Justin Sawon : Tate-Shafarevich twists in arithmetic and complex geometry

Posted by Laura Schaposnik , part of the Departmental Colloquium.

At

March 15, 2024, 3 p.m.

In

636 SEO

Abstract

Given an elliptic curve E defined over Q, its Tate-Shafarevich group classifies torsors over E. These are genus 1 curves that have points `locally' (p-adically) but not `globally' (rational points). The same ideas apply in complex geometry if we replace the field Q by the function field of a complex manifold, e.g., CP^1. Then E become an elliptic surface fibred over CP^1 which admits a section, and its torsors are elliptic surfaces that are isomorphic to E locally over the base, but not globally. In this talk we explain how Tate-Shafarevich twists can be applied in holomorphic symplectic geometry. Given a `Lagrangian fibration', i.e., a fibration X->B whose general fibres are Lagrangian abelian varieties, Tate-Shafarevich twists can produce new examples of holomorphic symplectic manifolds, and also reveal unexpected relations between these spaces.

There will be a colloquium dinner at 7:15. If you'd like to join please email schapo@uic.edu