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

In this paper, we study elementary abelian p-covers of P¹ totally ramified at exactly two points and unramified elsewhere. Such covers are isomorphic to the rank- Artin--Schreier curves Xf, : yp^ - y = f(x), where f ∈ F̄ₚ[x± 1] has a pole of order d₁>0 at $0$ and order d₂>0 at ∞. After fixing d₁ and d₂, the cover Xf, is generic if its Newton polygon is minimal as f varies over all Laurent polynomials with pole orders d₁ and d₂. For p sufficiently large relative to d₁ and d₂, we construct a combinatorial cost polygon GL(d₁,d₂,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 and depends only on d₁ and d₂, whereby for any f whose coefficients lie in U(̄), the reduction is generic at every prime ideal above sufficiently large rational primes, and for every rank permitted by the residue field. This result extends Zhu's result for one ramification point.