Asymptotic Variation of Elementary Abelian p-Extensions over $P^1$

Abstract

Let A^d denote the coefficient space of all degree-d polynomials f in one variable for some d\ge 3. For any \bar{f} in A^d(\bar\F_p), a rank-\ell Artin-Schreier curve X_{\bar{f},\ell}: y^{p^\ell}-y= \bar{f} is called ordinary if its normalized Newton polygon achieves the infimum in A^d(\bar\F_p). Given \ell and a number field K, we show that there exists a Zariski dense open subset U in A^d, defined over Q, such that if f in U(K) then X_{(f\bmod \wp),\ell} is ordinary for all primes $\wp|p$ with deg(\wp) in \ell\Z and p large enough.