Thanks a lot! I will wait a bit more see if someone knows something unconditional (and do some research myself). If nothing new comes out of it, I'll accept your answer.

Well, ditto for the choice of period then (and definition of the $p$-adic $L$-function): that is to say, in my opinion, this is more an indication that the choice of Neron period is not the correct one (replace it by the canonical period) than anything else. I want my conjectures on special values of $L$-functions to depend only on the motive, but I realize that this may be purely a question of taste.

By the way, this would be a very nice result, considering its application to the structure of Hecke algebras and to Iwasawa theory.

Though this is more an indication that the arithmetic $p$-adic $L$-function as defined in the article you quote is not the correct object than anything else. Add a few years and B.Perrin-Riou (among others) had an arithmetic $p$-adic function invariant under $p$-isogeny

I think the question about why so few students is easily answered: french mathematicians had very little incentives to take students at the time (check how many students had Serre). Nonetheless, I will ask Colliot-Thélène about it.

I am guessing that "doctorate" at the time for a student of ENS meant "Thèse d'État", so something closer to an habilitation today, and hence something you should get in around 8 years.

Its second cohomology group is very close to Sha, but there are differences linked to units in the ring of integers, Tamagawa numbers and (I think) the index of Z[G] in its normalisation. This is also discussed in C.Wuthrich and R.Pollack answers.

Thanks for the nice words, but I think you are giving too much credit. The crucial point here is "assuming all conjectures". Unless you can prove Stark's conjecture for $K$ (which I hear is accessible, if not known), all the above is pretty speculative.

Whether or not p divides a_p has a consequence on D_{pst} in terms of the classification of triangulable D by Colmez, so it can be read directly from the D. Perhaps this is too trivial a remark, though.