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 $p\equiv 1(\mathrm{mod}d)$, the family $C_{\alpha }$ is generically ordinary at every prime above $p$.

15 pages