Algebra & Computation: Elliptic Curves Talk 2: Group Laws, Torsion, Derivations

We always assume k = k to be an algebraically closed field with char(k) / ∈ {2, 3}. Let E be the elliptic curve over k defined by Y 2 − X 3 − AX − B. 1 Derivations We give a quick introduction to modules of differentials. The results are simplified versions of a more general theory which is treated very well and thoroughly in [Eis94]. Your motivation for this section should be that it will be a very handy technical asset. Definition 1.1. If S is a k-algebra, and M an S-module, a k-vectorspace ho-momorphism D : S → M is called a k-linear derivation if it satisfies the Leibnitz rule ∀f, g ∈ S : D(f g) = D(f) · g + f · D(g). (1) We denote by Der k (S, M) the S-module of all k-linear derivations from S to M with scalar multiplication defined by (f · D)(g) := f · D(g). We also write Der k (S) := Der k (S, S). Fact 1.2. Let D ∈ Der k (S).