Joint work with Bas Edixhoven. 

We present a generalization of Chabauty's method, that allows to compute the rational points on curves /$\mathbf{Q}$ when the Mordell-Weil rank is strictly smaller than $g+s-1$, where $g$ is the genus of the curve and $s$ is the rank of the Néron-Severi group of the Jacobian.

The idea is to enlarge the Jacobian by talking a $\mathbf{G}_m$-torsor over it and the algorithm ultimately consists in intersecting the integral points on the $\mathbf{G}_m$-torsor with (an image of) the $\mathbf{Z}_p$-points on the curve.

We can also view the method as a way of rephrasing the quadratic Chabauty method by Balakrishnan, Dogra, Muller, Tuitman and Vonk.