose exponent depends only on n and is subexponential in n.
  • The same bound applies to the number of degree-n number fields with discriminant at most X via the Weil-restriction embedding K ↦ R_{K/Q} G_m, since C(R_{K/Q}G_m) = D_K; this reproduces the known subexponential-in-n regime, though not the sharper c(log n)^2 exponent.
  • Each fixed finite subgroup H of GL_n(Z) gets a uniform count N_n^tor(X;H) ≪_H X^{Θ(H,V)} (log X)^{s(H,V)-1}, so the global theorem follows from a finite sum over conjugacy classes.
  • The proof yields an explicit recipe for the exponent of X for any H: decompose the natural representation into irreducible rational constituents, compute the minimal covector orbit size for each, and take the maximum weighted by multiplicities.
  • The theorem provides the first all-n upper bound for the torus-counting problem without relying on a classification of finite subgroups of GL_n(Z), so it is uniform in a way that case-by-case analyses cannot be.

Reading between the lines

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

  • A natural extension, not claimed by the paper: the same tuning-orbit mechanism should bound other Artin-conductor counting problems, such as counting Galois extensions with a fixed group and bounded conductor, whenever the relevant rational representation has a small covector orbit.
  • Going beyond the paper's statement, the exponent exp(C(log n)^2) is unlikely to be sharp; a sharper structure theorem for finite primitive linear groups, or a bound on the number of conjugacy classes of finite subgroups of GL_n(Z), could push it toward n^{O(log n)} or even polynomial in n.
  • The paper does not identify which subgroups H dominate the sum; if the maximum of μ_i/(2a_i) is usually attained by abelian or imprimitive constituents, the (log X)^{s-1} factor could be improved, approaching the conjectured X (log X)^{n-1} shape.
  • Computationally, for n=4 one could enumerate the finite subgroups of GL_4(Z), compute the minimal orbit μ_i for each irreducible constituent, and compare the predicted per-subgroup exponent with actual torus counts; this would test whether the subexponential bound is tight in practice.
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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

2 major / 4 minor

Summary. The paper proves an upper bound for the number of isomorphism classes of n-dimensional algebraic tori over Q with Artin conductor at most X: for an absolute constant C, N_n^tor(X) ≪_n X^{exp(C(log n)^2)}. The proof avoids classifying finite subgroups of GL_n(Z). It first reduces, for each finite subgroup H, to counting surjections G_Q ↠ H with bounded conductor (Lemma 2.1). The counting is then carried out by a tuning-lattice argument: Proposition 2.4 compares the tuning discriminant with the Artin conductor via Wood–Yasuda, Proposition 2.6 supplies a short vector by Minkowski, and Proposition 2.8 bounds the number of extensions by counting values of a finite H-invariant map. The key quantitative input is Theorem 1.2, asserting that for any faithful irreducible rational representation of dimension d there is a covector whose orbit has size at most exp(C(log 2d)^2). Theorem 1.2 is proved from Lemma 3.3, which in the primitive case uses a structure theorem for finite linear groups imported from an arXiv preprint of Weisfeiler (Lemma 3.2). Theorems 1.2 and 1.3 are then combined to give Theorem 1.1.

Significance. If the external group-theoretic input is valid, this is a substantial new uniform bound: it gives a subexponential-in-n exponent for a counting problem that generalizes number-field counting, without relying on an explicit classification of the finite subgroups of GL_n(Z). The counting chain from Lemma 2.1 through Theorem 1.3 appears internally sound: I traced the tuning-lattice argument, the conductor comparison, the invariant-polynomial count, and the dyadic summation, and found no fitted parameters or circular reasoning. The self-contained proof of Lemma 3.1 is a strong point. However, the final exponent rests on a single imported theorem, Lemma 3.2, whose stated reduction to alternating groups is not demonstrated in the manuscript. That point must be resolved before the main theorem can be regarded as established.

major comments (2)
  1. [§3, Lemma 3.2 and its use in Lemma 3.3 / Theorem 1.2] The subexponential exponent in Theorem 1.1 enters exclusively through Lemma 3.2, the bound [Γ:ZN] ≤ n^{2 log_2 n + 5} for finite primitive linear groups. This lemma is imported from the unpublished arXiv preprint [22], and the reduction to a direct product of alternating groups is justified only by the sentence 'the characteristic exponent of C is 1 and groups of Lie1-type are trivial'. That is not a proof, and as written it is cryptic: finite simple groups of Lie type over finite fields do have complex representations, so one cannot simply discard a 'Lie-type factor' without a precise statement of the theorem and an explanation of why the factor is absent in characteristic zero. The manuscript itself notes typographical errors in [22], which underscores that this is not a stable reference. Since Lemma 3.2 is load-bearing for Theorem 1.2 and hence for Theorem 1.1, the paper needs either
  2. [§2.1, Proposition 2.4] The conductor comparison is also load-bearing, and its justification for the 'balanced' hypothesis of Wood–Yasuda is one sentence: eigenvalues occur in inverse pairs away from ±1 because ρ is defined over Q⊂R. This may well be correct, but the paper should spell out how this verifies the exact condition in [23, Definition 4.4], especially since the local convention mismatch with Wood–Yasuda is handled separately. Adding the explicit definition and the verification would make the chain Proposition 2.4 → Lemma 2.5 → Proposition 2.6 self-contained for the reader.
minor comments (4)
  1. [Lemma 2.5] The displayed definition of W_c appears to have a typo: as printed, W_c := {z : z=ρ(c)z} would not equal U^+ ⊕ iU^-. The intended condition is z=ρ(c)\bar z (or equivalently ρ(c)\bar z = z). Please correct.
  2. [Lemma 3.2 preamble] The parenthetical remark that 'both occurrences of l in [22, p. 1] are typographical errors for the numeral 1' does not belong in a formal proof. If [22] contains such typos, the safe course is to state the theorem in full and prove it or cite a definitive version.
  3. [Throughout] The phrase 'the characteristic exponent of C is 1' is undefined in the manuscript. Even if it is standard in the theory of algebraic groups, it should be defined or replaced with the concrete statement about the absence of Lie-type factors.
  4. [Title/Abstract] The title contains a spacing artifact ('OVERQBY AR TIN'), and the abstract's AI-use disclosure is unusual but transparent. Neither affects the mathematics.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the exponent is derived from an independent group-theoretic input, not from the counting claim.

full rationale

The paper's derivation chain is self-contained apart from standard external results. Theorem 1.1 is reduced via Lemma 2.1 to counting surjections φ:G_Q ↠ H with bounded Artin conductor, and Theorem 1.3 then bounds this count by X^Θ(log X)^{s-1} using the tuning-module method of Dummit and Wood–Yasuda. The exponent Θ is expressed in terms of µ_i, the minimal covector orbit size, and Theorem 1.2 bounds µ_i using the structural Lemma 3.2 imported from Weisfeiler's arXiv preprint [22]. This import is external, not a self-citation, and it does not depend on the target counting statement; if the import fails, the exponent weakens but the logic is not circular. The quoted reductions (10), (12), and the proof of Theorem 1.3 are direct inequalities, not identities with the input. The author's earlier papers [14], [15] are cited only as context/conjectures and are never used to prove Theorem 1.1. The AI-use disclosure describes proof generation but does not indicate that any parameter was fitted to the result. Hence no step satisfies the criteria for circularity; the main concern about Lemma 3.2 is a correctness/verification risk, outside the circularity rubric.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

There are no empirical fits: all constants (c_0, c_1, c_2, C, and the finite local bounds b_{p,H,rho}) are existential or absolute constants whose values do not affect the theorem's shape. The paper's real debt is to external theorems: Weisfeiler's structure theorem for primitive linear groups (the bottleneck), the Wood-Yasuda conductor comparison, Dummit's tuning-lattice framework, and standard facts in representation theory and algebraic number theory, each listed above. The tuning submodule is a proved construction, not an invented entity.

assumptions (6)
  • domain assumption Weisfeiler's structure theorem for finite primitive linear groups (Lemma 3.2): Gamma contains a normal subgroup N, a direct product of alternating groups A_m (m ≥ 10), with [Gamma : Z(Gamma)N] ≤ n^{2 log_2 n + 5}.
    The subexponential exponent exp(C(log 2d)^2) in Theorem 1.2 flows entirely through this import from arXiv:1203.1960 [22, Theorem 1.3]. The paper also assumes its Lie-type factor is trivial over C with a one-sentence justification. This is the load-bearing external input.
  • domain assumption Wood-Yasuda comparison for tame local data: v_{Q_p,H,rho_p}(psi_p) = (1/2) a_{Q_p,H,rho_p}(psi_p), plus their v-invariant formalism (Definitions and Proposition 4.5 of [23]).
    Used in Proposition 2.4 to equate the p-adic valuation of the tuning determinant with the Artin conductor exponent for p not dividing |H|; requires the local representation to be balanced and the pointed torsor convention to match, which the paper argues in detail.
  • standard math Dummit's tuning-lattice framework for counting Galois extensions via invariant polynomials ([8, Chapter 3]).
    The tuning submodule, small-vector, and invariant-map counting structure of Propositions 2.4-2.8 is adapted from Dummit's thesis; treated as established background.
  • standard math Standard complex representation theory: classification of irreducible representations of symmetric and alternating groups ([13, Theorem 2.5.7]), Frobenius reciprocity, Clifford's theorem, Schur's lemma, tensor-product structure of irreducibles of direct products (Serre [20]).
    Used in Lemmas 3.1 and 3.3 to find small-stabilizer subgroups in alternating groups and to decompose irreducible restrictions of N on W.
  • standard math Galois descent for vector spaces ([18, Lemma 1.3.10]); Minkowski's first theorem; Noether normalization via linear projections (Eisenbud [9, Theorem 13.3]).
    Used in Lemma 2.2 (tuning module rank), Proposition 2.6 (short vectors), and Lemma 2.9 (construction of finite invariant maps).
  • standard math Background facts: Artin conductor is additive over direct sums; C(R_{K/Q}G_m) = D_K (cited to [15, Section 1.2]); finitely many conjugacy classes of finite subgroups of GL_n(Z) ([3, Corollary 4.8]).
    These frame the counting problem and Equation (2); the conductor-discriminant link is cited to the author's own earlier paper but is a standard fact.

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Pith. "Pith review of An upper bound for counting algebraic tori over $\mathbb{Q}$ by Artin conductor." pith.science (2026). https://pith.science/paper/A4ZUYK3H

@misc{pith2026260803651,
  author       = {Pith},
  title        = {Pith review of: An upper bound for counting algebraic tori over $\mathbbQ$ by Artin conductor},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A4ZUYK3H}},
  note         = {Machine review of arXiv:2608.03651}
}
abstract

Let $N_n^{\mathrm{tor}}(X)$ be the number of isomorphism classes of $n$-dimensional algebraic tori over $\mathbb{Q}$ whose Artin conductor is bounded by $X$. We prove that there is an absolute constant $C>0$ such that, for every positive integer $n \ge 2$, $N_n^{\mathrm{tor}}(X)\ll_n X^{\exp(C(\log n)^2)}$. The proofs of the main results were developed through an iterative dialogue with ChatGPT 5.6 Pro.

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