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

Abstract

The factorizations over $\mathbb{F}_p$ 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 $α= \pm 1/4, \pm 1/6$, respectively. The number of their linear factors (mod $p$) is determined in terms of class numbers of the imaginary quadratic fields $\mathbb{Q}(\sqrt{-dp})$, where $d \in \{1,2,3\}$. 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 $P_{(p-\bar e)/3}(x)$ 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 $h_p(x) \equiv P_n^{(\pm1/4,0)}(1-2x)$ were first discovered using artificial intelligence.