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Heights and morphisms in number fields

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Pith's one-line read For any nonconstant morphism between projective spaces over a number field K, the number of K-rational points with H(f(P)) ≤ X is c_K(f) X^{n(m+1)/d} plus an explicit power-saving error term whose constants are expressed through geometric…

desk verdict Main theorem is new and sound; the reported flaw in Proposition 2.11 does not survive reading the product over all places. read the letter →

arxiv 2411.13522 v1 pith:IGKRQVGY submitted 2024-11-20 math.NT math.DS

Matt Olechnowicz This is my paper · ORCID
classification math.NTmath.DS MSC 11G5014G0537P30
keywords heightsnumberfieldsmorphismsofprojectivespacescountingrationalpointsexpliciterrortermresultantidealcanonicalheightexcessdivisor
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 a counting formula with an explicit error term for the number of K-rational points P in projective m-space whose pullback height H(f(P)) under a morphism f of degree d to projective M-space over a number field K is at most X. The main term is $c_K(f) X^{n(m+1)/d}$, and the error term has order $X^{(n(m+1)-1)/d}$ up to logarithmic factors, with the constants depending on f and K through explicitly defined geometric data: the height of f, a bound on the resultant ideal, Lipschitz constants for the boundary of the local archimedean fundamental domains, and nonarchimedean local volumes. This is the pullback-height analogue of the classical asymptotic counting formula for points of bounded height in projective space, and it comes with a uniformly stated error term. The paper then derives counting formulae for the image f(P^m(K)) and for points of bounded canonical height in arithmetic dynamics.

What carries the argument

The load-bearing object is the excess divisor $\ell_f(x) = \langle F(x)\rangle / (\langle x\rangle^d \langle F\rangle)$, a fractional ideal that records the extra valuations picked up by $F(x)$ beyond the generic $\langle x\rangle^d \langle F\rangle$. The paper shows that $\ell_f$ is everywhere integral, divides the resultant ideal of f, and is periodic: it factors through the reduction map from $K^{m+1}\setminus\{0\}$ to the finite projective space $P^m(O_K/\operatorname{Res} f)$. Fibering the counting problem over ideal classes and over the finitely many possible excess divisors, the count of points with prescribed $\ell_f$ becomes a union of cosets of a lattice, and a generalized Chinese remainder theorem plus an explicit lattice-point counting principle with a Lipschitz boundary estimate for the expanding domain $D_{F,K}(T)$ yields the main term and the power-saving error. The nonarchimedean local constants $c_{K,v}(f)$ are computed as weighted sums of the local densities $\delta_{f,v}(i)$, and are strictly larger than the v-adic volume of $D_{f,v}$ in general.

What would settle it

Take the powering map $(x_0:\dots:x_m) \mapsto (x_0^d:\dots:x_m^d)$ over $\mathbb{Q}$, for which the formula is explicit ($c_{K,v}(f)$ known, $d=1$ recovers the classical count), and compute the exact difference $N_{f^*H,P^m(\mathbb{Q})}(X) - c_{\mathbb{Q}}(f)X^{n(m+1)/d}$ for a range of $X$; if the difference does not decay like $X^{(n(m+1)-1)/d}$ (up to log $X$), the error bound in Theorem 1.3 is wrong. For a sharper test, compute with a quadratic map over $\mathbb{Q}$ of the form given in Example 6.5, where the nonarchimedean local densities are known exactly, and check the predicted constant $c_{K,0}(f)$ against the formula $\sum_l \operatorname{Nm}(l)^{(m+1)/d} \delta_f(l)$.

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Extended reading notes

Core claim

The central discovery is that the counting function for the pullback height satisfies $N_{f^*H,P^m(K)}(X) = c_K(f) X^{n(m+1)/d} + E(X)$, with a fully explicit error bound of order $X^{(n(m+1)-1)/d}(1 + \log_+ X^{1/d})$ whose implicit constant depends on m and K and on the morphism through the finite list of invariants: $N_f$ and $L_f$, describing how the boundary of each archimedean fundamental domain $D_{f,v} = \{|F(z)|_v \le |F|_v\}$ can be covered by Lipschitz images of the unit cube; the constants $C^0_f$ and $C^\infty_f$ coming from the local comparison inequalities between $|F(z)|_v$ and $|z|_v^d$; the nonarchimedean local constants $c_{K,0}(f)$; and the height $H(f)$. The main constant factors as $c_K(f) = c_K(m) c_{K,\infty}(f) c_{K,0}(f)/H(f)^{n(m+1)/d}$, where $c_K(m)$ is the classical constant counting points of bounded height in projective space. The proof achieves this by fibering the count over ideal classes and over finitely many 'excess divisors' $\ell_f(x) = \langle F(x)\rangle / (\langle x\rangle^d \langle F\rangle)$, proving these divisors are bounded by the resultant ideal, are periodic modulo reduction to a finite projective ring, and have well-defined local densities, and then applying an explicit lattice-point counting principle to the resulting homogeneously expanding domains.

Load-bearing premise

The explicit power-saving error term depends on applying a lattice-point counting lemma whose stated hypothesis (a certain comparison constant at each archimedean place must not exceed 1) the paper declares superfluous without reproducing the proof of that relaxation; if the relaxation fails, the error bound as stated may not hold for arbitrary lifts of f.

Editorial extensions

If this is right

  • The number of K-rational points in the image $f(P^m(K))$ with Weil height at most $X$ is asymptotic to $(c_K(f)/\gamma) X^{n(m+1)/d}$ with power-saving error, where $\gamma$ is the number of K-rational mapping symmetries of f; the error has the same shape as in Theorem 1.3 up to the thin-set error coming from points with non-generic fibre size.
  • For an endomorphism f of degree $d \ge 2$, the number of points whose canonical height satisfies $\hat{h}_f(P) \le X$ is asymptotic to a constant times $X^{n(m+1)}$, where the constant is a limit of the constants $c_K(f^i \circ g)$ and admits an explicit formula in terms of the dynamical Green's functions of f.
  • The canonical-height constant is invariant under iterating f and under conjugation by automorphisms defined over K, but changes under conjugation defined over extensions of K, so the canonical height genuinely redistributes points compared to the Weil height.
  • The nonarchimedean local factors $c_{K,v}(f)$ are not the 'obvious' v-adic volumes: strict inequality $c_{K,v}(f) > \mu_v(D_{f,v})$ holds whenever some excess valuation is not divisible by d, as shown by explicit examples.
  • For the powering map and for maps with good reduction at all but finitely many places, the constants simplify to known values, recovering the classical projective-space counting formula when $d=1$.

Reading between the lines

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

  • The finiteness and periodicity of the excess divisor map means the constants $c_K(f)$ are computable in principle from finite data: residues modulo the resultant ideal together with the volume of the archimedean fundamental domains, so one could implement the formula for explicit morphisms and compare against direct point counts.
  • The method appears to transfer to counting points of bounded height in images of morphisms between more general varieties equipped with an equivariant height, since the only geometric input is the finite periodic structure of the excess divisors and the Lipschitz class of the archimedean level sets.
  • In the dynamical setting, the limit constant $\hat{c}_{K,f}(g)$ is a measure of how the canonical height's unit ball differs from the Weil height's unit ball; the paper's examples show this ratio is not 1 in general, and one could test numerically whether $\hat{c}_{K,f}(\mathrm{id})$ converges to the naive constant as the degree of f grows.
  • The strict inequality between the nonarchimedean local factor and the v-adic volume suggests that 'volume' of the pullback of the unit ball under a morphism is not the right invariant unless the excess valuations are multiples of the degree, which may be relevant when defining Tamagawa measures for pullback line bundles in more general height-counting conjectures.
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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

1 major / 4 minor

Summary. The paper studies, for a nonconstant morphism f:P^m→P^M defined over a number field K, the counting function N_{f*H,P^m(K)}(X)=#{P∈P^m(K):H(f(P))≤X}. Theorem 1.3 asserts an asymptotic with an explicit main term c_K(f)X^{n(m+1)/d} and a power-saving error term whose constants depend on f and K through explicit invariants, following Schanuel's strategy with a new 'excess divisor' machinery. The paper then derives Theorem 2.1, counting points in f(P^m(K)) with respect to H, and Theorem 2.7, a 'dynamical Schanuel' asymptotic for the Call-Silverman canonical height; Proposition 2.11 gives an explicit formula for the limit constant of that asymptotic.

Significance. If Theorem 1.3 is correct, it is a substantial quantitative contribution to the counting of rational points with respect to pullback heights, going beyond previously known Tamagawa-measure asymptotics by providing explicit error terms and explicit dependence on the morphism. The excess-divisor construction, the resultant ideal, and the local-density formalism are potentially useful new tools. Theorem 2.1's thin-set application is a natural and valuable consequence. The paper is carefully structured and contains detailed proofs of the main counting theorem. However, the explicit canonical-height formula in Proposition 2.11 is not supported by the given proof, and the advertised 'formula' for the canonical-height counting function must be corrected or substantially weakened.

major comments (1)
  1. [Section 9.4, Lemma 9.15] The proof of Lemma 9.15 asserts, after equation (83), that Widmer's hypothesis c_v≤1 is superfluous, with the justification 'inspecting his proof ... reveals that this requirement is superfluous'. Since Theorem 1.3's power-saving error term depends directly on applying [39, Lemma 7.1] with possibly c_v>1, this is a load-bearing step and should be justified explicitly, either by reproducing the argument or by quoting a version of Widmer's lemma that allows c_v>1. If the assertion is false, the stated error bound in Theorem 1.3 does not follow as written for arbitrary lifts of f.
minor comments (4)
  1. [Section 2.2, Proposition 2.11] The notation |\hat F∘G|_v := lim |F^i∘G|_v^{1/d^i} should be introduced before the 'In fact' paragraph, and the relation between this quantity and the archimedean threshold in the first display should be stated explicitly, since the two displays currently appear contradictory.
  2. [Section 7, Lemma 7.5] The function J_{OK,s}(R) is used before it is defined; it should be defined explicitly, for instance as the Jordan totient analogue J_{K,s}(R)=Nm R^s∏_{p|R}(1-Nm p^{-s}).
  3. [Section 6, Corollary 6.4] In part (iii), the equality condition 'δ(i)=0 for all i not divisible by d' should be stated together with the observation that this condition is not vacuous, as Example 6.5 demonstrates; the current wording could be misread as a definition.
  4. [Section 2, Example 2.14] In the formula for G_{S,2}(x,y), the Iverson bracket [|x|_2=|y|_2] is used without definition; a brief explanation of the notation and of why the formula holds would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main constant and error term are derived from first principles via explicit local integrals and external lattice-point counting results, not from the counting function being predicted.

full rationale

The central result (Theorem 1.3) is self-contained and non-circular. The main constant c_K(f) is defined by explicit products of archimedean volumes and nonarchimedean integrals over local fundamental domains, and the proof obtains the asymptotic by following Schanuel's framework with a homogeneous expanding domain, Möbius inversion over excess divisors, and Widmer's/Masser–Vaaler's lattice-point counting principles. No parameter is fitted to N_{f*H,P^m(K)}(X); the constant is computed independently and then shown by geometry of numbers to be the leading coefficient. The error term depends on explicit geometric invariants (N_f, L_f, C^0_f, C^∞_f) whose existence is proved from the Nullstellensatz and Lipschitz parametrizability, not assumed from the target estimate. The one self-citation, to the author's earlier work [27, Remark 8.6], is cited only as a strategy sketch and is not load-bearing; the proof itself invokes Schanuel, Masser–Vaaler, and Widmer as external, independently established results. The canonical-height application in Theorem 2.7 defines its constant as a limit of the previously derived constants c_K(f^i∘g) and proves the asymptotic by a standard squeezing argument; this does not assume the target conclusion. The skeptical concern about Proposition 2.11 omitting a coefficient-ideal norm factor is a potential correctness issue in the displayed formula, not a circularity: it does not make the derivation equivalent to its inputs. No step reduces, by construction or by self-citation, to the quantity being predicted. Accordingly the circularity score is 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 1 invented entities

The central term is defined by explicit local integrals with no fitted constants. The main structural input is the boundedness and periodicity of the excess divisor, which is proved. The one unproved technical assertion is the applicability of Widmer's lemma without the c_v≤1 hypothesis.

assumptions (3)
  • ad hoc to paper Widmer's counting principle and Lipschitz lemma apply to the domain D_{F,K}(1) even when the lower-bound constants c_v in (83) exceed 1.
    Stated around Lemma 9.15 and used in the proof of Theorem 1.3; the author asserts this without reproducing the proof from [39, Lemma 7.1].
  • standard math Schanuel's counting framework and the classical geometry-of-numbers results of Masser-Vaaler and Widmer are correct as black boxes.
    Relied on throughout Sections 8-10 to count lattice points in homogeneously expanding domains.
  • domain assumption The excess divisor ℓ_f(x) is bounded by the resultant ideal and is periodic modulo Res f.
    Proved in Section 5 via the new resultant ideal, but it is a key structural property needed for the Möbius inversion step in the proof of Theorem 1.3.
invented entities (1)
  • Excess divisor ℓ_f(x) independent evidence
    purpose: Tracks the nonarchimedean discrepancy v(F(x))-dv(x)-v(F) and lets the proof sum over possible divisors introduced by bad reduction.
    Bounded by Res f and periodic modulo Res f (Prop 5.11 and Cor 5.13), so it is not a free parameter or an unfalsifiable postulate.

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Cite this review

Pith. "Pith review of Heights and morphisms in number fields." pith.science (2026). https://pith.science/paper/IGKRQVGY

@misc{pith2026241113522,
  author       = {Pith},
  title        = {Pith review of: Heights and morphisms in number fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IGKRQVGY}},
  note         = {Machine review of arXiv:2411.13522}
}
abstract

We give a formula with explicit error term for the number of $K$-rational points $P$ satisfying $H(f(P)) \le X$ as $X \to \infty$, where $f$ is a nonconstant morphism between projective spaces defined over a number field $K$ and $H$ is the absolute multiplicative Weil height. This yields formulae for the counting functions of $f(\mathbb{P}^m(K))$ with respect to the Weil height as well as of $\mathbb{P}^m(K)$ with respect to the Call-Silverman canonical height.

Figures

Figures reproduced from arXiv: 2411.13522 by the authors.

Figure 1
Figure 1. Logical flow of the proof of Proposition 3.6. Proof. Suppose (i) holds. Then V (F)(K) ⊆ {0 m+1}, so by the semirational Nullstellensatz [6, Theorem 11.9] we have rad ⟨F0, . . . , FM⟩ = I(V (F)(K)) ⊇ I({0 m+1}) = ⟨X0, . . . , Xm⟩. Thus there exist integers ei such that X ei i ∈ ⟨F0, . . . , FM⟩ for all i. Letting e := maxi ei proves (ii), while letting D := 1 + P i (ei − 1) proves (iv).3 Of course, (iv) implies (ii) … view at source ↗
Figure 2
Figure 2. Diagram of the auxiliary maps involved in the proof of Theorem 1.3 [PITH_FULL_IMAGE:figures/full_fig_p060_2.png] view at source ↗

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