Generalized Howe curves of genus 4, 5, and 6 with completely decomposable Jacobians
Pith reviewed 2026-05-10 04:10 UTC · model grok-4.3
The pith
Generalized Howe curves with Jacobians splitting into four elliptic curves are superspecial for genus 4 in every prime between 20001 and 999999.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
By restricting attention to generalized Howe curves whose Jacobians are isomorphic to a product of four elliptic curves, superspeciality of the curve becomes equivalent to supersingularity of each elliptic factor. Computer enumeration under this equivalence confirms such superspecial genus-4 curves exist for every prime characteristic p satisfying 20000 < p < 10^6. The same approach supplies constructions of superspecial curves of genus 5 and genus 6 directly from supersingular elliptic curves, with existence verified computationally in the ranges 13 < p < 10^5 and 7 < p < 10^5.
What carries the argument
The completely decomposable generalized Howe curve, obtained by combining four elliptic curves so that the Jacobian factors as their product; the exact reduction of superspeciality to supersingularity of the four factors.
If this is right
- Superspecial genus-4 curves are now known to exist in every characteristic below 10^6.
- Superspecial curves of genus 5 and 6 can be built directly from supersingular elliptic curves alone.
- The method extends the verified range of existence far beyond the previous limit of p < 20000 for genus 4.
- These constructions supply explicit examples without requiring direct checks on higher-genus conditions.
Where Pith is reading between the lines
- The reduction may allow similar constructions for genera higher than 6 by increasing the number of elliptic factors.
- If the pattern of existence continues for all larger p, superspecial curves would exist in every characteristic for these genera.
- The efficiency of the elliptic-factor check could support systematic searches for other decomposable abelian varieties.
Load-bearing premise
Superspeciality of the generalized Howe curve holds if and only if each of its four elliptic curve factors is supersingular, and the computer enumeration finds every such curve without missing cases or implementation errors.
What would settle it
A single prime p between 20001 and 999999 for which no four supersingular elliptic curves combine to form a superspecial generalized Howe curve of genus 4 would disprove the claimed existence.
read the original abstract
Superspecial curves are important objects in number theory and algebraic geometry, and the existence in genus $g \geq 4$ remains an open problem for all but finitely many characteristics $p > 0$. As a computational approach to this problem, Kudo-Harashita-Howe (2020) showed that a superspecial curve of genus 4 exists in each characteristic $p$ with $7 < p < 20000$. Their method restricted attention to a specific class of curves, known as Howe curves, for which superspeciality is reduced to those of curves of genus at most 2. In this paper, we focus on a more specific class of curves, namely Howe curves whose Jacobians decompose into a product of four elliptic curves. By restricting our attention to such curves, the superspeciality reduces to the supersingularity of elliptic curves, which enables us to construct a superspecial curve of genus 4 more efficiently than Kudo-Harashita-Howe's method. As our first main result, we confirmed by computer the existence of such superspecial curves of genus 4 in characteristics $p$ with $20000 < p < 10^6$. Using a similar approach, we also propose constructions of superspecial curves of genera 5 and 6 from only supersingular elliptic curves. Furthermore, computational experiments establish the existence of superspecial curves of genus 5 (resp. genus 6) in characteristics $p$ with $13 < p < 10^5$ (resp. $7 < p < 10^5$).
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies generalized Howe curves of genus 4, 5, and 6 whose Jacobians are completely decomposable as products of elliptic curves. It reduces superspeciality of these curves to supersingularity of the elliptic factors and reports a computer-assisted verification that superspecial genus-4 examples exist for every prime p with 20000 < p < 10^6. Analogous constructions and computational checks are given for genera 5 and 6, establishing existence in the ranges 13 < p < 10^5 and 7 < p < 10^5 respectively.
Significance. If the computational enumerations are correct, the work substantially extends the known range of characteristics in which superspecial curves of genus 4 are known to exist, improving on the Kudo-Harashita-Howe bound of p < 20000. The reduction to supersingular elliptic curves is efficient and the explicit constructions for genera 5 and 6 are of independent interest for the open existence question in higher genus.
major comments (2)
- [§4 (computational results)] The central existence claim for genus 4 (abstract and §4) rests on exhaustive enumeration over supersingular elliptic quadruples for each p in (20000, 10^6). No description is given of the search strategy, pruning rules, arithmetic implementation in characteristic p, or termination/verification criteria, rendering the claim non-reproducible and only moderately supported.
- [§2] The reduction that superspeciality of the generalized Howe curve is equivalent to supersingularity of its four elliptic factors is stated without a self-contained proof or reference to the precise isomorphism criterion for the Jacobian (abstract and §2). While standard for abelian varieties, the precise statement used here should be recorded explicitly because it is load-bearing for all subsequent claims.
minor comments (2)
- [§1] Notation for the generalized Howe curve construction (e.g., the precise gluing data or the role of the four elliptic curves) is introduced without a diagram or explicit equations in the introductory sections.
- [§5] The ranges for genera 5 and 6 are stated as computational experiments rather than exhaustive searches; clarify whether the method is claimed to be exhaustive or merely heuristic in those cases.
Simulated Author's Rebuttal
We thank the referee for the careful reading and constructive comments, which have helped improve the clarity and reproducibility of the manuscript. We address each major comment below and have revised the paper accordingly.
read point-by-point responses
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Referee: [§4 (computational results)] The central existence claim for genus 4 (abstract and §4) rests on exhaustive enumeration over supersingular elliptic quadruples for each p in (20000, 10^6). No description is given of the search strategy, pruning rules, arithmetic implementation in characteristic p, or termination/verification criteria, rendering the claim non-reproducible and only moderately supported.
Authors: We agree that the original §4 provided insufficient implementation details. In the revised manuscript we have added a new subsection 4.1 that fully describes the algorithm: (i) supersingular j-invariants are enumerated via Bröker’s algorithm (time O(p^{1/2+ε})); (ii) all ordered quadruples are generated and pruned by the necessary trace condition on the product of Frobenius traces; (iii) for each surviving quadruple the existence of the generalized Howe curve is checked by solving the explicit system of equations given in §3 over F_p; (iv) all arithmetic is performed in SageMath with native finite-field operations. Termination is guaranteed by the finiteness of the supersingular set (size ∼p/12). Verification consists of matching our output for all p<20000 against the Kudo–Harashita–Howe tables and of independent re-runs on a second machine. The source code and raw data files have been deposited in a public repository linked from the revised paper. revision: yes
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Referee: [§2] The reduction that superspeciality of the generalized Howe curve is equivalent to supersingularity of its four elliptic factors is stated without a self-contained proof or reference to the precise isomorphism criterion for the Jacobian (abstract and §2). While standard for abelian varieties, the precise statement used here should be recorded explicitly because it is load-bearing for all subsequent claims.
Authors: We acknowledge that the reduction was presented too concisely. The revised §2 now contains an explicit statement together with a short self-contained argument: a principally polarized abelian variety A of dimension g is superspecial if and only if it is isogenous to a product of g supersingular elliptic curves (Oort, 1975; see also the Dieudonné-module criterion that the a-number equals g). For a generalized Howe curve C we prove that Jac(C) is isogenous to E1×E2×E3×E4 by exhibiting the explicit isogenies arising from the covering maps to the four elliptic curves; hence C is superspecial precisely when each Ei is supersingular. The necessary references and the two-line proof using the a-number have been inserted. revision: yes
Circularity Check
No circularity: computational verification is independent of inputs
full rationale
The paper's central results consist of explicit computer searches confirming existence of superspecial genus-4 generalized Howe curves for primes in (20000, 10^6) and analogous constructions for genera 5 and 6. Superspeciality follows directly from the Jacobian being a product of supersingular elliptic curves, a property of the curve construction itself rather than a fitted or self-referential definition. No equations or claims reduce a 'prediction' or existence statement to its own inputs by construction. Reliance on prior Howe-curve theory is external and not load-bearing in a circular sense; the enumerations for new p-ranges constitute independent checks. The derivation chain is self-contained against the stated computational benchmarks.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption The Jacobian of a curve over a field of characteristic p is an abelian variety whose superspeciality is preserved under isogeny and can be checked after decomposition into elliptic curves.
discussion (0)
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