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, Oc) ⊗k Ω1k/Z)δ(Γ(C, Ω1C/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 Ω1k/Z spanned by the absolute differential dj. Thus, we can interpret the logarithmic derivative as a map dlog : E(k) → Ω1k[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.
|Original language||English (US)|
|Number of pages||18|
|Journal||Journal of Pure and Applied Algebra|
|State||Published - May 7 1999|
ASJC Scopus subject areas
- Algebra and Number Theory