Say two points x, y on an algebraic curve are in the same torsion packet if [x - y] defines a torsion element of the Jacobian. In genus 0 and 1, torsion packets have infinitely many points. In higher genus, a theorem of Raynaud states that all torsion packets are finite. It is conjectured, but not known, that there is a uniform bound on the size of a torsion packet in terms of the genus of the underlying curve. We study the tropical analogue of this construction for a metric graph. On a higher genus metric graph, torsion packets are not always finite, but they are finite under an additional "genericity" assumption on the edge lengths. Under this genericity assumption, the torsion packets satisfy a uniform bound in terms of the genus of the underlying graph.
The talk will start with a pre-seminar at 2pm:
Title: What is the Jacobian of an algebraic curve?
Abstract: A cool fact about elliptic curves is that you can add points on them. On a higher-genus algebraic curve, we can't add points to get another point on the curve, but we can add points inside the Jacobian. I will describe the construction of the Jacobian of a complex algebraic curve from a topological perspective, along with a "tropical" version of this construction for metric graphs.
Organizer: Algebra and Algebraic Geometry Seminar