Together with colleagues at the Warwick Mathematics Institute, we have two fantastic fully funded collaborative projects for a Fall 2025 start as part of the HetSys CDT at Warwick, with different cross-disciplinary teams and a common theme of multi-physics interfacial fluid dynamics (links complete with embarrassing videos!):
16:00
3-descent on genus 2 Jacobians using visibility
Abstract
We show how to explicitly compute equations for everywhere locally soluble 3-coverings of Jacobians of genus 2 curves with a rational Weierstrass point, using the notion of visibility introduced by Cremona and Mazur. These 3-coverings are abelian surface torsors, embedded in the projective space $\mathbb{P}^8$ as degree 18 surfaces. They have points over every $p$-adic completion of $\mathbb{Q}$, but no rational points, and so are counterexamples to the Hasse principle and represent non-trivial elements of the Tate-Shafarevich group. Joint work in progress with Tom Fisher.