Abstract. We explicitly write down the Eisenstein elements inside the space of modular symbols for Eisenstein series with integer coefficients for the congruence subgroups Γ 0 (pq) with p and q distinct odd primes, giving an answer to a question of Merel in these cases. We also compute the winding elements explicitly for these congruence subgroups. Our results are explicit versions of the Manin-Drinfeld Theorem [Thm. 9].

We explicitly write the Eisenstein elements inside the space of modular symbols corresponding to each Eisenstein series for the congruence subgroup Γ 0 (p 2 ), answering a question of Merel. As a consequence, we also write the winding element explicitly for the congruence subgroup Γ 0 (p 2 ).

We compute an asymptotic expression for the Arakelov self-intersection number of the relative dualizing sheaf of Edixhoven's minimal regular model for the modular curve X 0 (p 2 ) over Q. The computation of the self-intersection numbers are used to prove an effective version of the Bogolomov conjecture for the semi-stable models of modular curves X 0 (p 2 ) and obtain a bound on the stable Faltings height for those curves in a companion article [6].