Construction of Generically Ordinary Families of Hyperelliptic Curves

Abstract

Katz conjectured in a 2018 lecture that the family of curves $y^2=x^d-dx+t$ over the $t$-line is generically ordinary for all sufficiently large primes $p$. We prove that, for every $g\ge 2$ and every nonzero algebraic integer $\alpha$, the genus-$g$ families $C_\alpha: y^2=x^d+\alpha x+t$ where $d\in\{2g+1, 2g+2\}$ are generically ordinary at every prime $p>P^+(d)$, provided that $\alpha$ is nonzero modulo every prime above $p$. The bound $P^+(d)=d^2-4d+2$ if $d$ is odd, and $P^+(d)=(d^2-3d+2)/2$ if $d$ is even. Finally, we show that for every algebraic integer $\alpha$ and every prime $p\equiv 1 \pmod d$, the family $C_\alpha$ is generically ordinary at every prime above $p$.