Computational Algebra Seminar: van Bommel -- Cluster pictures of hyperelliptic curves: introduction and implementation

Speaker: Raymond van Bommel  
Title: Cluster pictures of hyperelliptic curves: introduction and implementation  
Time & Place: 15.00-16.00, Thursday 30 April, SMRI Seminar Rm   
Abstract: Consider a hyperelliptic curve given by an equation y^2 = f(x) over
a local field of odd residue characteristic.  The so-called cluster picture of this
curve keeps track of the p-adic distances between the roots of f(x).  This framework was
originally introduced by Dokchitser, Dokchitser, Maistret, and Morgan, and has since
been further developed by numerous other people.  

On the one hand, cluster pictures can be used as a way to get a conceptual understanding
of the geometry and arithmetic of the curve.  On the other hand, they also provides a
convenient way to construct many interesting examples of hyperelliptic curves.  

In this talk, I will introduce the theory of cluster pictures and discuss
implementations in Magma (ongoing, joint with Martin Azon and Himanshu Shukla, based on
a previous implementation by Tim Dokchitser) and in SageMath (joint with Alex J.
Best).