A certified census of the superspecial generalized Howe curves of genus 4, 5 and 6, and the cell (g,p)=(6,13)
Abstract
A recent preprint of Ohashi (arXiv:2604.18074) constructs, in every characteristic p>5, three explicit families of curves of genus 4, 5 and 6 whose Jacobians are completely decomposable, and reduces the question of whether a member is superspecial to whether three elliptic curves are supersingular — that is, to the vanishing of an explicit polynomial (Deuring's for genus 4, the Ibukiyama–Katsura–Oort polynomials for genus 5 and 6) at points of 𝔽ₚ². We take that finite core as the object of study. For every prime 7 ≤ p ≤ 157 (7 ≤ p ≤ 113 in genus 4) we compute, and certify with a proof assistant, the cardinalities of the root sets of the three polynomials in 𝔽ₚ² and the exact numbers N₄(p), N₅(p), N₆(p) of ordered parameter pairs that pass the criterion — numbers the preprint never prints, since its algorithms stop at the first witness. The zero sets reproduce the preprint's exception lists for genus 4 and genus 5 exactly. In genus 6 they reproduce the part of its list that lies in range, with one difference: the genus-6 family contains no superspecial member in characteristic 13, whereas the preprint's Theorem 6.4 asserts one for every prime 11<p<10⁵ outside a list that does not contain 13, and nothing else in the preprint covers p=13. The author's own public computation log records the same failure at p=13, so the discrepancy is a transcription slip in the exception list, and (g,p)=(6,13) joins (5,13) and (6,7) as a cell the preprint leaves open. We also observe that the genus-6 count is a solution count, N₆(p)=#{(x,y,z) ∈ T₆(p)³: xyz=1}, which carries an action of a group of order 12 that the preprint's algorithm does not use, and that 6 | N₆(p) at every prime in range. Every statement about 𝔽ₚ² is machine-checked in Lean 4; the statements about p=13 are checked by the kernel alone.
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Archived files
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Source snapshot 2026-09-07 03:53 UTC
File fingerprint
a94c9a6ea36ca23031b7c15bd285c6be197d41b2468dc28eb005cf89e2c189f8
Claim ledger
Stated results
HS0measurement2026-09-07
Lean core's Nat.choose is the naive Pascal recursion, so evaluating it costs Theta(C(n,k)) additions and neither kernel nor interpreter finishes Nat.choose 78 39 (about 10²2 steps); the multiplicative recurrence fastChoose n (k+1) = fastChoose n k * (n-k) / (k+1) is k bignum operations and is proved equal to Nat.choose for all n, k
HS1routine2026-09-07
The hand-built model of F_(p²) = Fₚ[w]/(w² - n) as ZMod p x ZMod p: fadd/fmul are commutative, associative and distributive for EVERY p and n (proved by ring on the components, no finite check); 2 is a quadratic non-residue mod 13, 19, 11 and 3 mod 7, 17; leastNR returns those; every non-zero element of F_(13²) is invertible; and with the square n = 3 instead exactly 24 non-zero elements have no inverse
HS2known data2026-09-07
The IKO polynomial of source equation (5.2) at p = 13 is g₁3(alpha) = 2 + 3a + 4a² + 3a³ + 2a⁴ over F₁3, and its root set in F₁3[w]/(w²-2) is exactly 5, 8, 9+w, 9+12w: four distinct roots, equal to deg g₁3 = floor(13/3), so g₁3 splits completely over F_(13²)
HS3correction2026-09-07
CORRECTION to arXiv:2604.18074v3 Theorem 6.4: there is NO superspecial curve Z_(s,t) of genus 6 in characteristic 13 – no two distinct roots alpha₁, alpha₂ of g₁3 have alpha₁/alpha₂ again a root (exhaustive over F_(13²), 12 ordered pairs of distinct roots, 48 triples). Theorem 6.4 states existence for every prime with 11 < p < 10⁵ outside an explicit 19-element list that does not contain 13, so it is false at p = 13; Appendix B omits 13 as well, so the circle at (g,p) = (6,13) in the source's Table 1 and Theorem 1.3's range 7 < p < 10⁵ have no support at p = 13 anywhere in the paper, and (6,13) is a third open cell beside (5,13) and (6,7)
HS4known data2026-09-07
Inside the source's genus-6 family Z_(s,t) there is also nothing at p = 7 (the source's own open cell (6,7), where g₇ = 3 + 2a + 3a² has exactly 2 = floor(7/3) roots in F₄₉) and nothing at p = 11 (3 roots), the prime below the range of its Theorem 6.4 that its Appendix B supplies by a different construction
HS5routine2026-09-07
Genus-6 controls: at p = 17 the family is NOT empty – alpha₁ = 4, alpha₂ = 13 and alpha₃ = 16 are roots of g₁7 with alpha₂*alpha₃ = alpha₁, and N₆(17) = 6; and 13 is not the last failure, N₆(19) = 0 with |T| = 6
HS6known data2026-09-07
Deuring's polynomial at p = 13 is H₁3(lambda) = 1 + 10L + 4L² + 10L³ + 4L⁴ + 10L⁵ + L⁶ over F₁3, with exactly 6 = (13-1)/2 roots in F_(13²)
HS7known data2026-09-07
Inside the source's genus-4 family X_(s,t) there is no superspecial member at p = 7, p = 13 or p = 19: exhaustively over pairs from the Deuring root set in F_(p²), with the four non-vanishing conditions (3.2) and lambda₄ given by (3.3)
HS8routine2026-09-07
Genus-4 controls: N₄(11) = 8 with |T| = 5, so the genus-4 family is not always empty; and the four conditions (3.2) are not vacuous – at p = 11 they reject exactly 5 of the 25 ordered pairs of Deuring roots
HS9known data2026-09-07
The IKO polynomial of source equation (4.3) at p = 13 is h₁3(beta) = 7 + 12b + 12b² + 7b³, with root set 8+5w, 8+8w, 12 in F₁3[w]/(w²-2): three roots, equal to deg h₁3 = floor(13/4)
HS10known data2026-09-07
Inside the source's genus-5 family Y_(s,t) there is nothing at p = 13 – the source's own open cell (5,13) – and nothing at p = 7, 11, 17, 19, the four primes below the range of its Theorem 6.3 for which its Appendix A supplies individually constructed curves instead. The genus-5 test is stated inversion-free: E₃: y² = (x²-b₁)(x²-b₂) is supersingular iff the Hasse invariant, the coefficient of xᵖ⁻¹ in ((x²-b₁)(x²-b₂))^((p-1)/2), vanishes, which is the Deuring form homogenised at (b₂, b₁)
HS11routine2026-09-07
Genus-5 controls: N₅(23) = 6 with |T| = 5, so the genus-5 family is not always empty; and N₅(37) = 0 with |T| = 9, the first prime of the source's Theorem 6.3 exception list, so 19 is not the last failure
HS12routine2026-09-07
N₆(p), the number of ordered parameter pairs (s⁶, t⁶) for which the source's genus-6 curve Z_(s,t) is superspecial, for every prime 7 <= p <= 157: 0,0,0,6,0,6,6,24,0,30,0,24,12,30,0,0,48,24,0,42,36,0,48,24,24,0,60,0,72,6,24,18,0,0. The zeros are 7,11,13,19,37,43,61,67,79,97,109,127,151,157
HS13routine2026-09-07
N₅(p), the number of ordered pairs (s², t²) for which the source's genus-5 curve Y_(s,t) is superspecial, for every prime 7 <= p <= 157: 0,0,0,0,0,6,8,4,0,4,8,24,0,16,16,16,32,16,10,24,0,0,0,34,32,16,16,40,36,0,12,8,60,8. The zeros are exactly the complete exception list 37,53,89,97,101,137 of the source's Theorem 6.3 together with every prime below its range, 7,11,13,17,19
HS14routine2026-09-07
N₄(p), the number of ordered pairs (lambda₁, lambda₃) for which the source's genus-4 curve X_(s,t) is superspecial, for every prime 7 <= p <= 113: 0,8,0,8,0,32,26,36,12,56,12,80,38,110,30,24,242,0,54,110,122,36,224,24,116,84,116. The zeros are exactly 7, 13, 19, 73, the complete exception list of the source's Theorem 1.1 plus p = 7
HS15known data2026-09-07
The three cardinality controls the source quotes – |T₄| = (p-1)/2 (Silverman Theorem V.4.1), |T₅| = floor(p/4) and |T₆| = floor(p/3) (Ibukiyama-Katsura-Oort Proposition 1.14 plus the degrees) – hold at every prime 7 <= p <= 157, together with the matching degrees of Hₚ, hₚ and gₚ; each also shows the polynomial splits completely over F_(p²), which makes the sweeps exhaustive without appeal to the source's Lemma 5.1
HS16candidate2026-09-07
The genus-6 census is a solution count: N₆(p) = #(x,y,z) in T₆³: xyz = 1, where T₆ is the root set of the IKO polynomial gₚ in F_(p²), by the bijection (a₁,a₂) -> (a₁, a₂⁻¹, a₂/a₁). The right-hand side carries an S₃ x <inversion> action of order 12 that the source's Algorithm 5.3 does not exploit, and 6 divides N₆(p) at every prime 7 <= p <= 157
HS17routine2026-09-07
All three root sets T₄, T₅, T₆ are closed under inversion at every prime 7 <= p <= 31 (stated without inverses: every root has a partner in the set whose product with it is 1), and consequently the genus-5 census is swap-invariant, so 2 divides N₅(p) at every prime 7 <= p <= 157
Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- Superspecial generalized Howe curves with completely decomposable Jacobians: the finite Fₚ² core of arXiv:2604.18074v3, its certified census, and the (6,13) correction.
- Snapshot
- 2026-09-07 03:53 UTC
- Ledger commit
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