Diagonal cycles, triple product L-functions and rational points on elliptic curves (Salle 1)

The theme of this talk is the connection between the pro-unipotent fundamental group $\pi_1(X; o)$ of a pointed modular curve $X$, algebraic cycles, and special values of $L$-functions. The extension of mixed Hodge structures arising in the second stage in the lower central series of $\pi_1(X; o)$ gives rise to a supply of points on the Jacobian $\mathrm{Jac}(X)$ of $X$, indexed by Hodge cycles on the surface $X^2$. I will explain how these points can be computed in practice and how are related to the image of the diagonal in $X^3$ under the (complex, étale or $p$-adic de Rham) Abel-Jacobi map. When combined with a formula of Gross-Zagier type for triple product $L$-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).