Asymptotic variation of elementary abelian p-extensions over P¹

Let A^d denote the coefficient space of all degree-d polynomials f in one variable for some d≥3. For any f‾∈A^d(F‾_p), a rank-ℓ Artin-Schreier curve X_f‾,ℓ:y^pℓ−y=f‾ is called ordinary if its normalized Newton polygon achieves the infimum in A^d(F‾_p). Given ℓ 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∈U(K) then X_{(fmod℘),ℓ} is ordinary for all primes ℘|p with deg⁡(℘)∈ℓZ and p large enough.