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Explicit Geometry on a Family of Curves of Genus 3
Author(s) -
Guàrdia J.
Publication year - 2001
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610701002538
Subject(s) - mathematics , jacobian matrix and determinant , bijection , quotient , elliptic curve , genus , homogeneous space , pure mathematics , fermat's last theorem , family of curves , geometry , mathematical analysis , combinatorics , botany , biology
An explicit geometrical study of the curves C a : Y 4 = X 4 −( a 2 + a −2 ) X 2 +1 a ∈R, a ≠0,±1 is presented. These are non‐singular curves of genus 3, defined over Q( a ). By exploiting their symmetries, it is possible to determine most of their geometric invariants, such as their bitangent lines and their period lattice. An explicit description is given of the bijection induced by the Abel–Jacobi map between their bitangent lines and odd 2‐torsion points on their jacobian. Finally, three elliptic quotients of these curves are constructed that provide a splitting of their jacobians. In the case of the curveC 1 ± 2, which is isomorphic to the Fermat curve of degree 4, the computations yield a finer splitting of its jacobian than the classical one.