Reduction Types of Genus 2 Curves

Edwina Aylward; Elvira Lupoian; Lilybelle Cowland Kellock; Vladimir Dokchitser

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Reduction Types of Genus 2 Curves

Edwina Aylward, Lilybelle Cowland Kellock, Vladimir Dokchitser, Elvira Lupoian

Number Theory · Submitted 8 Jul 2026

Pith's verdict: Genus 2 analogue of Tate's table: 106 reduction type families with all arithmetic invariants tabulated; new monodromy method for component groups; resolves Lorenzini's Z/2Z⊕Z/4Z question. Central bijections rest on an explicit but unverifiable-from-the-paper case check.

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arXiv ID2607.07558 (version 2)

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CitationEdwina Aylward, Elvira Lupoian, Lilybelle Cowland Kellock, Vladimir Dokchitser. "Reduction Types of Genus 2 Curves." arXiv:2607.07558 (2026). https://pith.science/paper/OPN2D53Q

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Editor's note

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This is the genus 2 analogue of Tate's table for elliptic curves — 106 families of reduction types with component groups, conductor exponents, minimal discriminants, BSD fudge factors, and cluster pictures all listed together. That alone makes it a reference people will use. The monodromy method for component groups (Theorem 4.4) is a genuinely new technique: it keeps you at 2g×2g Smith normal forms instead of matrices that grow with parameters, and the explicit normal forms are worked out and presented in full. The resolution of Lorenzini's Z/2Z⊕Z/4Z question (Theorem 1.10) is a clean bonus that falls out of the tables. The cluster picture equivalence relation (Definition 3.9) and the proof that equivalence classes biject with (MRM type, potential stable type) pairs is the conceptual backbone, carried through carefully — reversibility of each move is explicit, and the Möbius realisations in Proposition 3.14 are concrete. Theorem 4.1, identifying monodromy with tame inertia, is a nice observation that I suspect will be cited independently. The generalised Ogg's formula (Theorem 5.12) ties the invariants together cleanly. The soft spot is Proposition 2.6. The central 1-to-1 correspondences between MRM types, MRNC types, and cluster picture equivalence classes all flow from it, and its proof is an explicit verification against the 106 Namikawa–Ueno families. The paper does not ship the verification code or a systematic case-by-case table, so a reader cannot independently confirm completeness without redoing the computation. That said, the concern is bounded: the cross-references to [7, Table G2] for MRNC types and [2, App. C] for admissible cluster pictures provide independent partial checks, and the authors note SageMath/Magma usage throughout. The risk is that individual table entries could contain errors — particularly in the parameter-dependent sub-cases in Tables 7.2–7.4 and the duplicated II*_t-m / 2I*_m-t families — but the overall framework does not depend on any single case being right. The stress-test concern about Proposition 2.6 is legitimate but I would not overweight it. This is standard practice for classification work in this area, and the cross-checks are real. The paper would benefit from supplementary code or a machine-readable appendix, but that is a refinement, not a prerequisite. This paper is for arithmetic geometers working on local invariants of curves and for anyone computing BSD data or component groups for genus 2 curves in practice. It deserves a serious referee who can spot-check a sample of table entries against independent computations.

Referee report

Major objections (3)

Proposition 2.6 and its proof: The completeness of the central classification (Theorem 1.1) and the 1-to-1 correspondences (Corollary 2.7, Theorem 3.4(2)–(3)) rest on an 'explicit verification' across 106 families that is not reproduced in the paper. The proof states: 'We have explicitly computed the cluster picture in each case.' While the paper cross-references [7, Table G2], [2, App. C], and [22], a reader cannot independently verify completeness or check for gaps/duplicates without repeating the computation. The concern is bounded — the cross-references provide independent checkpoints — but it is load-bearing for the central claim. Providing the verification code (Remark 1.11 mentions SageMath/Magma were used) or a supplementary table of the 106 cases would substantially strengthen the paper and allow independent verification.
Theorem 4.1 (and its proof): The identification of monodromy with tame inertia is the foundation of the Néron component group technique. The proof uses the profinite completion of π₁(D,t) and the comparison theorem. The step where the topological generator [γ] of π₁(D,t) is mapped to a generator of tame inertia is standard, but the paper should clarify whether the comparison isomorphism identifies the geometric monodromy with the arithmetic tame inertia generator directly, or only up to an automorphism of Ẑ (i.e., up to a unit). Since the Smith normal form of M−I is invariant under replacing the generator by a power coprime to its order, this does not affect the component group computation, but the current phrasing 'coincides with the action of some generator' in the statement could be made more precise in the proof.
Theorem 3.4(1): The proof states 'A finite comparison of the conditions in [2, App. C] with the conditions displayed in Tables 7.1–7.7 shows that the degree 6 pictures in the tables are exactly the admissible degree 6 genus 2 cluster pictures with tame reduction.' This is another finite but large verification that is not shown. Given that Theorem 3.4(3) (the bijection between equivalence classes and (MRM type, potential stable type) pairs) depends on this, the paper would benefit from at least sketching the comparison or providing code.

Minor objections (7)

The monodromy matrix table in the proof of Theorem 4.5: the notation 'ℓ' appears in the B matrix for 2I*_l-t (l odd) — this should presumably be 'l'.
Table 7.8: the arrows and loops labelled A, B, C, D are described in the text, but a legend on the table itself (or its caption) would improve readability for readers using the tables as a reference.
§1.6 notation: 'ω_min' is defined as dx/y ∧ xdx/y for a minimal Weierstrass model, but the sign convention for v(ω_min/ω₀) (which is stated to be negative in the table notes) could be stated more prominently to avoid confusion.
Lemma 2.5: the case-by-case check through [22] is described as 'simple' but involves non-trivial singularity analysis. A brief indication of the method (e.g., 'by comparing the intersection matrices and singularity types in [22] with Definition 2.1') would help the reader.
Remark 1.12(2): the count '106 families' vs '104 families of MRNC types' vs '105 families' in Namikawa–Ueno is explained, but the arithmetic could be summarised more cleanly (e.g., '104 MRNC types + 2 duplicated MRM types = 106; 105 NU families with 2I_{l-t} split = 106').
The paper references [7] as 'Available at: arxiv:2512.09082' — this appears to be a future-dated preprint; please verify the identifier.
Lemma 5.17: 'changes of model modify the valuation of the discriminant by a factor of 10' should read 'by a multiple of 10' (i.e., v(Δ) changes by 10n, not by a factor of 10).

Simulated authors' rebuttal

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We thank the referee for their careful reading and positive assessment. The three major comments all concern the transparency of finite verifications that underpin the classification. We address each below.

Referee: Proposition 2.6: completeness rests on an unshown 'explicit verification' across 106 families. Request for verification code or supplementary table.

Authors: The referee is right that the verification is load-bearing and not currently reproducible by a reader. We will include the verification code (SageMath/Magma) as a supplementary file and add a more detailed description of the verification procedure in the proof of Proposition 2.6. Specifically, the code takes each Namikawa–Ueno family, computes the cluster picture from the explicit polynomial, determines the MRNC type via the recipe in [11], and checks the result against Tables 7.1–7.7 and [7, Table G2]. We will also add a brief summary table in an appendix indicating, for each of the 106 families, the Namikawa–Ueno label, the MRNC label, and the cluster picture of the representative, so that a reader can spot-check individual cases without running code. revision: yes
Referee: Theorem 4.1: clarify whether the comparison isomorphism identifies geometric monodromy with the tame inertia generator directly or only up to a unit in Z-hat.

Authors: The referee's observation is correct: the comparison identifies the topological generator [γ] with a generator of tame inertia only up to a unit in Ẑ, not canonically with a specific generator. As the referee notes, this does not affect the component group computation, since the Smith normal form of M−I is invariant under replacing the generator by a coprime power. We will revise the proof of Theorem 4.1 to state this explicitly: the image of [γ] under the comparison isomorphism is a generator of tame inertia, well-defined up to multiplication by a unit in Ẑ (i.e., up to an automorphism of Ẑ). We will also adjust the phrasing in the statement from 'coincides with the action of some generator' to 'coincides with the action of a generator of tame inertia, up to a unit in Ẑ.' revision: yes
Referee: Theorem 3.4(1): another finite but large verification not shown; request to sketch the comparison or provide code.

Authors: This is the same type of verification as in Proposition 2.6, and we agree that it should be made transparent. The verification compares the admissibility conditions from [2, App. C] (which lists all admissible degree 6 genus 2 cluster pictures with tame reduction) against the conditions displayed in Tables 7.1–7.7. We will include this verification in the same supplementary code file mentioned above: the code enumerates all cluster pictures satisfying the admissibility conditions of [2, App. C] and checks that each appears in exactly one column of the tables, and conversely that every displayed cluster picture satisfies the admissibility conditions. We will also add a sentence in the proof of Theorem 3.4(1) sketching the nature of the comparison (checking cluster shapes, depth constraints, and parity conditions column by column) and pointing to the supplementary code. revision: yes

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The paper

The paper claims

Tate's elliptic curve tables, generalized to genus 2

The reading

For elliptic curves over local fields, Tate's classical table lets you read off the Néron component group, conductor exponent, discriminant valuation, and other invariants directly from the Kodaira reduction type — a single combinatorial label. This paper produces the analogous tables for curves of genus 2. The authors classify all 106 families of reduction types (building on the Namikawa–Ueno classification) and show that each type determines, in a purely combinatorial way, the Néron component group of the Jacobian, the tame conductor exponent, whether the reduction is tame, and — when the residue characteristic is not 2 and the reduction is tame — also the potential stable type, the minimal discriminant valuation, the BSD fudge factor, and the complete list of cluster pictures (a combinatorial encoding of how the roots of the defining polynomial cluster under the valuation). The central mechanism is that these invariants, which classically required algorithmic computation from a specific equation of the curve, are in fact determined by combinatorial data: the configuration of components, singularities, and intersection pairings on the special fibre of a regular model. For the Néron component group specifically, the authors introduce a new method that replaces large intersection-pairing matrices with fixed 2×2 monodromy matrices, using the fact that component groups can be recovered from the torsion of co-invariants of the monodromy action on homology.

What carries the argument

The argument rests on three pillars. First, the Namikawa–Ueno classification provides 105 families of degenerations of genus 2 curves, which the authors split into 106 families (duplicating two types that share a special fibre but have different potential stable types). Second, cluster pictures — combinatorial encodings of the p-adic distances between roots of the defining polynomial — serve as a replacement for the j-invariant and discriminant valuations that classify elliptic curve reduction. The authors define an equivalence relation on cluster pictures (Definition 3.9) and prove it is in bijection with pairs (MRM reduction type, potential stable type). Third, a new method for computing N

Extended reading notes

Core claim

The paper's central result is Theorem 1.1, which states that Tables 7.1–7.7 list all possible reduction types of the minimal regular model of genus 2 curves over a complete discrete valuation ring, and that each reduction type determines eight families of invariants: (1) the MRNC reduction type, (2) the Néron component group, (3) the tame conductor exponent, (4) whether the reduction is tame, and — for tame reduction with p≠2 — (5) the potential stable type, (6) the minimal discriminant valuation, (7) the BSD fudge factor v(ω_min/ω_0), and (8) the complete list of cluster pictures. A key structural finding is that for genus 2 curves, the MRM and MRNC reduction types are in one-to-one-corre s

Load-bearing premise

The completeness of the classification and the correspondence between reduction types, MRNC types, and cluster picture equivalence classes rest on Proposition 2.6, which is verified by an explicit case-by-case check through the Namikawa–Ueno examples. If any case in this finite but large verification was mishandled, the one-to-one correspondences could have gaps or duplicates.

Editorial extensions

If this is right

Given a genus 2 curve over a local field, one can read off its Néron component group, conductor exponent, minimal discriminant, and BSD fudge factor from a table lookup once the reduction type is identified — no algorithmic computation from the curve's equation is needed.
The cluster picture of a tame genus 2 curve (p≠2) completely determines its reduction type and potential stable type, and the tables provide explicit recipes for finding minimal Weierstrass equations and tracing how reduction types change under field extensions.
Theorem 1.10 resolves an open question of Lorenzini by showing that Z/2Z ⊕ Z/4Z cannot occur as the prime-to-p part of the Néron component group of a principally polarised abelian surface with potentially good reduction.
The one-to-one correspondence between MRM and MRNC reduction types for genus 2 curves (Theorem 1.2) clarifies the relationship between two competing notions of reduction type that have appeared in recent literature.

Reading between the lines

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

The analogy with Tate's table suggests that similar combinatorial classification tables could in principle be constructed for higher-genus hyperelliptic curves, though the number of reduction type families and the complexity of cluster picture equivalence relations would grow substantially.
The new monodromy-based method for computing Néron component groups (Theorem 1.3) is stated for curves of arbitrary genus and could be applied beyond genus 2, potentially simplifying component group computations for higher-genus Jacobians where intersection pairing matrices become unwieldy.
The dependence on p≠2 for the cluster picture results reflects a genuine obstruction: genus 2 curves in residue characteristic 2 need not be hyperelliptic in a useful sense, and the paper notes it is not even known whether all reduction types are realised in characteristic 2.

Original abstract

Tate produced a table for elliptic curves over local fields that beautifully summarises their arithmetic invariants in terms of the Kodaira type. We present an analogous set of tables for curves of genus 2. The invariants addressed include: reduction type of the minimal regular model, reduction type of the minimal regular model with normal crossings, N\'eron component group of the Jacobian, conductor exponent, valuation of the discriminant of a minimal Weierstrass model, and cluster pictures.

How the literature uses it

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Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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One can however also choose to classify curves by the special fibre of a minimal regular normal crossings model, see for example the work of Dokchitser [ref] and the genus 2 classification of Aylward, Kellock, Dokchitser and Lupoian [ref] .

Defining reduction types of curves via minimal regular and minimal normal crossings models · Introduction · 0.88

What it cites · 7 sentences without a confident label

Cites as background 1 paper

(See [ref] or [ref] for the definition and an overview of MRNC models.)

2512.09082 · Notation, terminology and conventions · 0.96

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