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.
Cite this article
Minhyong Kim, The Unipotent Albanese Map and Selmer Varieties for Curves. Publ. Res. Inst. Math. Sci. 45 (2009), no. 1, pp. 89–133DOI 10.2977/PRIMS/1234361156