The unipotent Albanese map and Selmer varieties for curves

We study the unipotent Albanese map that associates the torsor of paths for p-adic fundamental groups to a point on a hyperbolic curve. It is shown that the map is very transcendental in nature, while standard conjectures about the structure of mixed motives provide control over the image of the map. As a consequence, conjectures of ‘Birch and Swinnerton-Dyer type ’ are connected to finiteness theorems of Faltings-Siegel type. In a letter to Faltings [16] dated June, 1983, Grothendieck proposed several striking conjectural connections between the arithmetic geometry of ‘anabelian schemes ’ and their fundamental groups, among which one finds issues of con-siderable interest to classical Diophantine geometers. Here we will trouble the reader with a careful formulation of just one of them. Let F be a number field and f: X→Spec(F) a smooth, compact, hyperbolic curve over F. After the choice of an algebraic closur