Elliptic curves and logarithmic derivatives

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 (H-1(C, C-C(n)) x(k) Omega(k/Z)(1))/delta(Gamma(C, Omega(C/k)(1))), where delta 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 delta is the fi-vector subspace of Omega(k/Z)(1) spanned by the absolute differential dj. Thus, we can interpret the logarithmic derivative as a map dlog: E(k) --> Omega(k[j]/Z)(1) Finally, we compute the kernel of this morphism explicitly. To describe the main theorem, write the Weierstrass equation of E in the form y(2) = x(3) + a(4)x + ns. Let k(o) be the prime field of k and let F be the algebraic closure in ii of the field k(0)(a(4),a(6)). 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. (C) 1999 Elsevier Science B.V. All rights reserved.