The Poincaré torsor and the quadratic Chabauty method
Guido Lido (Università di Roma "Tor Vergata" e Univ. Leiden)

Let C be a curve of genus g>1 defined over the rationals and let J be its Jacobian. Faltings's theorem states that C has only finitely many rational points, but in practice there is no general procedure to provably compute the set C(Q).

When the rank of J(Q) is smaller than g we can use Chabauty's method: embedding C in J the set C(Q) is a subset of the intersection of C(Qp) and the closure of J(Q) inside the p-adic manifold J(Qp); since this intersection isfinite and computable up to finite precision we can use it to compute C(Q).

Minhyong Kim has generalized this method inspecting the (Qp-prounipotent etale) fundamental group of C and his ideas have been made effective in some new cases by Balakrishnan, Dogra, Muller, Tuitman and Vonk: their "quadratic Chabauty method" works when the rank of J(Q) is smaller than g+s-1 (with s the rank of the Neron-Severi group of J).

In this seminar we describe a reinterpretation of the quadratic Chabauty method that does not need the fundamental group of C but uses only some geometry over the integers and the Gm-torsor associated to the Poincaré bundle over J.

This work is in collaboration and under the supervision of Bas Exidhoven.