Iterated integrals, diagonal cycles and rational points on elliptic curves

The theme of this article is the connection between the pro-unipotent fundamental group π1(X; o) of a pointed algebraic curve X, algebraic cycles, and special values of Lfunctions. The extension of mixed Hodge structures arising in the second stage in the lower central series of π1(X; o) gives rise to a supply of complex points on the Jacobian Jac(X) of X, indexed by Hodge cycles on X ×X. The main result of this note relates these points to the Abel-Jacobi image of Gross-Kudla-Schoen’s modified diagonal ∆ in X. When combined with the recent formula 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 when X is a modular or Shimura curve.