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EXPLICIT GEOMETRY ON A FAMILY OF CURVES OF GENUS 3

Published online by Cambridge University Press:  30 October 2001

J. GUÀRDIA
J. GUÀRDIA
Affiliation:
Departament de Matemàtica Aplicade IV, Escola Universitaria Politècnica de Vilanova i la Geltrú, Avenida Víctor Balaguer s/n, E-08800 Vilanova i la Geltrú, Spain; guardia@mat.upc.es
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Abstract

An explicit geometrical study of the curves

[formula here]

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 curve [Cscr ]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.

Type
Research Article
Information
Journal of the London Mathematical Society , Volume 64 , Issue 2 , October 2001 , pp. 299 - 310
Copyright
The London Mathematical Society 2001

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