Quadratic Chabauty and cubical torsors

The geometric quadratic Chabauty method of Edixhoven—Lido provides a p-adic obstruction to rational points coming from the geometry of $G_m$-torsors over abelian varieties. However, the locus cut out by geometric quadratic Chabauty can be smaller than the locus cut out by the original "cohomological" quadratic Chabauty method. In this talk, I will sketch some new ideas for how to modify the Edixhoven—Lido construction so as to recover exactly the cohomological quadratic Chabauty method. Time permitting, I will also explain what this tells us about the structure of the quadratic Chabauty locus when the Jacobian has rank 0.