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

  • Date: 03/29/2007
Lecturer(s):

Nils Bruin (Simon Fraser University)

Location: 

University of British Columbia

Topic: 

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

Sponsor: 

pims