Plane curves with ordinary multiple points and variables ordinary nodes

5.1 Ordinary and Supersingular Curves

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...

On Finding Ordinary Intersection Points∗

An algorithm is demonstrated that finds an ordinary intersection in an arrangement of n lines in R, not all parallel and not all passing through a common point, in time O(n logn). The algorithm is then extended to find an ordinary intersection among an arrangement of hyperplanes in R, no d passing through a line and not all passing through the same point, again, in time O(n logn). Two additiona...

Rational Curves and Ordinary Differential Equations

Complex analytic 2nd order ODE systems whose solutions close up to become rational curves are characterized by the vanishing of an explicit differential invariant, and form an infinite dimensional family of integrable systems.

A THEORY OF ORDINARY P-ADIC CURVES A Theory of Ordinary p-adic Curves

Ordinary plane models and completely split divisors

Let C be a smooth, non-hyperelliptic curve over an algebraically closed field of genus g ≥ 4. We show that the projection from the canonical model of C through (g−3) generic points on C is a birational morphism to a plane curve which has only finitely many non-ordinary tangents, that is, flexor bitangents. For smooth, non-hyperelliptic curves of a fixed genus g ≥ 4 over finite fields, we show t...