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 (H¹(C, O_c) ⊗_k Ω¹_k/Z)δ(Γ(C, Ω¹_C/k)), 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 Ω¹_k/Z spanned by the absolute differential dj. Thus, we can interpret the logarithmic derivative as a map dlog : E(k) → Ω¹_k[j]/Z. 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.