Iterated integrals, diagonal cycles and rational points on elliptic curves

Torsion properties of modified diagonal classes on triple products of modular curves

“…Darmon, Rotger, and Sols [7] have studied such points, in the broader context of Shimura curves over totally real fields, notably by computing their images under the complex Abel-Jacobi map in terms of iterated integrals. Methods have been developed by Darmon, Daub, Lichtenstein, Rotger, and Stein [5] to numerically calculate such points in the case of modular curves.…”

“…We exhibit a correspondence mapping ∆ F GKS (e) to P f g (e). When the global root number W (F) is −1, Darmon, Rotger, and Sols [7] have studied the nontorsion properties of P f g (ξ ∞ ), building on [28]. In the complementary situation when W (F) = +1, we use Theorem 1.1 and functoriality of Abel-Jacobi maps with respect to correspondences to deduce the following: Theorem 1.3 Let f and g be as above, and let F = g ⊗ g ⊗ f .…”

“…This convention is borrowed from [7] and differs from the more classical notation of [10]. Given two varieties X and Y over K, we write Corr r (X, Y ) := CH dim X+r (X × Y ).…”

“…Techniques were developed in [5] to numerically calculate Chow-Heegner points associated with modified diagonal cycles. The algorithms are based on a formula for the image of these cycles under the complex Abel-Jacobi map (4.1) proved in [7]. Most of the examples calculated in [5] concern the situation where the elliptic curve E f has algebraic rank equal to 1.…”

“…Building on the work of Yuan, Zhang, and Zhang [28], Darmon, Rotger, and Sols proved the following in [7]:…”