On arithmetic intersection numbers on self-products of curves

We give a close formula for the N\'eron-Tate height of tautological integral cycles on Jacobians curves over number fields as well new lower bound arithmetic self-intersection $\hat{\omega}^2$ dualizing sheaf curve in terms Zhang's invariant $\varphi$. As an application, we obtain effective Bogomolov-type result cycles. deduce these results from more general combinatorial computation intersection numbers adelic line bundles higher self-products curves, which are linear combinations pullbacks and diagonal bundle.