The theme of this talk is the connection between the pro-unipotent fundamental group of a pointed modular curve , algebraic cycles, and special values of -functions.
The extension of mixed Hodge structures arising in the second stage in the lower central series of gives rise to a supply of points on the Jacobian
of , indexed by Hodge cycles on the surface .
I will explain how these points can be computed in practice and how are related to the image of the diagonal in under the (complex, étale or -adic de Rham) Abel-Jacobi map.
When combined with a formula of Gross-Zagier type for triple product -functions obtained by X. Yuan, S. Zhang and
W. Zhang, this yields a criterion, in terms of the leading terms of certain L-series attached
to modular forms, for these points to be of infinite order.
This reports on a joint work with H. Darmon (partly in collaboration with M. Daub, S. Lichstenstein, I. Sols and W. Stein).