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Invariants recovering the reduction type of a hyperelliptic curve

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Pith's one-line read Valuations of a finite invariant list recover hyperelliptic reduction type.

desk verdict Genuinely new and mostly solid: first uniform Tate-analogue for reduction types of hyperelliptic curves of all genera, plus a good negative theorem for the non-semistable case; the tree-reconstruction lemma (Prop. 7.6-7.8) is where a referee should look hardest. read the letter →

arxiv 2502.08487 v3 pith:JNHPB7CZ submitted 2025-02-12 math.NT

Lilybelle Cowland Kellock , Elisa Lorenzo This is my paper · ORCID
classification math.NT MSC 11G2014D1014G2014H4514Q05
keywords hyperellipticcurvesreductiontypeabsoluteinvariantsclusterpicturesstablemodeltreevaluationtheorylocalfieldsdualgraph
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 claims that the reduction type of a hyperelliptic curve of genus $g \ge 2$ over a local field of odd residue characteristic is determined by the valuations of an explicit finite list of absolute invariants, one list for each genus. If the curve is semistable, these valuations recover the dual graph of the special fibre of the minimal regular model; if not, they recover the dual graph of the potential stable model. This generalises the elliptic-curve fact that $\mathrm{ord}(j_E)$ and $\mathrm{ord}(\Delta_E)$ encode the Kodaira type. The proof passes through a new model-independent object, the stable model tree, and gives an explicit inductive algorithm that reads the tree off from invariant valuations.

What carries the argument

The stable model tree $T_C$: a weighted tree whose vertices are the clusters of a Weierstrass equation's roots, with edge lengths the relative depths, and with the top-cluster vertex removed when the root set splits into two. It is model independent and in bijection with the BY tree, so recovering $T_C$ from invariants recovers the dual graph. The invariants themselves are built from the cross-ratio building block $(ij,kl)=\frac{(X_i-X_k)(X_i-X_l)(X_j-X_k)(X_j-X_l)}{(X_i-X_j)^2(X_k-X_l)^2}$, which has valuation $-2\delta(v,w)$ exactly when the paths between $s_i,s_j$ and $s_k,s_l$ are disjoint with separating distance $\delta(v,w)$; symmetrising products of these blocks yields absolute invariants whose valuations encode the ordered distances $\delta_n(C)$.

What would settle it

For a large random sample of semistable genus 2 curves over $\mathbb{Q}_p$, run the algorithm of Theorem 8.1 to obtain a candidate dual graph from the 24 invariant valuations, and compare each output with the dual graph computed by an independent regular-model routine; any disagreement would refute the recovery claim for the semistable case.

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Extended reading notes

Core claim

The central discovery is that the cluster picture of a hyperelliptic curve, which records the $\pi$-adic distances between roots of a Weierstrass equation, can be repackaged as a weighted tree — the stable model tree $T_C$ — that is independent of the chosen model, and that the valuations of carefully symmetrised cross-ratio expressions recover this tree. For each genus $g$ there is a finite set $\mathcal{T}_g$ of possible stable model trees with orderings on vertex distances; for each $(T,I) \in \mathcal{T}_g$ and each $n \le n_{T,I}$ the paper defines an absolute invariant $\mathrm{Inv}_{T,I,n}$ whose valuation, evaluated on $C$, is $-2 \sum_{m=1}^n K_m(C)(n+1-m)\delta_m(C)$, where the $\delta_m(C)$ are the ordered distances in $T_C$ and $K_m(C)$ counts the pairs of leaf-pairs realising $\delta_m$. By comparing these valuations via an averaging function $B_n$, the summands $R_{T,I,n}$ are identified inductively, culminating in $T_C$ itself. The BY tree of [12] is then read off from $T_C$, and [12] Theorem 5.18 turns it into the desired dual graph. A complementary theorem shows that, for $g \ge 2$ and without the semistability assumption, no such list can recover the minimal regular model's dual graph: the paper exhibits semistable and non-semistable curves that have the same values for every absolute invariant and the same valuation for every weighted invariant, yet different dual graphs.

Load-bearing premise

The recovery procedure assumes the full correctness of the cluster-picture theorems of [12] — that the BY tree determines the dual graph of the minimal regular model for semistable hyperelliptic curves and that semistability is detected by the cluster picture; those theorems are used as black boxes and are not reproved here.

Editorial extensions

If this is right

  • If the theorem is correct, computing the reduction type of a semistable hyperelliptic curve becomes a finite valuation check on invariants written in Weierstrass coefficients, with no need to compute roots or run a full regular-model algorithm.
  • The same finite list also gives the dual graph of the potential stable model in the non-semistable case, so it serves as a reduction-type oracle for all genus $g$ hyperelliptic curves over such fields.
  • The genus 2 case is worked out completely: 24 absolute invariants and a table and algorithm that output the special fibre from their valuations, including the lengths of chains of $\mathbb{P}^1$'s.
  • The negative theorem shows the semistability assumption is essential: for genus $\ge 2$ no invariant-valuation list can distinguish the minimal-regular-model dual graph for arbitrary curves.
  • Over fields of odd residue characteristic this gives a uniform, genus-dependent replacement for the role of $j$ and $\Delta$ for elliptic curves.

Reading between the lines

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

  • Since $T_C$ is recovered completely, other cluster-picture encoded invariants, such as conductor exponents or Tamagawa numbers, are plausibly computable from the same absolute invariant valuations.
  • The author states that the construction should extend to general discretely valued fields with perfect residue field of characteristic $\ne 2$; proving Theorem 1.1 in that generality is a direct testable extension of the paper's methods.
  • Remark 4.10 suggests that the possible orderings on tree distances can be axiomatised using the four-point condition, which could turn the inductive recovery into a purely combinatorial classification of hyperelliptic reduction types.
  • For genus $g \ge 3$ the list is much larger; finding smaller generating sets of invariants (for instance via transvectants) that still separate the relevant stable model trees would make the algorithm practically useful.
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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

3 major / 5 minor

Summary. This paper generalises Tate's-algorithm-type statements for elliptic curves to hyperelliptic curves of genus g ≥ 2 over local fields of odd residue characteristic. The author defines, for each genus g, a finite list of absolute invariants associated to all possible 'stable model trees' with orderings on vertex distances, and proves (Theorem 5.2) an exact formula for the valuation of the invariant attached to the actual tree of a curve in terms of the ordered distances and the counts K_m(C). Section 6 gives an averaging function B_n that compares competing candidate trees, and Section 7 gives an inductive procedure to recover the full stable model tree, and hence the dual graph of the relevant special fibre, from the valuations. A converse-type theorem (Theorem 2.7) exhibits two genus-g curves, one semistable and one not, that are isomorphic over the algebraic closure, so that all absolute invariants and all invariant valuations agree, while the dual graphs differ; this rules out an analogue of the non-semistable elliptic-curve statement for hyperelliptic curves. Section 8 and the appendix give an explicit genus-2 list, a table, a worked example, and Magma/Sage code expressing the invariants in terms of Igusa invariants.

Significance. If the main theorem is correct, it provides a genuine structural generalisation of the elliptic-curve result: reduction type is readable from finitely many invariant valuations, with a list depending only on the genus. A particular strength is that the invariants are constructed, not fitted: the valuation formula is proved from the definition via the rearrangement inequality, and the genus-2 table is concrete and testable. The paper also gives an explicit negative result showing that, without the semistability hypothesis, no such list can work, which clarifies the boundary of the phenomenon. The appendix supplies computational verification of the genus-2 invariant list in terms of Igusa invariants and an implementation of the algorithm; these are useful and reproducible contributions. The main risk is the purely combinatorial reconstruction step in Section 7, which is load-bearing for uniqueness and is not machine-checked.

major comments (3)
  1. [§7, Prop. 7.6 and Lemmas 7.7–7.8] The central uniqueness claim that the final summand R_{T,I,n_{T,I}} determines the ordered tree (T,I) is proved by a long combinatorial induction whose key steps are only sketched. In particular, the deletion argument in Lemma 7.8 asserts that the renormalised expression R'' equals R_{T_N,I_N,n_{T_N,I_N}} after arguing factor-by-factor that distances and exponents coincide; the step 'by the definition of R_{T,I,n_{T,I}}, if v is the vertex ... then δ(v,w)=δ_m(T,I) if and only if the exponent ...' is plausible but quite compressed. Since this proposition is the only point at which the invariant data are shown to separate distinct reduction graphs, the paper should either give a more formal and complete proof, or supply an independent computational verification for a nontrivial genus (for example, exhaustive check for genus 3 over the possible orderings). Without this, the recovery theorem rests on an unverified combinatorial assertion.
  2. [§2, Theorem 2.7, Cases 2 and 3] In both of the odd-genus cases, the proof states 'One can verify from the cluster picture that ord(Δ_C) = ord(Δ_C')' and 'one can verify from [12] Theorem 1.8 that C/K is semistable and C'/K is not semistable', but the verifications are not carried out and no cluster depths or the relevant semistability criterion from [12] are displayed. Because Theorem 2.7 is the logical basis for the paper's central negative claim (no list of invariant valuations works without the semistability assumption), these two assertions are load-bearing and should be demonstrated explicitly or replaced with a fully detailed computation.
  3. [§6, Theorem 6.3 and §7, Theorem 7.1] The tie-breaking rule in Theorem 7.1(ii) selects, among the candidates maximising B_{n+1}(T,I,C), one with largest K_{n+1}(T,I). The proof of Theorem 6.3(ii) justifies the strict inequality in the case K_{n+1}(T,I) ≥ K_{n+1}(C), but the equality case K_{n+1}(T,I)=K_{n+1}(C) is handled by asserting that equality in the valuation bound would force R_{T,I,n+1}=R^σ_{T_C,I_C,n+1} 'for some σ'; this is the key point that makes the tie-break work, and the argument is terse. Please expand this step so that the reader can verify that no other summand can achieve the same valuation without being the same summand up to permutation.
minor comments (5)
  1. [Throughout] There are several typos and small textual slips: 'chatacteristic' in Corollary 7.2, 'semsitable' in Proposition 3.12(ii), 'gen us' in the abstract, and 'W eierstrass' in the header. These should be corrected in revision.
  2. [§6, first paragraph] The opening sentence says 'The main theorem we prove in this section is Theorem 5.2'; the intended reference is Theorem 6.3, since Theorem 5.2 is proved in the preceding section. Please correct the cross-reference.
  3. [§4, Definition 4.1] The displayed ordering I uses an unusual indexing (for example δ(v_q,v_{q+1}) followed by δ(v_{q+3},v_{q+4}) and repeated indices), immediately after 'there will be v_i=v_j for some i and j'. The notation is confusing and should be clarified, for instance by writing the ordering as a list δ_1 > δ_2 > ... with each δ_l representing an equivalence class of distances.
  4. [§8, Example 8.3] The formula for the chain length a contains an unmatched parenthesis: it reads 'v(DA(C)' instead of 'v(DA(C))' in the line 'a = 1/2 (−2v(D(C)) + 3v(DA(C) − v(DAC(C)))'. The same typo appears in the table row for D DE DEC in the formula for a.
  5. [§2, Theorem 2.7, Case 1] In the discriminant computation, the displayed formula for ord(disc) is plausible but the derivation is not shown; since the equality of discriminants is stated to follow from the cluster picture, a short explanation of how the displayed sum is read from the picture would improve the exposition.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the invariant valuations are universal functions, and the recovery argument relies on external DDM cluster-picture theorems rather than on the paper's own conclusions.

full rationale

The paper's derivation is not circular. The absolute invariants Inv_{T,I,n}(C) in Definition 4.4 are explicit universal symmetric functions in the root differences, indexed by all possible stable model trees (T,I) of genus g; they are constructed once and for all from the combinatorics of Tg, not fitted to any particular curve or to the reduction graph they recover. Theorem 5.2 then proves a valuation formula for the true invariant, and Theorem 7.1 gives an inductive procedure whose base step (the empty product) does not presuppose TC; at each later step only the already-recovered summands R_{TC,I_C,n} are used to select the next summand, so the reconstruction is an algorithm rather than a tautology. The decisive external input is the Dokchitser–Dokchitser–Maistret–Morgan theory ([12] Theorems 1.8, 5.18, 5.24), which is published work by different authors and is used as a black-box theorem; this is a domain assumption but not a self-referential input. The only self-citation, [8], occurs in the introduction as a contrast ('if one is not restricted to using invariants...') and is not load-bearing. The un-machine-checked combinatorial reconstruction Lemma 7.8 is a proof-completeness or correctness risk, not a circular step: nothing in it equates the conclusion with an input by definition.

Assumptions & free parameters 0 free parameters · 6 assumptions · 1 invented entities

The central claim rests on the cluster picture and BY tree theory of [12] as external domain assumptions; the paper contributes the invariant construction and recovery algorithm on top of that theory. There are no free parameters fitted to data and no physically invented entities.

assumptions (6)
  • domain assumption Dokchitser-Dokchitser-Maistret-Morgan theory: the BY tree of the cluster picture determines the dual graph of the special fibre of the minimal regular model of a semistable hyperelliptic curve over a local field of odd residue characteristic ([12] Theorem 5.18).
    Invoked in Proposition 3.12 and Corollary 7.2 to pass from stable model tree to dual graph; not reproved in this paper.
  • domain assumption Semistability and potential good reduction criteria for hyperelliptic curves in terms of cluster pictures ([12] Theorem 1.8).
    Used in proof of Theorem 2.7 and Proposition 3.2 to assert which constructed curves are semistable or have potentially good reduction.
  • domain assumption Equivalent cluster pictures have the same BY tree ([11] Theorem 1.1).
    Used in Corollary 3.5 to show the stable model tree is independent of the chosen Weierstrass model.
  • standard math The rearrangement inequality (Fact 5.7): a sum of products is maximized when the largest weights are matched to the largest values.
    Core of the valuation comparison in Theorems 5.2 and 6.3; the paper proves it by reference to the classical rearrangement inequality.
  • standard math Fundamental theorem of symmetric rational functions: symmetric rational expressions in the roots of f can be written in terms of the coefficients of f.
    Used to justify that absolute invariants can be evaluated from the Weierstrass coefficients without listing roots, stated in Section 1 and used throughout.
  • domain assumption Finiteness of the set of stable model trees and orderings for fixed genus (Lemma 4.7).
    The proof notes that there are finitely many cluster pictures of a degree 2g+2 polynomial; this is a combinatorial finiteness fact rather than a derived theorem.
invented entities (1)
  • Stable model tree T_C (Definition 3.1) independent evidence
    purpose: A model-independent weighted tree encoding the p-adic distances between clusters; the intermediary from which the BY tree and dual graph are recovered.
    Proposition 3.10 establishes a bijection with BY trees and Proposition 3.12 ties it to dual graphs via [12], giving a checkable external connection rather than a purely internal postulate.

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Pith. "Pith review of Invariants recovering the reduction type of a hyperelliptic curve." pith.science (2026). https://pith.science/paper/JNHPB7CZ

@misc{pith2026250208487,
  author       = {Pith},
  title        = {Pith review of: Invariants recovering the reduction type of a hyperelliptic curve},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JNHPB7CZ}},
  note         = {Machine review of arXiv:2502.08487}
}
abstract

Tate's algorithm tells us that for an elliptic curve $E$ over a local field $K$ of residue characteristic $\geq 5$, $E/K$ has potentially good reduction if and only if $\text{ord}(j_E)\geq 0$. It also tells us that when $E/K$ is semistable the dual graph of the special fibre of the minimal regular model of $E/K^{\text{unr}}$ can be recovered from $\text{ord}(j_E)$. We generalise these results to hyperelliptic curves of genus $g\geq 2$ over local fields of odd residue characteristic $K$ by defining a list of absolute invariants that determine the potential stable model of a genus $g$ hyperelliptic curve $C$. They also determine the dual graph of the special fibre of the minimal regular model of $C/K^{\text{unr}}$ if $C/K$ is semistable. This list depends only on the genus of $C$, and the absolute invariants can be written in terms of the coefficients of a Weierstrass equation for $C$. We explicitly describe the method by which the valuations of the invariants recover the dual graphs. Additionally, we show by way of a counterexample that if $g \geq 2$, there is no list of invariants whose valuations determine the dual graph of the special fibre of the minimal regular model of a genus $g$ hyperelliptic curve $C$ over a local field $K$ of odd residue characteristic when $C$ is not assumed to be semistable.

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