Computational Algebra Seminar: Bruin -- Bounding Mordell-Weil ranks of Jacobians of smooth plane quartics

Speaker: Nils Bruin
Title: Bounding Mordell-Weil ranks of Jacobians of smooth plane quartics
Time & Place: 3-4pm, Thursday 9 July, Carslaw 535

Abstract:
Many methods for solving diophantine questions rely on knowing
the group of rational points on an abelian variety. In only very few cases
this is know to be a solvable problem at all. We do not even have an
algorithm that is guaranteed to work for all elliptic curves (one
dimensional abelian varieties).

We do have quite a collection of methods for elliptic curves that work
quite well in many practical cases. We can also handle Jacobians of
hyperelliptic curves in a fair number of practical settings. This
motivated us to try Jacobians of non-hyperelliptic curves. The simplest
examples occur as genus 3, smooth plane quartic curves.

We have mixed success. There are several theoretical and computational
obstacles, although most theoretical obstructions can be replaced with
much worse computation ones.

Ingredients we need:
 - Groebner basis and resultant computations (these are not a bottleneck)
 - Rings of integers of degree 28 algebras with very large discriminants
 - S-unit groups of number fields up to degree 28
 - Identification of galois groups and decomposition groups in Sp(6,GF(2))
 - Cohomology of several Sp(6,GF(2))-modules over GF(2)
 - Computations in p-adic extensions

In other words, we are using a very large part of magma and in many cases
(judging from the bugs we ran into), drive it a little beyond its comfort
range.

This will be more a progress report than a polished presentation and I
will try to emphasise the interaction with magma we experienced.

This is joint work with Victor Flynn, Bjorn Poonen and Michael Stoll.