Remark on fundamental groups and effective Diophantine methods for hyperbolic curves

In a few earlier papers ([8], [9], [10]) attention was called to the striking parallel between the ideas surrounding the well-known conjecture of Birch and Swinnerton-Dyer for elliptic curves, and the mysterious section conjecture of Grothendieck [6] that concerns hyperbolic curves. We wish to explain here some preliminary ideas for ‘effective non-abelian descent’ on hyperbolic curves equipped with at least one rational point. We again follow in an obvious manner the method of descent on elliptic curves and, therefore, rely on conjectures. In fact, the main point is to substitute the section conjecture for the finiteness of the Shafarevich group. That is to say, the input of the section conjecture is of the form