Computational Algebra Seminar

Frederik Vercauteren
(Katholieke Universiteit Leuven)
Forms of Elliptic Curves and their Arithmetic
3pm–4pm, Thursday 5th February, 2009
Carlaw 535
In this talk we will review some toric geometry and how different models of elliptic curves proposed in the literature fit in this framework. Then we describe an algorithm to find good models admitting efficient arithmetic.

Gavin Brown
(University of Kent, UK)
Introduction to toric geometry: Part 2: The Cox Ring
4pm–5pm, Thursday 5th February, 2009
Carlaw 535
Part 2: Homogeneous coordinate rings of toric varieties.

It turns out that a toric variety has a naturally-associated ring of functions – it's just a polynomial ring, sometimes called the Cox ring, but it has a grading by a finitely-generated abelian group (i.e. a few ℤ-gradings and then perhaps a little cyclic group grading too). We can use this ring to define varieties inside toric varieties (just as in the usual case of varieties in affine or projective space), to define maps, and anything else you'd expect.