Computations in the Poincaré torsor and the quadratic Chabauty method

Abstract:

(Joint work with Bas Edixhoven.)
Falting's theorem states that a curve C of genus g>1 defined over the rationals has a finite number of rational points. In practice anyway there is no general procedure to provably compute the set C(Q). When the rank of the Mordell-Weil group J(Q) (with J the Jacobian of C) is smaller than g we can use Chabauty method, i.e. we can embed C in J and, after choosing a prime p, we can view C(Q) as a subset of the intersection of C(Q_p) and  the closure of J(Q) inside the p-adic manifold J(Q_p); this intersection is always finite and computable up to finite precision.
Minhyong Kim has generalized this method by inspecting (possibly non-abelian) quotients of the fundamental group of C. 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 the Mordell-Weil group is strictly less than g + s -1 (with s the rank of the Neron-Severi group of J).
In the seminar we will give a reinterpretation of the quadratic Chabauty method, only using the Poincaré torsor of J and a little of formal geometry, and we will show how to make it effective.

Il seminario si terrà in presenza. Chi fosse interessato a partecipare in presenza deve scrivere a andrea.ferraguti@sns.it entro le 10:00 di Giovedì 10 marzo 2022.