On different expressions for invariants of hyperelliptic curves of genus 3
Pith reviewed 2026-05-24 22:21 UTC · model grok-4.3
The pith
Passage formulas relate Tsuyumine and Shioda invariants for genus 3 hyperelliptic curves.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The central claim is that a passage formula exists between Tsuyumine and Shioda invariants of genus 3 hyperelliptic curves, and that Shioda invariants admit an expression in terms of differences of roots; the latter yields a criterion for the type of bad reduction whenever the Jacobian meets stated conditions.
What carries the argument
The passage formula that translates between the two sets of invariants and thereby produces modular expressions and root-difference forms.
If this is right
- Modular expressions for Shioda invariants become available.
- Verification of numerically computed equations for CM genus 3 hyperelliptic curves is supported.
- Shioda invariants are available in terms of root differences.
- A criterion identifies the type of bad reduction under the given Jacobian conditions.
Where Pith is reading between the lines
- The root-difference expressions could be used to study arithmetic properties of the curves beyond reduction type.
- Similar translation formulas between invariant sets may be sought for hyperelliptic curves of other genera.
- The modular expressions could streamline the search for curves with prescribed invariants in computational settings.
Load-bearing premise
The derivations rest on the standard definitions of Tsuyumine and Shioda invariants together with the assumption that the stated Jacobian conditions are satisfied for the reduction criterion.
What would settle it
An explicit genus 3 hyperelliptic curve for which the passage formula produces values that disagree with independently computed invariants would falsify the claimed relation.
read the original abstract
In this paper we give a passage formula between different invariants of genus 3 hyperelliptic curves: in particular between Tsuyumine and Shioda invariants. This is needed to get modular expressions for Shioda invariants, that is, for example, useful for proving the correctness of numerically computed equations of CM genus 3 hyperelliptic curves. On the other hand, we also get Shioda invariants described in terms of differences of roots of the equation defining the hyperelliptic curve, that has applications for studying the reduction type of the curve. Under certain conditions on its jacobian, we give a criterion for determining the type of bad reduction of a genus 3 hyperelliptic curve.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper derives explicit passage formulas relating Tsuyumine invariants to Shioda invariants for genus-3 hyperelliptic curves, expressing the latter both in modular form and in terms of root differences of the defining equation. It additionally supplies a criterion, conditioned on properties of the Jacobian, for identifying the type of bad reduction.
Significance. The relations would allow modular expressions for Shioda invariants, supporting verification of numerically computed CM curves, and the root-difference form would aid arithmetic studies of reduction types. Both directions build directly on existing Igusa-type invariant literature without introducing new ad-hoc parameters.
minor comments (3)
- The abstract asserts the existence of the passage formulas and the reduction criterion but does not indicate the algebraic manipulations or intermediate identities used; the introduction or §2 should contain at least one explicit low-degree example relating the two sets of invariants.
- Notation for the root differences and the precise statement of the Jacobian conditions in the reduction criterion should be introduced with a numbered equation or displayed formula to avoid ambiguity when the criterion is applied.
- The paper cites the original definitions of Tsuyumine and Shioda invariants; a short table comparing the two sets (degrees, number of generators) would improve readability for readers unfamiliar with the genus-3 case.
Simulated Author's Rebuttal
We thank the referee for the positive summary and recommendation of minor revision. No specific major comments were provided in the report, so we have no points requiring rebuttal or revision at this stage. The manuscript stands as submitted.
Circularity Check
No significant circularity identified
full rationale
The paper's central derivations consist of explicit algebraic passage formulas relating Tsuyumine and Shioda invariants via root differences and modular expressions, together with a reduction-type criterion conditioned on Jacobian properties. These steps rely on standard prior definitions of the invariants and perform direct algebraic manipulations without reducing any claimed result to a fitted parameter, self-definition, or load-bearing self-citation chain. The derivations remain independent of the target identities and are self-contained against external algebraic benchmarks.
discussion (0)
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