Patrick Morton

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

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 $P_n^{(\alpha,0)}(1-2x)$, for $\alpha = \pm 1/4, \pm 1/...