This question was discussed on E. Kowalski's blog.

Here is a comment I made:

Dear Emmanuel,

You have raised an interesting issue, with (I believe) no simple answer. I think that Terry’s suggestion on how to deal with the situation is a sensible one. I might add another piece of advice. (Note that, as with Terry’s advice, this is not advice on how to address this issue in one’s writing, but rather, how one should proceed when confronted with this situation in one’s research, so as to avoid blunders.)

Most pieces of mathematics (including Weil II, for example) fit into a framework (and I don’t here mean a logical framework, but rather a narrative framework), with illustrative analogies to other parts of mathematics (in the case of the Weil conjectures, there are important analogies with algebraic topology and Hodge theory), interconnections between various results in the area, key motivations and heuristics, and so on, and one can often learn these even if learning the actual details of the arguments is out of the question.

If there is such a narrative that one can learn, I would say it is normally a good idea to learn it, since it will give one a better feeling for the results being cited, and a better feeling for how to apply them correctly. On the other hand, if such a narrative structure isn’t available, it will probably be harder to test the correctness of one’s understanding of the results, since (short of actually reading the proof), there is nothing to check against. Perhaps in such a situation, it is probably a good idea, if possible, to verify with an expert that one is really applying the result in a correct manner. Good expository literature can also help a lot (both to learn the narrative, if one is available, or at least to learn one’s way around the results that one wants to apply).

On the question of how one should phrase the citation in such a situation (of citing a result whose proof one doesn’t know): I think that having a good understanding of how to apply a result is itself a valid and important skill, whether or not one knows how to prove the result. (Similarly, we value good drivers/pilots of vehicles, as well as the engineers who build the vehicles themselves.) I don’t think that there is any intellectual dishonesty in citing a result with confidence, if one is genuinely confident that it is true (and trust in a group of established experts is a genuine and legitimate source of confidence) and one is genuinely confident that one understands the statement and the ways in which it can be applied.

On the other hand, if one doesn’t have this genuine confidence with regard to a result that one is applying in some argument, then one could be heading for a blunder, and I would say that caution is required, not just in the citation, but in the construction of the argument itself.

Just to add to this: if a result is generally certified by experts, is well-established, and widely used and understood (even if not by you personally), then there is surely no problem in quoting it, applying it, and relying on it. (As Andrew notes in his answer, if such a result does somehow collapse in the future, you will have plenty of good company with whom to commiserate about the collapse of you own work.)

On the other hand, if a result is not like this, you should be more cautious in applying it, not for any ethical reason (at least in my view), but so as to avoid having your own work built on an unstable foundation. As I write above, when you can't verify the result yourself, do your best at least to see that it fits into a reasonable narrative framework, and also try to find experts that you trust who can certify the results correctness, and that you are applying it correctly.