Elliptic curves and logarithmic derivatives Original Research Article

Let C be a curve with Jacobian variety J defined over an arbitrary field k. In this paper, we show that the logarithmic derivative induces a natural homomorphism from the group J(k) of k-rational points on J into the group image, where δ is a connecting homomorphism in a natural sequence of Zariski cohomology groups. When C = E is an elliptic curve with j-invariant equal to j, we show that the image of δ is the k-vector subspace of Ωk/z1 spanned by the absolute differential dj. Thus, we can interpret the logarithmic derivative as a map dlog :E(k) → Ωk[j]/z1. Finally, we compute the kernel of this morphism explicitly. To describe the main theorem, write the Weierstrass equation of E in the form y2 = x3 + a4x + a6. Let k0 be the prime field of k and let F be the algebraic closure in k of the field k0(a4, a6). We show that the kernel of dlog can be identified with the group E(F) of F-rational points on E. In particular, notice that when k = C is the field of complex numbers, then the kernel of dlog is countable, and its image must be uncountable.