The aim of this paper is to prove that an algebraic variety over a field can be desingularized locally along a valuation after a purely inseparable alteration. Zariski was first to study the problem of desingularizing algebraic varieties along valuations. He called this problem local uniformization of valuations and observed that it should be considered as the local part of the desingularizatio...

Theorem 15.1. Let E/Fq be an elliptic curve over a finite field, and let πE be the Frobenius endomorphism of E. Then E is supersingular if and only if trπE ≡ 0 mod p. Proof. Let q = pn and let π be the p-power Frobenius map π(x, y) = (xp, yp) (note that π is an isogeny, but not necessarily an endomorphism, since E need not be defined over Fp). We have π̂π = [p], where [p] denotes the multiplicat...