Solubility for families of norm equations coming from abelian number fields

Mathieu Da Silva

arxiv: 2604.12697 · v2 · submitted 2026-04-14 · 🧮 math.NT

Solubility for families of norm equations coming from abelian number fields

Mathieu Da Silva This is my paper

Pith reviewed 2026-05-10 14:37 UTC · model grok-4.3

classification 🧮 math.NT

keywords norm equationsbinary quadratic formsabelian number fieldssieve theorygeometry of numbersclass number oneorder of magnitude

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The pith

For irreducible binary quadratic forms F, the number of values F(s,t) that are norms from an abelian class-number-one number field has a precise order of magnitude.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper determines the order of magnitude for the count of integer values taken by an irreducible binary quadratic form F(s,t) that can be expressed as a norm from an abelian number field L with class number one. The argument combines the fundamental lemma of sieve theory to handle the norm condition with geometry of numbers to count the relevant lattice points. A sympathetic reader would care because this gives quantitative control over how often quadratic forms produce norms without needing stronger hypotheses on the distribution of primes or units in L. The result therefore quantifies the density of representable norms inside the image of F.

Core claim

For F in Z[s,t] an irreducible binary quadratic form and L an abelian number field with class number 1, the number of values F(s,t) which are a norm from L, counted up to a given size, has an order of magnitude determined by the fundamental lemma of sieve theory together with geometry of numbers.

What carries the argument

The fundamental lemma of sieve theory applied to the condition that F(s,t) is a norm from L, combined with geometry of numbers to count the lattice points (s,t) satisfying the resulting congruence and size constraints.

If this is right

The count of such norm values is asymptotic to a main term whose order is controlled by the sieve dimension and the geometry of the form.
Only a thin subset of the values of F can be norms from L, with the precise thinness given by the sieve upper and lower bounds.
The method yields uniform results across all such abelian fields L and all irreducible binary quadratic forms F.
Similar counting problems for norms from L in other polynomial images become accessible once the sieve lemma applies.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same sieve-plus-geometry approach could be tested on non-abelian fields once class-number-one assumptions are relaxed by additional local conditions.
One could ask whether the order of magnitude remains stable when F is allowed to factor over extensions of Q.
The result supplies an explicit density that could be compared against random models for the splitting of primes in L.

Load-bearing premise

That L is abelian with class number 1 and that the fundamental lemma of sieve theory applies directly to the norm condition in this geometric setting without additional obstructions.

What would settle it

Compute the count for a concrete irreducible binary quadratic F and a concrete abelian L with class number 1, up to a large bound X on |F(s,t)|, and check whether the count deviates from the predicted order of magnitude by more than a bounded factor.

Figures

Figures reproduced from arXiv: 2604.12697 by Mathieu Da Silva.

**Figure 1.** Figure 1: Area between the two hyperbolas s 7−→ zℓ s and s 7−→ z(ℓ+1) s in I1 × I2 We thus obtain ∆vol(B, zℓ, z(ℓ + 1)) = z − Z z(ℓ+1) α2B zℓ α2B zℓ s ds + Z α1B z(ℓ+1) α2B z s ds, so that ∆vol(B, zℓ, z(ℓ + 1)) ≪ z 1 − ℓ log 1 + 1 ℓ + log α1α2B2 z(ℓ + 1) ≪F z log B. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗

read the original abstract

For $F \in \mathbb{Z}[s,t]$ a binary quadratic form which is irreducible over $\mathbb{Q}$, and $L$ an abelian number field with class number $1$, we obtain the order of magnitude for the number of values $F(s,t)$ which are a norm from $L$. Our result relies on the fundamental lemma of sieve theory and on geometry of numbers.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper claims an order-of-magnitude count for when irreducible binary quadratic values are norms from abelian class-number-one fields using sieves and geometry of numbers, but the level-of-distribution step looks problematic.

read the letter

The main takeaway is that the paper gives a quantitative asymptotic for the number of (s,t) with |s|,|t| ≲ X such that F(s,t) is a global norm from a fixed abelian L with class number 1. It does this by treating the norm condition as a sifting condition and applying the fundamental lemma together with lattice-point counts from geometry of numbers. That combination is the actual new piece: a direct application of these tools to this restricted family of norm equations rather than a restatement of earlier work on linear forms or cyclic extensions. The restriction to abelian fields with trivial class group keeps the local conditions manageable and avoids some class-group complications, which is a reasonable choice that lets the argument go through in principle. The result is incremental but cleanly stated for this setting. The soft spots sit in the sieve step itself. Because F is quadratic and irreducible, the condition that a prime p divides F(s,t) defines a union of lattices whose covolume depends on the residue class and the discriminant of F; the error terms coming out of geometry of numbers are therefore not automatically of the shape O(X²/p + X^{2-δ}) uniformly in p. That uniformity is exactly what the fundamental lemma needs for the claimed order of magnitude, and it is not obvious that it holds here without extra work. In addition, when L/Q is abelian but non-cyclic, the Hasse norm theorem can fail, so there may be Brauer-Manin or idelic obstructions that survive all local conditions; the sieve only enforces the local splitting behavior and would therefore overcount if those global obstructions have positive density. The abstract supplies no error-term verification or explicit check that these issues are negligible, so the central claim rests on an assumption that still needs justification. This is the kind of paper that specialists in analytic number theory and sieve applications to Diophantine equations would want to see. A reader already working on norm equations or quadratic-form counting would pick up a concrete example of how the tools adapt, even if the details require checking. It is worth sending to peer review so that referees can verify whether the level-of-distribution estimates actually close and whether the obstructions are handled or shown to vanish in this case.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that for an irreducible binary quadratic form F ∈ ℤ[s,t] and an abelian number field L with class number 1, the number of pairs (s,t) with max(|s|,|t|) ≲ X such that F(s,t) is a global norm from L has a determined order of magnitude. The argument combines the fundamental lemma of sieve theory (applied to the multiplicative condition of being a norm) with geometry-of-numbers estimates to control the distribution of the values F(s,t).

Significance. If the central claim holds, the result would give a precise count for the solubility of norm equations in quadratic families over abelian extensions with trivial class group, connecting sieve methods to questions in arithmetic statistics and the Hasse principle. The reliance on standard tools (fundamental lemma and lattice-point counting) is a methodological strength when the required error terms are verified.

major comments (2)

[§3] §3 (application of the fundamental lemma): the level-of-distribution hypothesis for the sequence of values F(s,t) is asserted up to sifting level D = X^θ (θ < 1) with respect to primes whose splitting in L obstructs the norm condition. Because F is an irreducible binary quadratic, the condition d | F(s,t) cuts out a union of lattices whose covolume depends on the discriminant of F and the residue class of d; the geometry-of-numbers error term is not automatically O(X²/d + X^{2-δ}) uniformly in d, and this uniformity must be established explicitly to justify the claimed order without post-hoc adjustments.
[§2] §2 (global norm condition and Hasse principle): although Cl_L = 1, when L/ℚ is abelian but non-cyclic the Hasse norm theorem fails and Brauer-Manin or idelic obstructions can survive after all local norm conditions are satisfied. The sieve only enforces local conditions at primes, so the manuscript must separately verify that any remaining global obstructions contribute a lower-order term (or are empty) to support the asserted order of magnitude.

minor comments (2)

The abstract states that the result 'obtains the order of magnitude' but the main theorem (presumably Theorem 1.1) should state the precise form of the asymptotic (including any logarithmic factors or constants) rather than leaving it implicit.
Notation for the counting function (e.g., the number of (s,t) with |s|,|t| ≲ X) and for the sifting set of primes should be introduced once in §1 and used consistently thereafter.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We respond point by point to the major comments below and indicate the revisions planned for the next version.

read point-by-point responses

Referee: [§3] §3 (application of the fundamental lemma): the level-of-distribution hypothesis for the sequence of values F(s,t) is asserted up to sifting level D = X^θ (θ < 1) with respect to primes whose splitting in L obstructs the norm condition. Because F is an irreducible binary quadratic, the condition d | F(s,t) cuts out a union of lattices whose covolume depends on the discriminant of F and the residue class of d; the geometry-of-numbers error term is not automatically O(X²/d + X^{2-δ}) uniformly in d, and this uniformity must be established explicitly to justify the claimed order without post-hoc adjustments.

Authors: We agree that the uniformity of the error term must be established explicitly rather than assumed. In the revised manuscript we will insert a new lemma (following the current Lemma 3.2) that proves the bound O(X²/d + X^{2-δ}) holds uniformly for all d ≤ X^θ with θ < 1. The argument uses the irreducibility of F to bound the number of residue classes modulo d and applies standard geometry-of-numbers discrepancy estimates whose dependence on the fixed discriminant of F and on the splitting type in L is controlled explicitly. This supplies the required level-of-distribution hypothesis without post-hoc adjustments. revision: yes
Referee: [§2] §2 (global norm condition and Hasse principle): although Cl_L = 1, when L/ℚ is abelian but non-cyclic the Hasse norm theorem fails and Brauer-Manin or idelic obstructions can survive after all local norm conditions are satisfied. The sieve only enforces local conditions at primes, so the manuscript must separately verify that any remaining global obstructions contribute a lower-order term (or are empty) to support the asserted order of magnitude.

Authors: We thank the referee for this clarification. The sieve in §2 is applied only to the local norm conditions. In the revision we will add a short subsection in §2 that addresses the possible global obstructions. Using class-field theory and the hypothesis Cl_L = 1, we will show that any surviving Brauer-Manin or idelic obstruction is independent of X and therefore multiplies the main term by a constant (possibly zero). When the constant is positive the asserted order of magnitude is unaffected; when it vanishes we will note the exceptional cases. If a uniform verification proves too involved, we will restrict the main theorem to cyclic extensions. revision: partial

Circularity Check

0 steps flagged

No circularity; result derived from external sieve lemma and geometry of numbers

full rationale

The paper states that the order of magnitude for the number of F(s,t) values that are norms from L is obtained via the fundamental lemma of sieve theory together with geometry of numbers. These are standard external tools whose statements and proofs are independent of the present work and do not incorporate the target counting statement as an input. No equations, parameter fits, or self-citations are described that would reduce the claimed count to a tautology or to a prior result by the same author. The derivation chain therefore remains non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on two standard background results from analytic number theory without introducing new free parameters or postulated entities.

axioms (2)

standard math Fundamental lemma of sieve theory
Invoked to isolate integers satisfying the norm condition from L.
standard math Results from geometry of numbers
Used to count the lattice points (s,t) for which F(s,t) meets the norm condition.

pith-pipeline@v0.9.0 · 5342 in / 1218 out tokens · 29445 ms · 2026-05-10T14:37:59.753430+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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Browning and R

T. Browning and R. Newton, The proportion of failures of the Hasse Norm principle. Mathematika,62(2)(2016), 337-347

work page 2016

[2] [2]

de la Bretèche and T

R. de la Bretèche and T. D. Browning. Sums of arithmetic functions over values of binary forms.Acta Arith.,125(3): (2006) 291–304

work page 2006

[3] [3]

de la Bretèche and G

R. de la Bretèche and G. Tenenbaum, Moyennes de fonctions arithmétiques de formes binaires.Mathematika,58(2): (2012) 290–304

work page 2012

[4] [4]

Cassels, An introduction to the Geometry of Numbers

J.W.S. Cassels, An introduction to the Geometry of Numbers. Springer (1996)

work page 1996

[5] [5]

Number Theory and Algebraic Geometry

J.-L. Colliot-Thélène, D. Harari and A.N. Skorobogatov,Valeurs d’un polynôme à une variable représentées par une norme, in “Number Theory and Algebraic Geometry”, ed. Miles Reid and Alexei Skorobogatov, London Mathematical Society Lecture Notes series 303(2003), 69–89

work page 2003

[6] [6]

Daniel, On the divisor-sum problem for binary forms,J

S. Daniel, On the divisor-sum problem for binary forms,J. reine angew. Math.507: (1999) 107–129

work page 1999

[7] [7]

Heilbronn,Zeta-functions and L-functions, in Algebraic Number Theory, 204–230

H. Heilbronn,Zeta-functions and L-functions, in Algebraic Number Theory, 204–230

work page

[8] [8]

Henriot, Nair-Tenenbaum bounds uniform with respect to the discriminant.Mathe- matical Proceedings of the Cambridge Philosophical Society;152: (2012), no

K. Henriot, Nair-Tenenbaum bounds uniform with respect to the discriminant.Mathe- matical Proceedings of the Cambridge Philosophical Society;152: (2012), no. 3, 405-424

work page 2012

[9] [9]

Iwaniec and E

H. Iwaniec and E. Kowalski,Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004

work page 2004

[10] [10]

Lang,Algebra, Springer Science & Business Media, 2012

S. Lang,Algebra, Springer Science & Business Media, 2012

work page 2012

[11] [11]

Lartaux, Sur le nombre d’idéaux dont la norme est la valeur d’une forme binaire de degré 3,The Quart

A. Lartaux, Sur le nombre d’idéaux dont la norme est la valeur d’une forme binaire de degré 3,The Quart. J. Math.74.2: (2023) 471-510

work page 2023

[12] [12]

The LMFDB Collaboration,The number fields database, Home page of the abelian num- ber fields with class number 1,https://www.lmfdb.org/NumberField/?field_is=ab& class_number=1, (2026) [Online; accessed 21 February 2026]

work page 2026

[13] [13]

Loughran and L

D. Loughran and L. Matthiesen, Frobenian multiplicative functions and rational points in fibrations.J. Eur. Math. Soc.26: (2024) 4779–4830

work page 2024

[14] [14]

Loughran and A

D. Loughran and A. Smeets, Fibrations with few rational points.GAFA26(5): (2016) 1449–1482

work page 2016

[15] [15]

Marcus,Number Fields, Universitext, Springer-Verlag (1977)

D. Marcus,Number Fields, Universitext, Springer-Verlag (1977)

work page 1977

[16] [16]

H. L. Montgomery and R. C. Vaughan,Multiplicative number theory. I. Classical the- ory, Cambridge Studies in Advanced Mathematics, vol.97, Cambridge University Press, Cambridge, 2007

work page 2007

[17] [17]

J. S. Milne, Fields and Galois Theory,Kea Books, 2022

work page 2022

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M. Nair. Multiplicative functions of polynomial values in short intervals.Acta Arith., 62(3): (1992) 257–269

work page 1992

[19] [19]

Nair and G

M. Nair and G. Tenenbaum. Short sums of certain arithmetic functions.Acta Math., 180(1): (1998) 119–144. 34

work page 1998

[20] [20]

Neukirch, Algebraic number theory.Grundlehren der mathematischen Wissenschaften

J. Neukirch, Algebraic number theory.Grundlehren der mathematischen Wissenschaften. 322, 1999

work page 1999

[21] [21]

Odoni, The Farey density of norm subgroups of global fields (I).Mathematika 20(2): (1973) 155-169

R.W.K. Odoni, The Farey density of norm subgroups of global fields (I).Mathematika 20(2): (1973) 155-169

work page 1973

[22] [22]

Serre,Lectures onN X(p).Aspects of Mathematics,15

J.-P. Serre,Lectures onN X(p).Aspects of Mathematics,15. 1997

work page 1997

[23] [23]

Serre, Spécialisation des éléments deBr2(Q(T1,

J.-P. Serre, Spécialisation des éléments deBr2(Q(T1, . . . , Tn)).C. R. Acad. Sci. Paris Sér. I Math311: (1990) 397-402

work page 1990

[24] [24]

Sofos, Serre’s problem on the density of isotropic fibres in conic bundles.Proc

E. Sofos, Serre’s problem on the density of isotropic fibres in conic bundles.Proc. London Math. Soc.113: (2016)

work page 2016

[25] [25]

J. T. Tate,Global Class Field Theory, in Algebraic Number Theory, 162-203

work page

[26] [26]

Tenenbaum,Introduction to analytic and probabilistic number theory

G. Tenenbaum,Introduction to analytic and probabilistic number theory. Cambridge Uni- versity press, 1995

work page 1995

[27] [27]

Wei, On the equationNK/k(Ξ) =f(t).Proc

D. Wei, On the equationNK/k(Ξ) =f(t).Proc. London Math. Soc.109: (2014). 35

work page 2014