(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.