Dear Loic,

I just noticed an assertion, which is attributed to you, in a recent 
paper of Frey and Muller: "Arithmetic of Modular Curves and Applications"
(http://www.exp-math.uni-essen.de/zahlentheorie/preprints/Index.html)
I think the statement is false, so I wanted to point it out to you in
case someone asks you about in the future.   The statement is:

  "One can show that the relative homology group splits into three
   direct summands [Mer94]
          H_1(X_0(N),cusp,Z) = Eis(N) oplus S_2(N) oplus Sbar_2(N)."

You show this, but with ring Z replaced by the complex numbers. 
I think it is false over Z.  For example, let N=37.  Then 
  [H_1(X_0(N),cusp,Z) : Eis(N) + S_2(N) + Sbar_2(N)] = 6.
In general, the index is always divisible by the number of real 
components of J_0(N).

Best,
  William