## Abstract

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, O_{c}) ⊗_{k} Ω^{1}_{k/Z})δ(Γ(C, Ω^{1}_{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 Ω^{1}_{k/Z} spanned by the absolute differential dj. Thus, we can interpret the logarithmic derivative as a map dlog : E(k) → Ω^{1}_{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 y^{2} = x^{3}+ a_{4}x + a_{6}. Let k_{0} be the prime field of k and let F be the algebraic closure in k 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.

Original language | English (US) |
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Pages (from-to) | 21-38 |

Number of pages | 18 |

Journal | Journal of Pure and Applied Algebra |

Volume | 138 |

Issue number | 1 |

DOIs | |

State | Published - May 7 1999 |

Externally published | Yes |

## ASJC Scopus subject areas

- Algebra and Number Theory