The shapes of Galois quartic fields

Piper H; Robert Harron

REVIEW 1 major objections 4 minor 21 references

The shapes of Galois quartic fields

T0 review · 1 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper classifies the shapes of all Galois quartic fields and proves that V4 shapes equidistribute in orthorhombic lattice spaces with explicit densities, while C4 shapes form discrete families with counted asymptotics.

desk verdict Completes the Galois quartic shape classification with a clean equidistribution proof; minor sieve terseness and a typo in the introduction don't affect the main results. read the letter →

arxiv 1908.03969 v1 pith:U37QK254 submitted 2019-08-11 math.NT

Piper H , Robert Harron This is my paper · ORCID

classification math.NT MSC 11R1611R4511E1211P21

keywords shapeofnumberfieldGaloisquarticfieldsequidistributionorthorhombiclatticestetragonalcarefreetuplesdiscriminantasymptotics

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 determines the shape invariant for every degree-four number field that is Galois, and it does so completely: fields with Galois group $V_4$ have orthorhombic shapes, and fields with group $C_4$ have tetragonal shapes. For $V_4$-quartic fields the shape is the lattice whose side ratios are $\sqrt{|\Delta_1|}:\sqrt{|\Delta_2|}:\sqrt{|\Delta_3|}$, where the $\Delta_i$ are the discriminants of the three quadratic subfields, and the paper proves that these shapes are equidistributed, in a regularized sense, in the two-dimensional spaces of base-centered and body-centered orthorhombic lattices as the discriminant grows. The limiting densities are explicit Euler products: $C_{\mathrm{wild}} = \frac{5}{48}\prod_{p \text{ odd}}(1 - 6p^{-2} + 8p^{-3} - 3p^{-4})$ and $C_{\mathrm{tame}} = \frac{1}{6}$ times the same product. For $C_4$-quartic fields the shape is a tetragonal lattice determined by a ramification ratio, and the paper proves asymptotic formulas for the number of fields of a given shape. A corollary is that the shape is a complete invariant within the totally real $V_4$-quartic fields and within the tamely ramified $V_4$-quartic fields.

What carries the argument

The argument is carried by the conorm diagram of [CS92], a labeling of the points of the Fano plane that encodes a rank-3 lattice up to isomorphism and exposes the combinatorial type of its Voronoi cell. Combined with the integral-basis data of [Wil70] for biquadratic fields, this identifies the $V_4$ shape as the appropriate orthorhombic lattice. The counting proof bijects $V_4$-quartic fields with $\ast$-strongly carefree triples, estimates the number of such triples in the relevant region with the lattice-point counting lemma of [Bha05], and passes from a finite sieve to all primes by the method of [DH71], producing the Euler product $\prod_{p\text{ odd}}(1-6p^{-2}+8p^{-3}-3p^{-4})$. For $C_4$ fields, the parametrization $K = Q(\sqrt{A(D+B\sqrt{D})})$ from [HHR+86], together with the integral-basis results of [HW90] and [SW06], reduces the fixed-shape count to sums-of-two-squares representations, handled by a Dirichlet-series factorization and a Tauberian theorem.

What would settle it

Enumerate all $V_4$-quartic fields with discriminant below a large $X$ (for instance $X = 10^{12}$), count those whose shape falls in a fixed box such as $1 \le x \le y \le 2$, and test whether $N_{\mathrm{wild}}(X,W)X^{-1/2}$ approaches $C_{\mathrm{wild}}\,\mu_{oC}(W)$; any systematic deviation beyond the claimed $o(1)$ would disprove Theorem C.

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Extended reading notes

Core claim

The central claim is that the shape of a Galois quartic field falls into one of four infinite families and that, in the $V_4$ case, these shapes behave like random points in their natural two-dimensional spaces. Precisely, if $K$ is a $V_4$-quartic field with quadratic subfields $Q(\sqrt{\Delta_i})$, then its shape is a base-centered orthorhombic lattice with side ratios $\sqrt{|\Delta_1|}:\sqrt{|\Delta_2|}:\sqrt{|\Delta_3|}$ when the prime $2$ ramifies, and a body-centered orthorhombic lattice with the same ratios when $2$ is unramified. The main theorem states that for compact continuity sets $W$, the counts satisfy $N_{\mathrm{wild}}(X,W)/X^{1/2} \to C_{\mathrm{wild}}\,\mu_{oC}(W)$ and $N_{\mathrm{tame}}(X,W)/X^{1/2} \to C_{\mathrm{tame}}\,\mu_{oI}(W)$, with the constants given above. For $C_4$-quartic fields, the shape is a primitive tetragonal lattice with side ratio $\sqrt{2}\cdot r_{\mathrm{rat}}^{-1/4}$ if $2$ ramifies and a body-centered tetragonal lattice with side ratio $r_{\mathrm{rat}}^{-1/4}$ otherwise, where $r_{\mathrm{rat}} = N/|\Delta_2|$ is the ramification ratio; these shapes sit as discrete points, and the paper gives $X^{1/3}$-asymptotics for the number of fields with a specified shape.

Load-bearing premise

The load-bearing premise of the $V_4$ counting theorem is that imposing the squarefree and pairwise-coprime conditions at all primes simultaneously changes the count from the finite-sieve count by only $o(N)$; if this error were larger, the explicit constants in the equidistribution theorem would not be the true densities.

Editorial extensions

If this is right

Within the totally real $V_4$-quartic fields, and within the tamely ramified $V_4$-quartic fields, the shape determines the field uniquely.
For $V_4$ fields with shape in a compact set, the count grows like $X^{1/2}$ times the measure of the shape set; both logarithmic factors in the total count $X^{1/2}\log^2 X$ come from the unbounded shape parameters.
For $C_4$ fields, the discriminant determines the shape, and the number of fields with a fixed shape and discriminant at most $X$ is asymptotic to a constant times $X^{1/3}$, where the constant depends on the shape through the ramification ratio and the set of primes dividing it.
The proportion of $C_4$ fields with ramification ratio $r_{\mathrm{rat}}$ versus $p\, r_{\mathrm{rat}}$ depends on whether $p \equiv 1$ or $3 \pmod 4$, so the distribution of these shapes is not governed by a real Lie group action.

Reading between the lines

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

The wild families of $C_3^2$-octic fields, mentioned in the paper only in the tame totally real case, should exhibit the same regularized equidistribution if the same conorm-diagram and sieve machinery is applied, with the density constants depending on the local behavior at the ramified prime.
The ratio $C_{\mathrm{wild}}/C_{\mathrm{tame}} = 5/2$ suggests a purely local explanation at the prime $2$; interpreting this as the ratio of admissible congruence patterns at $2$ could predict similar constants for other Galois groups with a single wildly ramified prime.
A numerical test of Theorem C at moderate $X$ is feasible because the field count is $X^{1/2}$ up to logs; computing the shape box count for $X \sim 10^{12}$ would separate the claimed Euler-product constant from nearby alternatives and provide a concrete check of the sieve step.
The same shape-space viewpoint could resolve the general question of which $S_n$-fields have 'random' shapes versus rigid ones: Galois groups with large automorphism groups impose symmetries that shrink the shape space, and the dichotomy seen here between $V_4$ (equidistributed) and $C_4$ (discrete) should persist for other abelian Galois groups.

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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 short version: this paper completes the shape classification for Galois quartic fields and gives the first nontrivial equidistribution result for shapes in degree 4. V4 shapes live in two two-dimensional families (base-centered and body-centered orthorhombic lattices); the authors prove regularized equidistribution with explicit constants 5/48 and 1/6 times an Euler product. C4 shapes are discrete tetragonal lattices, parameterized by a ramification ratio, and the paper gives asymptotic counts for each shape.

What's genuinely new: Terr's thesis handled Galois cubics, where all shapes collapse to hexagonal. Here the Galois quartic case has infinite families, and the equidistribution statement (Theorem C) is not in the prior literature. The proof is a clean reduction to counting strongly carefree triples with congruence conditions, using the Principle of Lipschitz and a Davenport–Heilbronn-style sieve. The explicit integral bases and conorm diagrams are worked out carefully; the constants emerge from the case analysis rather than being fitted. The C4 counting via sums of two squares and Wirsing–Odoni is standard but effective, and the arithmetic proportions (depending on p mod 4) are a nice observation that the distribution is not coming from a Lie group action.

The soft spots are mostly about terseness and imported facts. Section 5.2 jumps through the sieve limit interchange quickly; the stress-test note says the missing step is a standard union bound, and I agree the argument is fine, but a referee will want that written out. The paper also leans on earlier classifications (Williams integral bases, HHR+86 parametrization, Hudson–Williams, Spearman–Williams) without proof; that's normal for these results but means the reader has to trust a chain of external references. Minor blemishes: the introduction's Theorem E has a sign error in the Euler product (1 − f_A(p)/p instead of 1 + f_A(p)/p), which would make the constant zero for infinitely many p; the body uses the correct plus sign, so it's a typo. Also the reference to arXiv:11907.07209 looks like a typo for 1907.07209. None of this affects the main results.

Who is this for: anyone working on shapes of number fields, equidistribution of lattice shapes, or counting number fields by discriminant. The paper deserves a serious referee; the central claims are explicit and checkable. I'd send it out and expect minor revisions.

Referee Report

1 major / 4 minor

Summary. The paper determines the shapes of all Galois quartic number fields, splitting into four families according to Galois group C4 or V4 and tame versus wild ramification. In the V4 case, shapes are shown to form two-dimensional families of orthorhombic lattices, with explicit side ratios in terms of the three quadratic subfield discriminants, and the main Theorem C establishes a regularized equidistribution statement with explicit constants: the number of fields with shape in a compact continuity set is asymptotic to C_wild μ_oC(W) X^{1/2} or C_tame μ_oI(W) X^{1/2}, with C_wild = 5/48 and C_tame = 1/6 times the displayed Euler product over odd primes. In the C4 case, shapes are discrete tetragonal lattices determined by a ramification ratio, and Theorem E/Theorem 7.1 provides asymptotic counts for fields of a given shape. The proofs are built on Williams' integral bases, Conway-Sloane conorm diagrams, a bijection with strongly carefree triples, the Principle of Lipschitz, a sieve following Davenport-Heilbronn, and for the C4 case the Wirsing-Odoni and Wiener-Ikehara Tauberian methods.

Significance. If the results hold, this is a substantial contribution to the study of shape distributions of number fields, extending the cubic and S_n results of Terr and Bhargava-Harron to the Galois quartic setting. The paper contains no fitted parameters: all constants are derived from explicit integral bases, Gram matrices, and sieve densities, and the V4 result is tested against Baily's independent count of V4 fields. The regularized equidistribution result gives a concrete explanation for the appearance of log-squared terms in counting V4 fields, and Theorem A's complete-invariant statements are of independent interest. The analytic number theory is standard but carefully assembled, with the reduction to strongly carefree triples and the case-by-case integral-basis computations forming a solid and reproducible core.

major comments (1)

[§1.2, Theorem E] The Euler product defining CΣ_A is printed as ∏_p (1 − f_A(p)/p)(1 − 1/p). For primes p ≡ 1 (mod 4) with p ∤ A this factor equals (1 − 2/p)(1 − 1/p), so the product over these primes diverges to 0; this contradicts the positive constants stated in Theorem 7.1 and equation (7.18), where the factor is (1 + f_A(p)/p)(1 − 1/p). Theorem E should be corrected to match (7.18).

minor comments (4)

[§5.1] The line 'Let s(ii) = 2 and s(i) = s(ii) = 1' is internally inconsistent; it should read s(i) = s(iii) = 1 and s(ii) = 2, as used in Theorem 5.10 and in the subsequent substitutions for the counting regions.
[§1.1, Remark 1.1(b)] The name 'Revera-Guaca' is a typo for 'Rivera-Guaca' as in the reference list; please correct it in the remark.
[References, [Har19]] The arXiv identifier for [Har19] is printed as 'arXiv:11907.07209'; this should be 'arXiv:1907.07209'.
[§3.1] The term 'subperbase' appears to be a typo for 'superbase': the sentence should say that a quadruple not necessarily satisfying the obtuse condition is called a superbase.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the main constants and measures are derived from explicit counting, not fitted or imported as conclusions.

full rationale

I walked the derivation chain for both the V4 and C4 halves of the paper. In the V4 case, the shape classification in Theorem 4.2 is obtained directly from Williams' integral bases and explicit Gram-matrix computations; the side ratios sqrt(|Delta_1|):sqrt(|Delta_2|):sqrt(|Delta_3|) are computed, not assumed. The equidistribution result in Theorem C rests on counting strongly carefree triples with congruence conditions: Corollary 5.9 gives the finite-sieve main term with explicit densities, and the passage from the finite sieve to all primes is justified by a union-bound over p>Y with the stated O(N/p^2) bound. The constants 5/48 and 1/6 are then arithmetical products of the case constants 1/96 and 1/24 with the box-measure factor 1/2; they are not fitted to data or to the claimed limit. The measure mu_oC and mu_oI are defined naturally from the diagonal group action before any counting, and the box volumes are computed independently in Lemmas 3.11 and 3.13, so the equidistribution statement is not true by construction. In the C4 case, Theorem 7.1 is derived from the external Hardy-Hudson-Richman-Williams-Holtz parametrization plus Hudson-Williams and Spearman-Williams integral-basis data, and the asymptotics come from Wirsing-Odoni and Wiener-Ikehara applied to the sums-of-two-squares counting function; no parameter is adjusted to force the stated answer. The paper does cite the second author's earlier work [Har17, Har19] for the idea of regularized equidistribution and for context on log terms, but these citations are not load-bearing: the definitions, bijections, and analytic estimates that produce the theorems are all given in this paper. I found no step where a fitted parameter is renamed a prediction, no definition of an input in terms of the claimed output, and no self-citation used as the sole justification of a central premise. The only notable defect is a sign typo in the introduction's Theorem E, where the Euler product is printed with (1 - f_A(p)/p) instead of the positive (1 + f_A(p)/p) factor proved in Theorem 7.1; that is a correctness or typographical issue, not a circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No parameter is fitted to data; all asymptotic constants emerge from Euler products and counting arguments. The regularized sense of equidistribution is a definitional choice, not a free parameter. External classifications and analytic-number-theory tools are clearly cited; the main burden lies in those imports, listed above.

assumptions (5)

standard math Conway-Sloane conorm diagram classification of rank 3 lattices by non-negative labels on the Fano plane.
Used throughout Sections 3, 4, and 6 to identify tetragonal, orthorhombic, hexagonal, and cubic lattice types from computed Gram matrices; cited to [CS92].
domain assumption Williams' classification of integral bases of biquadratic (V4) fields into three cases, with the corresponding discriminants.
Theorem 4.1 is taken from [Wil70] and is the starting point for all V4 shape computations in Section 4; if a case were missing or misstated, Theorem 4.2 would fail.
domain assumption Parametrization of C4 fields as Q(sqrt(A(D+B sqrt(D)))) with A squarefree odd, D = B^2 + C^2 squarefree, gcd(A,D) = 1, together with the five-case discriminant and integral-basis data.
Imported from [HHR+86], [SW06], and [HW90] in Section 6; underpins Theorems D and E. The paper does not prove this classification.
domain assumption Davenport-Heilbronn sieve principle: the count of triples satisfying all infinitely many squarefree and coprimality conditions is the limit of finite-sieve counts with the stated constants and error o(N).
Used in Section 5.2 after equation (5.2); the interchange of limits and the O(p^-2) bound for large primes is cited to [DH71] and asserted rather than proved in detail.
standard math Wirsing-Odoni method and Wiener-Ikehara Tauberian theorem yield the stated main terms for the arithmetic sums F_Sigma and F_Sigma_U.
Used in Section 7.3 to derive the C4 counting asymptotics; cited to [FMS10] and [Mur08].

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Pith. "Pith review of The shapes of Galois quartic fields." pith.science (2026). https://pith.science/paper/U37QK254

@misc{pith2026190803969,
  author       = {Pith},
  title        = {Pith review of: The shapes of Galois quartic fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U37QK254}},
  note         = {Machine review of arXiv:1908.03969}
}

abstract

We determine the shapes of all degree $4$ number fields that are Galois. These lie in four infinite families depending on the Galois group and the tame versus wild ramification of the field. In the $V_4$ case, each family is a two-dimensional space of orthorhombic lattices and we show that the shapes are equidistributed, in a regularized sense, in these spaces as the discriminant goes to infinity (with respect to natural measures). We also show that the shape is a complete invariant in some natural families of $V_4$-quartic fields. For $C_4$-quartic fields, each family is a one-dimensional space of tetragonal lattices and the shapes make up a discrete subset of points in these spaces. We prove asymptotics for the number of fields with a given shape in this case.

Figures

Figures reproduced from arXiv: 1908.03969 by the authors.

**Figure 1.** Conorm diagram of an obtuse superbase. Conway and Sloane develop an algorithm they call Voronoi reduction which transforms a putative conorm diagram for Λ into a conorm diagram for Λ. We note that the above results show that two lattices have the same shape if and only if there is an automorphism of the Fano plane that brings one conorm diagram to a scaled version of the other. 3.2. Combinatorial type of a lattice. … view at source ↗

**Figure 2.** Conorm diagrams of the 5 families of Voronoi cells. 0 P1 P2 P1 P1 P2 P1 (a) c a < √ 2, P1 = c 2 4 , P2 = 2a 2−c 2 4 0 P1 0 P1 P1 P2 P1 (b) c a > √ 2, P1 = a 2 2 , P2 = c 2−2a 2 4 [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Conorm diagrams of body-centered tetragonal lattices. The body-centered cubic lattice is obtained by taking P1 = P2 in the diagram on the left, while the facecentered cubic lattice is obtained by taking P2 = 0 in either diagram. Proof. Independent of the value of c/a, the vectors w1 = (c, 0, 0), w2 = (0, a, 0), w3 = (0, 0, a) form a basis of a primitive tetragonal lattice. The associated body-centered lattice is th… view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: The primitive cubic lattice is obtained by taking a = c. 0 0 0 0 a 2 c 2 a 2 [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

**Figure 5.** Figure 5: Conorm diagram of the base-centered orthorhombic lattice. The primitive hexagonal lattice is obtained by taking b = √ 3a. When a = b, we obtain a primitive tetragonal lattice with base of side a/√ 2. Proof. An obtuse superbase is given by v0 = (−a, 0, −c) v1 = 1 2 (a, …

**Figure 6.** Figure 6: 0 a 2 2 0 b 2 2 a 2 2 c 2 − (a 2 + b 2 ) 4 b 2 2 (a) a 2 + b 2 ≤ c 2 0 P2 P1 P3 P3 P1 P2 (b) a 2 + b 2 ≥ c 2 [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Reine Angew

Andrew Marc Baily, On the density of discriminants of quartic fields, J. Reine Angew. Math. 315 (1980), 190--210. 564533

work page 1980

[2] [2]

6, 1111--1120

Manjul Bhargava and Piper H, The equidistribution of lattice shapes of rings of integers in cubic, quartic, and quintic number fields, Compositio Mathematica 152 (2016), no. 6, 1111--1120. 3518306

work page 2016

[3] [3]

Manjul Bhargava, The density of discriminants of quartic rings and fields, Ann. of Math. (2) 162 (2005), no. 2, 1031--1063. 2183288

work page 2005

[4] [4]

Wilmar Bola \ n os and Guillermo Mantilla-Soler, The trace form over cyclic number fields, 2019, preprint, available at https://arxiv.org/abs/1904.10080v2 arXiv:1904.10080v2 [math.NT]

work page Pith review arXiv 2019

[5] [5]

J. H. Conway and N. J. A. Sloane, Low-dimensional lattices. VI . V orono reduction of three-dimensional lattices , Proc. Roy. Soc. London Ser. A 436 (1992), no. 1896, 55--68. 1177121

work page 1992

[6] [6]

Davenport and H

H. Davenport and H. Heilbronn, On the density of discriminants of cubic fields. II , Proc. Roy. Soc. London Ser. A 322 (1971), no. 1551, 405--420. 0491593

work page 1971

[7] [7]

E. S. Fedorov, The symmetry of regular systems of figures, Zapiski Mineralogi c eskih Ob s c estva (2) 28 (1891), 1--146, English translation in Symmetry of crystals, ACA Monograph no.\ 7, pp. 50--131, New York (1971)

work page 1971

[8] [8]

, Na c ala u c eniya o figurah [ E lements of the study of figures] , Izdat. Akad. Nauk SSSR, Moscow, [1885] 1953, Orig.\ published in Zapiski Mineralogi c eskih Ob s c estva (2) 21 (1885), 1--279. 0062061

work page 1953