A Global Generic Locus for Elementary Abelian p-covers of P^1 with two Ramification Points

Abstract

In this paper, we study elementary abelian $p$-covers of $P^1$ totally ramified at exactly two points and unramified elsewhere. Such covers are isomorphic to the rank-$\ell$ Artin--Schreier curves $ X_{f, \ell} : y^{p^\ell} - y = f(x), $ where $f \in \bar{F}_p[x^{\pm 1}]$ has a pole of order $d_1>0$ at $0$ and order $d_2>0$ at $\infty$. After fixing $d_1$ and $d_2$, the cover $X_{f, \ell}$ is generic if its Newton polygon is minimal as $f$ varies over all Laurent polynomials with pole orders $d_1$ and $d_2$. For $p$ sufficiently large relative to $d_1$ and $d_2$, we construct a combinatorial cost polygon $GL(d_1,d_2,p)$ and prove that its attainment is characterized by the nonvanishing of a homogeneous polynomial in the coefficients of $f$. Moreover, we prove that these conditions stabilize: there exists a Zariski-open dense subset $U$ in the coefficient space that is defined over $\Q$ and depends only on $d_1$ and $d_2$, whereby for any $f$ whose coefficients lie in $U(\overline{\Q})$, the reduction is generic at every prime ideal above sufficiently large rational primes, and for every rank $\ell$ permitted by the residue field. This result extends Zhu's result for one ramification point.