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Lower bounds on heights of odd degree points of hyperelliptic curves

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Pith's one-line read For 100% of hyperelliptic curves, every odd-degree point of degree at most 2g−1 has height at least (1+1/(2g+2)−ε) log Ht(f), and the same bound holds for degree-(2g−1) divisors.

desk verdict A density-1 height lower bound for odd-degree points on even-degree hyperelliptic curves, with a long but coherent proof; one omitted case in Lemma 5.6 should be expanded before publication. read the letter →

arxiv 2507.08652 v1 pith:GIWSPZQN submitted 2025-07-11 math.NT math.AG

Jef Laga , Jack A. Thorne This is my paper · ORCID
classification math.NTmath.AG MSC 11G3011G5014H10
keywords hyperellipticcurvesWeilheightreductiontheorybinaryformsarithmeticinvariantequidistributiondivisorsodd-degreepoints
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, in a density-1 family of hyperelliptic curves, odd-degree points cannot be too small: their heights must be at least a positive fraction of the logarithm of the curve's defining binary form. Concretely, for 100% of integral binary forms of degree 2g+2 with nonzero discriminant, ordered by height, every effective divisor of degree 2g−1 has logarithmic height at least (1+1/(2g+2)−ε) log Ht(f). For genus 1, this says that every rational point on 100% of curves z²=f(x,y) with f a quartic form has height at least (5/4−ε) log Ht(f). The proof develops a new reduction theory for the action of SL_n on pairs of symmetric n×n matrices and shows that the associated reduction covariant equidistributes over integral orbits.

What carries the argument

The load-bearing object is the reduction covariant R: SL_n(Z)\V(Z)_{Δ≠0} → SL_n(Z)\X, where X=SL_n(R)/SO_n(R) is the symmetric space of inner products on R^n up to scaling. For a pair (A,B) of symmetric matrices one forms the pencil Ax−By, simultaneously diagonalizes it over C, and builds a positive definite inner product H_{A,B} by taking the maximum of the absolute diagonal values; this construction is equivariant and passes to a map on integral orbits. Odd-degree divisors on X_f produce integral orbits for this representation through the correspondence between orbits and ideal classes in the ring attached to f, using coherent Grothendieck duality; the divisor height controls the norm of a primitive vector in Z^n with respect to H_{A,B}. The proof then shows that these reduction covariants equidistribute over all integral orbits of bounded height, while lattices admitting a very short vector form a set of small measure; combining these facts gives the density-1 theorem.

What would settle it

Take n=4 and m=1, pick two elements of the same fundamental set L(1,τ) using the block forms (5.1)–(5.2), and compute their reduction covariants; if the classes in SL_4(Z)\X differ, Lemma 5.6 is false and the equidistribution theorem fails.

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Extended reading notes

Core claim

The central claim is Theorem 6.5: for every ε>0, the set S_ε of binary forms f∈Z[x,y] of degree 2g+2 and nonzero discriminant such that every effective divisor D on X_f: z²=f(x,y) of odd degree ≤2g−1 satisfies h(D) ≥ (1+1/(2g+2)−ε) log Ht(f) has density 1 when forms are ordered by height. Equivalently, asymptotically all forms belong to S_ε, so small-height divisors are rare: the count of forms admitting a degree-(2g−1) divisor whose height violates the bound is o($X^{{2g+3}}$). The same statement transfers to algebraic points: for a Galois orbit P of odd degree m≤2g−1, the quantity m·h(π(P)) obeys the same lower bound.

Load-bearing premise

The proof relies on the claim that the normalization procedure gives the same output for every representative in certain standard slices of the matrix space, but the paper checks this only for the slices where all blocks are real and says the mixed complex case is similar without giving details.

Editorial extensions

If this is right

  • For genus 1, the result gives that on 100% of integral binary quartics, every rational point P satisfies h((x:y)) ≥ (5/4−ε) log Ht(f).
  • For every genus g≥1, every effective divisor of degree 2g−1 on 100% of hyperelliptic curves satisfies h(D) ≥ (1+1/(2g+2)−ε) log Ht(f).
  • For every algebraic point of odd degree m≤2g−1 on 100% of such curves, m·h(π(P)) obeys the same lower bound.
  • The equidistribution theorem covers all integral orbits of nonzero discriminant, not only irreducible orbits, so the density-1 conclusion follows without a separate reducible-orbit argument.
  • The reduction covariant provides a uniform method for reducing pairs of symmetric matrices, which the paper expects to be useful for computational arithmetic invariant theory.

Reading between the lines

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

  • The proof of Lemma 5.6 explicitly leaves the case m<n/2 unverified: it checks constancy of the reduction covariant on the fundamental sets only when all blocks are real and says the mixed complex case is similar. The equidistribution theorem and therefore the density-1 height bound depend on this omitted case, so a reader checking the proof should verify it before treating the theorem as complete.
  • The construction of integral orbits from divisors via coherent Grothendieck duality is likely reusable in other arithmetic invariant theory settings, including the odd hyperelliptic family treated in the authors' earlier work.
  • A direct numerical check of the reduction covariant on the fundamental sets L(m,τ) for small n and m<n/2 would provide a concrete test of the omitted case.
  • The height bound concerns the naive divisor height h(D); it does not by itself yield statements about canonical heights or about proportions of curves with points of larger odd degree.
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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

0 major / 6 minor

Summary. This paper develops a reduction theory for the representation of SL_n on pairs of symmetric n×n matrices, constructs a reduction covariant R, proves its equidistribution over integral orbits, and uses this to show that in a density-1 family of integral binary forms of degree 2g+2, every Q-rational effective divisor of odd degree at most 2g−1 on the associated hyperelliptic curve has logarithmic Weil height at least (1+1/(2g+2)−ε) log Ht(f). For g=1 this gives a rational-point height lower bound of (5/4−ε) log Ht(f) for 100% of quartic forms. The proof combines a construction of integral orbits from divisors via coherent duality (Theorems 4.1 and 4.5), a norm computation (Proposition 4.7), an equidistribution theorem for the reduction covariant (Theorem 5.9), and a discriminant-based density-1 family (Proposition 6.1).

Significance. If correct, the main theorems give the first density-1 height lower bounds for odd-degree points on the full family of hyperelliptic curves, sharp up to ε, with the natural exponent 1+1/(2g+2). The method introduces a reduction covariant for a representation outside Vinberg theory and proves an equidistribution statement of independent interest, and the construction of integral orbits via Grothendieck duality is a technical innovation. The argument is internally consistent; the one flagged omission in Lemma 5.6 (the case m<n/2) is filled by a direct matrix calculation, so I do not regard it as a threat to the main claim. No free parameters are fitted to data, and the density-1 family is a standard sieve input.

minor comments (6)
  1. [§5.1, Lemma 5.6] The proof of Lemma 5.6 handles only the case m = n/2 and states that m < n/2 is similar without details. Because this constancy is used to normalize R(L(m,τ)) = H0 and consequently in Proposition 5.10 and Theorem 5.9, the omitted case should be written out; it follows by simultaneously diagonalizing each complex 2×2 block ψ(λ), ψ(μ) with the fixed basis (1,i)^t and (1,-i)^t, which yields the standard Hermitian form after the normalization max(|λ|,|μ|)=1.
  2. [§6.1, proof of Theorem 6.4] The statement 'D does not intersect the irreducible Weierstrass locus S_f' is used to apply the orbit construction of §4, but the reason is not given: because f is irreducible over Q, any Q-rational effective divisor supported on S_f has degree at least 2g+2, so a degree 2g−1 divisor cannot meet S_f. Please state this explicitly.
  3. [§6.2, proof of Theorem 6.4] In the final sentence of the proof, 'n log(w,w)_{HA,B} − det H_{A,B}' should read 'n log(w,w)_{HA,B} − log det H_{A,B}'.
  4. [§6.3, Theorem 6.5] The statement of Theorem 6.5 (and similarly the abstract and Theorem 6.4) should specify that the divisors are effective Q-rational divisors, since the height h(D) is defined only for such divisors.
  5. [§5.1, V(0,τ)] In the definition of V(0,1+) and V(0,1−), the text writes 'Let V(0,τ) denote the set...' but τ is not a free parameter there; the two cases should be named explicitly.
  6. [§5.2, Lemma 5.8] The formula '#F(X) = 2n+1X^{n+1} + O(X^n)' appears with a formatting error; it should be 2^{n+1}X^{n+1}.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the density-1 height bound is derived from an independently constructed reduction covariant and equidistribution theorem, with no fitted inputs or self-citation chain forcing the conclusion.

full rationale

The paper's derivation chain is self-contained and does not reduce to its inputs. The main theorem (Theorem 6.5) combines: (i) an explicit construction of integral orbits from effective odd-degree divisors (Theorem 4.5, Proposition 4.6); (ii) a direct norm computation for the reduction covariant (Proposition 4.7 with Lemma 3.11); (iii) an equidistribution theorem for the reduction covariant over all integral orbits of nonzero discriminant (Theorem 5.9), proved by adapting Bhargava's geometry-of-numbers method; and (iv) a density-1 family F_delta(X) defined by irreducibility and a mild discriminant lower bound (Proposition 6.1). No parameter is fitted to the target family or to the divisor in a way that forces the bound: delta is chosen arbitrarily small before X tends to infinity, epsilon is arbitrary, and the constant 1 + 1/(2g+2) emerges from the asymptotic count #F(X) ~ c X^{n+1} combined with the volume of the small-vector neighbourhood in Corollary 5.12. The reduction covariant R is defined constructively in Section 3.4 (Construction 3.9) and its key constancy on Bhargava fundamental sets (Lemma 5.6) is asserted with the m < n/2 case omitted; this is an exposition gap rather than a circular step, since the complex-block case follows by direct simultaneous diagonalization. The self-citations [19], [29], [30] appear for comparison or for elementary semialgebraic lemmas (Lemmas 5.1, 5.3) and are not load-bearing for the main theorem; the central equidistribution argument is proved in the paper, following Bhargava [5]. No known result is merely renamed, and no uniqueness theorem from the authors' prior work is invoked to forbid alternative constructions. The paper's claim is therefore a genuine derivation rather than a circular one.

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

The paper introduces no free parameters and no new entities. All inputs are standard theorems from the literature or the paper's own proved lemmas. The single item of concern is Lemma 5.6, which is ad hoc to this paper and has an omitted case; it functions as an unproved premise for the equidistribution theorem. The remaining axioms are standard background mathematics (duality, geometry of numbers, sieve inputs), cited with locations.

assumptions (6)
  • domain assumption Bhargava's counting and geometry-of-numbers for SL_n-orbits on V ([5, Section 4]), including the partition of V(R) into fundamental sets L(m,tau) and the asymptotic N(V(Z);X) = cX^{n+1} + o(X^{n+1}).
    Used in Section 5.2; Theorem 5.9 is an adaptation of these results to the reduction covariant, and the constants c_{m,tau} in (5.5) are taken from Bhargava.
  • ad hoc to paper Lemma 5.6, including the omitted case m < n/2: the reduction covariant is constant on each fundamental set L(m,tau).
    The proof covers only m = n/2 and omits details for m < n/2; constancy is needed to set R(L(m,tau)) = H0, which is essential to Proposition 5.10 and Theorem 5.9.
  • standard math Grothendieck duality and Hartshorne's theory of generalized divisors and reflexive sheaves ([13], [17]) used to construct O_X(D) and prove H^1(S_f, M) = 0.
    Sections 2 and 4; underpins the integral orbit construction (Theorem 4.5) and Proposition 4.6, including the hyperelliptic ribbon argument in Lemma 4.4.
  • domain assumption Wood's parametrization of ideal classes for rank-n rings by SL_n(Z)-orbits on pairs of symmetric matrices ([32]), with the orbit geometry of [8] as corrected by Swaminathan [27, Appendix 4A].
    Sections 2 and 3; the paper notes [8, Theorem 15] had gaps fixed in [27] and builds its own construction on Wood's bijection.
  • standard math Standard density-1 inputs: Hilbert irreducibility and the discriminant bound of [10, Lemma 6.1] give #F_delta(X) = (2X)^{n+1} + o(X^{n+1}).
    Proposition 6.1; guarantees the height machinery applies to 100% of forms, and the condition |disc(f)| >= X^{2n-2-delta} feeds into Proposition 6.2.
  • standard math Soundararajan's root-separation bound [25, Theorem 1] bounding the leading coefficient c of f in terms of its coefficients.
    Proposition 6.2; yields log|c| <= 0.5 log(n(n+1)) + log X, which produces the (n+1)log X term in Theorem 6.4 and hence the constant 1 + 1/(2g+2).

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Pith. "Pith review of Lower bounds on heights of odd degree points of hyperelliptic curves." pith.science (2026). https://pith.science/paper/GIWSPZQN

@misc{pith2026250708652,
  author       = {Pith},
  title        = {Pith review of: Lower bounds on heights of odd degree points of hyperelliptic curves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GIWSPZQN}},
  note         = {Machine review of arXiv:2507.08652}
}
abstract

We develop a reduction theory for the representation of $\mathrm{SL}_n$ on pairs of symmetric $n\times n$ matrices. We apply this theory to the pencils of quadrics arising from divisors on hyperelliptic curves. We use these results to show that, in a density $1$ family, an odd degree point $P$ of degree at most $2g-1$ on the hyperelliptic curve $z^2 = f_0x^{2g+2} + f_1 x^{2g+1} y + \cdots + f_{2g+2}y^{2g+2}$ cannot have small Weil height.

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