Comments on your prop, in 30 minutes flat!

1. Now that the Shimura-Taniyama-Weil conjecture has been proved,
                                ^^^^^^

Including Weil is "politically risky"!   What if Serge Lang is deciding
on whether or not you get an NSF?    Maybe you are including Weil because
you're in France right now.  The NSF is in the US.

2. A lot of progress has been made on 
   ^^^^^^

Replace by "Much progress"

By the way, nice intro paragraph!

3. "et al.; but a"

"but" is a correlative conjunction so the semicolon can be replaced by
a comma.

4. Your definition of Tamagawa number is a little misleading, because
you only define | G | to mean "order" for finite groups.  The
component group is a group SCHEME and the order of that group is
#Phi_p(A)(Fbar_p).

5. "It is known that $L_A(1)/\Omega_A$ is a 
   rational number~\cite{shimura:onperiods}"

Where, EXACTLY, in that paper does he deduce that result?  He seems
to deduce lots of related things, e.g., in his Section 2, but I don't
see your assertion.   It might be better to say that results of Shimura
imply that L/Om in Q, unless I'm missing something.

Skipping to sections 1.3 and 1.5:

6. I don't see Te/I_fe = Phi_{A_f}(e), i.e., why 
       ker (Te ----> Phi_{A_f}(Te)) is
   exactly I_fe.  It's probably true, but I don't see it immediately.


7. "Is it a good idea to state such results 
    as a theorem in a research proposal?}"

   I think that it is.

8. "... we prove that"
 
   I usually write something more like either
   "the applicant and Agashe proved that"...
   "Agashe and I proved that"...

9. "As~$c_\infty$ is known to be a power of~$2$, this 
    is strong evidence towards formula~(\ref{af2})."

   HUH!?  We only know that c_inf is a 2-power in case the
   level is SQUARE-FREE!!!!

10. "and the latter lattice index is itself the order
     of a module over $\T / I_f$;"
  
   HUH!?  Which module?  A fractional ideal maybe.

11. "...to check with of the above two possibilities..."
             ^^^^^^

    It's OK to put my name there.   Yes, you get Hecke operators
    from the Method of Graphs.   It would be interesting to investigate
    whether your module (see 10 above) is principal or not.  
    Computing the T-module structure of Sha might be possible 
    *when Sha is visible* (new idea!).

12. > \subsubsection{Is $\Mid \Sha_{A_f} \miD$ a perfect square?} 
  
    a) "Later B.~Poonen communicated to us an example where this gcd is
       not one. However in that example, the endomorphism ring of the
       abelian variety was~$\Z$, whereas Shimura quotients have large
       endomorphism rings."
       I think I found a quotient of J_1(43*7^2) with non-Z endomorphism
       ring where Bjorn proved that the gcd is divisible by 3.
    b) I have found an example of a quotient of J_1(43*7^2) for which 
       Sha is finite and BSD ==> #Sha[3] = 3, hence oddpart(#Sha) is 
       not square. 
    c) I do not conjecture that oddpart(#Sha(A_f)) is square for optimal 
       quotients of J_0(N) of rank 0, in general, even though I've
       not found a counterexample yet.    However, it's still possible
       that oddpart() is square.
    d) See my paper
          http://modular.fas.harvard.edu/papers/bigsha/


13."Our strategies mentioned above  might show that the 
conjectural order of $\Sha_{J_e}$ as predicted by the BSD formula"
    It would be more accurate to say 
"might show that the BSD conjecture about rank implies the
 conjectural order of $\Sha_{J_e}$ is as predicted by the BSD formula."

Skipping to 1.5:

14. "It is known that that $c_A$ is an"
                ^^^^^^^^^^^

15. "One still does not have a direct generalization of
the result of~\cite{abbes-ullmo} which would say..."
    You are specifically referring to the prime p=2 here.  Say so.


OK.  Overall, it looks good.  If you keep working at polishing your
research proposal, I bet you'll get a grant.