The modulo eight behavior of common factors of two hypergeometric polynomials arising from a genus three heperelliptic family

We study a one-parameter family of smooth projective genus-three hyperelliptic curves over algebraically closed fields of odd characteristic, with the parameter different from zero and one. We translate superspeciality into the simultaneous vanishing of two truncated hypergeometric polynomials arising from the Cartier operator. For their monic greatest common divisor, we prove squarefreeness and show that every irreducible factor has degree at most two. We then determine its behavior according to the characteristic modulo eight: in one residue class every nonconstant irreducible factor is quadratic; in two residue classes the greatest common divisor is trivial; and in the remaining residue class a distinguished linear factor always occurs.