Arakelov theory, modular curves and effective Bogomolov conjecture

After we recall some background on Arakelov theory and modular curves, we discuss the computation of the self-intersection of Arakelov canonical sheaf for modular curves. We enter in more detail in the case of the modular curve X_0(N) giving, for some models, the asymptotic behaviour of the Arakelov canonical sheaf. Finally, we explain how and when this information gives an effective version of Bogomolov conjecture for curves.