AI 中文总结
研究奇特征代数闭域上单参数亏格为三超椭圆曲线族，将超特殊性质转化为超几何多项式消失，证明其首一最大公因式无平方因子且不可约因子次数至多为二，确定其模八特征下的行为。
AI 中文摘要
我们研究在奇特征代数闭域上的一个单参数光滑射影亏格为三的超椭圆曲线族，参数不为零和一。我们将超特殊性质转化为卡蒂尔算子产生的两个截断超几何多项式的同时消失。对于它们的首一最大公因式，我们证明其无平方因子，并表明每个不可约因子的次数至多为二。然后我们根据模八的特征确定其行为：在一个剩余类中每个非常数不可约因子是二次的；在两个剩余类中最大公因式是平凡的；在其余剩余类中总是出现一个特殊的线性因子。
英文摘要
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.
