Non-conservative function fields of genus one II

Classification of Enriques surfaces with finite automorphism group in characteristic 2

“…Note that the case (3) in Queen [33,Theorem 2] does not occur in our case, because the transcendental degree of K = k(s) over k is 1. As for the relative generalized Jacobians of these quasi-elliptic surfaces, Queen [34,Theorem 1] showed the following:…”

Mordell's equation in characteristic three

“…Before stating our theorem let is introduce some notation. Let C be the curve y 2 -x 3 + a, a e K\K 3 and denote by C(L) the L-rational points of C for L D K, any field. Since C is a cubic we can define a group law on the nonsingular (projective) points of C(-ft") by the usual chord and tangent method.…”

“…Since C is a cubic we can define a group law on the nonsingular (projective) points of C(-ft") by the usual chord and tangent method. The singular point of C is (-a'' 3 ,0) which is not on C(K), thus C(K) is a group with identity O, the point at infinity. Finally, since a £ K 3 Differentiating the defining equation y 2 = x 3 +a, we get 2ydy/da = 1, so /x(P) ^ 0 for P ^ O, and n is injective.…”

“…The singular point of C is (-a'' 3 ,0) which is not on C(K), thus C(K) is a group with identity O, the point at infinity. Finally, since a £ K 3 Differentiating the defining equation y 2 = x 3 +a, we get 2ydy/da = 1, so /x(P) ^ 0 for P ^ O, and n is injective. Now, let v be a place of K and P -(x,y) G C(K).…”