I‑AIMS Institute for Artificial Intelligence
and Mathematical Sciences
/

Detecting Split Jacobians in Genus-Two Isogeny Graphs: The Humbert Form of Discriminant 36

Tanush Shaska

I-AIMS preprint sh-129 · 2026

Explicit equations for the locus ℒₙ of genus-two curves with a maximal degree-n elliptic subcover have been computed only for n ≤ 5, and by elimination, whose cost is not predictable in advance. A recent degree formula changes this. It gives deg_w Fₙ = k(H_(n²)) − 10 ν(n) in closed form, and with it a reduced monomial support for the associated Humbert modular form G_(n²), so that the reconstruction becomes a determined linear problem whose every dimension is known before any computation begins. We carry this out for the first new case it opens, discriminant 36: we determine k(H₃₆) = 720, ν(6) = 12, deg_w F₆ = 600 and an admissible modular basis of size 41962, and we compute the four Siegel generators explicitly along Kumar’s rational parametrisation of H₃₆, obtaining in particular a complete factorisation of χ₁₀ whose factors recover the degenerate loci of the family from the modular side. We then draw the consequences for isogeny-based cryptography. The equation Fₙ decides optimal (n,n)-splitting at every point of the moduli space in characteristic p, with no hypothesis on the Newton polygon; this is asserted but not proved in the cryptographic literature, and the available proof excluded exactly the superspecial locus where the question is asked. The weight k(H_(n²)) bounds, unconditionally and with explicit constants, the number of superspecial nodes that detection at level n adds to the target set of the best known attack in dimension two. And the same torsion count ν(n) that appears in the degree formula governs the cost of both known detection routes, which places a barrier on raising the level and singles out n = 6 as the one case where a new equation can matter in practice.

MSC classes: Primary 14K02; Secondary 11F46, 14H40, 94A60

References

  1. N. Bruin and K. Doerksen. The arithmetic of genus two curves with (4,4)-split Jacobians. Canad. J. Math. 63 (2011), 992–1024.
  2. W. Castryck, T. Decru and B. Smith. Hash functions from superspecial genus-2 curves using Richelot isogenies. J. Math. Cryptol. 14 (2020), 268–292.
  3. A. Clingher, A. Malmendier and T. Shaska. Kummer surfaces, isogenies, and theta functions. NATO Sci. Peace Secur. Ser. D 66 (2025), 1–55.
  4. M. Corte-Real Santos, C. Costello and S. Frengley. An algorithm for efficient detection of (N, N)-splittings and its application to the isogeny problem in dimension
  5. PKC 2024, LNCS 14603, Springer, 157–189.
  6. M. Corte-Real Santos, C. Costello and S. Frengley. Efficient algorithms for the detection of (N, N)-splittings and endomorphisms. J. Cryptology 39 (2026), no. 1, Paper No. 2.
  7. C. Costello and B. Smith. The supersingular isogeny problem in genus 2 and beyond. PQCrypto 2020, LNCS 12100, Springer, 151–168.
  8. G. van der Geer and S. Harashita. The cycle class of the supersingular locus of principally polarized abelian varieties. Selecta Math. 31 (2025).
  9. E. Z. Goren and K. E. Lauter. Evil primes and superspecial moduli. Int. Math. Res. Not. (2006), Art. ID 53864.
  10. A. Grothendieck. Eĺéments de géométrie algébrique IV. Publ. Math. IH EŚ 32 (1967).
  11. D. Gruenewald. Computing Humbert surfaces and applications. AGCT 2009, Contemp. Math. 521, Amer. Math. Soc., 59–69.
  12. T. Ibukiyama, T. Katsura and F. Oort. Supersingular curves of genus two and class numbers. Compositio Math. 57 (1986), 127–152.
  13. E. Kani. Elliptic curves on abelian surfaces. Manuscripta Math. 84 (1994), 199–223.
  14. E. Kani. The number of curves of genus two with elliptic differentials. J. Reine Angew. Math. 485 (1997), 93–122.
  15. E. Kani. Curves of genus 2 on abelian surfaces. Manuscripta Math., to appear.
  16. T. Katsura and F. Oort. Families of supersingular abelian surfaces. Compositio Math. 62 (1987), 107–167.
  17. E. Kırımlı and G. Korpal. Refined Humbert invariants in supersingular isogeny degree analysis. Cryptology ePrint Archive, Paper 2025/1605.
  18. E. Kırımlı and C. Martindale. The computational refined Humbert invariant problem is equivalent to the computational isogeny problem. Cryptology ePrint Archive, Paper 2025/1295.
  19. A. Kumar. Hilbert modular surfaces for square discriminants and elliptic subfields of genus 2 function fields. Res. Math. Sci. 2 (2015), Art. 24.
  20. A. Malmendier and T. Shaska. A universal genus-two curve from Siegel modular forms. SIGMA 13 (2017), Paper No. 089.
  21. K. Magaard, T. Shaska and H. Völklein. Genus 2 curves that admit a degree 5 map to an elliptic curve. Forum Math. 21 (2009), 547–566.
  22. T. Shaska. Curves of genus 2 with (N, N)-decomposable Jacobians. J. Symbolic Comput. 31 (2001), 603–617.
  23. T. Shaska. Degrees of genus-two split-Jacobian loci and Humbert-form reconstruction. Preprint, 2026. arXiv:2608.16426.

All preprints