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Legendre polynomials and complex multiplication, II: class numbers of quadratic fields and genus 2 supersingular polynomials

  • Andrew Karl
  • Patrick Morton
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Abstract

The factorizations over 𝔽p of two supersingular polynomials hp(x) and gp(x) for genus $2$ curves, discussed by Ibukiyama, Katsura and Oort in their 1986 paper, are investigated. These polynomials are congruent modulo p to the Jacobi polynomials Pn(α,0)(1−2x), for α= ± 1/4, ± 1/6, respectively. The number of their linear factors (mod p) is determined in terms of class numbers of the imaginary quadratic fields ℚ(√(−dp)), where d ∈ {1,2,3}. The proofs use a quadratic transformation relating these polynomials to the Legendre polynomials Pn(x); previous results on linear and binomial quadratic factors of P(p−e)/4(x) and P(p−e̅)/3(x) proved by Brillhart and Morton; and properties of the irreducible quadratic factors of the class equations H−3p(X) and H−12p(X) (mod p). The quadratic transformation and linear factor results for hp(x) ≡ Pn(±1/4,0)(1−2x) were first discovered using artificial intelligence.

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Cite

BibTeXRIS
BibTeX
@misc{karl2026legendre,
  title = {{Legendre polynomials and complex multiplication, II: class numbers of quadratic fields and genus 2 supersingular polynomials}},
  author = {Andrew Karl and Patrick Morton},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.21198},
  eprint = {2608.21198},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.21198},
}
RIS
TY  - UNPB
TI  - Legendre polynomials and complex multiplication, II: class numbers of quadratic fields and genus 2 supersingular polynomials
AU  - Andrew Karl
AU  - Patrick Morton
PY  - 2026
DA  - 2026/08/31
PB  - arXiv
DO  - 10.48550/arxiv.2608.21198
UR  - https://arxiv.org/abs/2608.21198
AB  - The factorizations over 𝔽_p of two supersingular polynomials h_p(x) and g_p(x) for genus $2$ curves, discussed by Ibukiyama, Katsura and Oort in their 1986 paper, are investigated. These polynomials are congruent modulo p to the Jacobi polynomials P_n^(α,0)(1−2x), for α= ± 1/4, ± 1/6, respectively. The number of their linear factors (mod p) is determined in terms of class numbers of the imaginary quadratic fields ℚ(√(−dp)), where d ∈ {1,2,3}. The proofs use a quadratic transformation relating these polynomials to the Legendre polynomials P_n(x); previous results on linear and binomial quadratic factors of P_(p−e)/4(x) and P_(p−e̅)/3(x) proved by Brillhart and Morton; and properties of the irreducible quadratic factors of the class equations H_−3p(X) and H_−12p(X) (mod p). The quadratic transformation and linear factor results for h_p(x) ≡ P_n^(±1/4,0)(1−2x) were first discovered using artificial intelligence.
ER  -