Geometric interpretation of quadratic Chabauty

Joint work with Guido Lido. Chabauty’s method to find all rational points on a curve C over Q of genus g > 1 is to intersect, for a suitable prime p, inside the p-adic Lie group J(Q_p) (with J the jacobian of C), the 1-dimensional p-adic manifold C(Q_p) with the closure of J(Q). This closure is a p-adic Lie group of dimension at most r, the rank of J(Q). If r < g then this works well. Minhyong Kim has a program called `nonabelian Chabauty', where deeper quotients of the fundamental group of C are exploited (J corresponds to the abelianisation). The recently developed `quadratic Chabauty method' (Balakrishnan, Dogra, Muller, Tuitman, Vonk) can treat cases where r is larger and J has sufficiently many symmetric endomorphisms, notably the `cursed curve'. In this lecture I will give a geometric description of the quadratic Chabauty method in terms of the Poincare torsor on J times its dual.