Fake 2-descent on the Jacobian of a genus-3 curve

  • Date: 03/29/2007

Nils Bruin (Simon Fraser University)


University of British Columbia


For many questions in explicit arithmetic geometry of curves,
one needs detailed information on the rational points of the Jacobian
of the curve. A first step is to bound the free rank of the finitely
generated group that they form. For hyperelliptic curves [curves
admitting a model of the form y2 = f(x)], we have fairly good methods
for producing bounds, and curves of genus 2 are always hyperelliptic.
Curves of genus 3 (for instance smooth plane quartics) are generally
not hyperelliptic. A straightforward generalization of the standard
methods to these curves would lead to infeasible computational tasks
involving number fields up to degree 756. We propose a modification,
which requires number fields up to degree 28 and is sometimes just
about feasible.

Other Information: 

SFU/UBC Number Theory Seminar 2007