Algorithms for Algebraic Curves

Definition Let K be a field. An elliptic curve over K is a pair (X, O) where X is a smooth absolutely integral proper curve over K with genus one and O is a K-rational point on X. Terminology (see Liu [Liu] and Stichtenoth [Sti]) A curve over K is an equidimensional variety over K of dimension one. A variety over K is a separated scheme of finite type over Spec(K). Examples A triangle with equation XY Z = 0, a singular cubic with equation Y 2 Z = X 2 (X − Z), a conic with equation X 2 − Y 2 = Z 2 , two disjoint lines X 2 − Z 2 = 0, a double line Y 2 = 0. Counter example : the union of a point and a line. Terminology A morphism of schemes is proper iff it is separated, of finite type, universally closed. A scheme is proper if it is proper over Z. Example : the projective line. Counter example : the affine line. Integral means that O X (U) is an integral domain for every non-empty open set U. This is equivalent to irreducible and reduced. Counter example : the double line, the triangle. Absolutely integral means that X remains integral after base change to ¯ K. Counter example : x 2 + y 2 = 1. The arithmetic genus p a (X) is 1 − χ(O X) = dim H 1 (O X). The geometric genus g is dim H 0 (X, ω X/K). For an absolutely integral smooth proper curve, these two spaces are dual. So they have the same dimension. See Stichtenoth [Sti] for the duality between adeles and differential forms. A map of ringed spaces if a continuous map f : X → Y plus a map of sheaves of rings O Y → f * (O X) such that the induced map on stalks O Y,f (x) → O X,x is a morphism of local rings. That is f (m f (x)) ⊂ m x or f −1 (m x) = m f (x). An open immersion f is a topological open immersion such that f x is bijective. A closed immersion f is a topological closed immersion such that f x is surjective. Equidimensional means that all the irreducible components (maximal irreducible subsets) have the same dimension (length of longest chain of closed irreducible subsets). …