## 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.

SFU/UBC Number Theory Seminar 2007