I agree with GH. An approach to a problem which misses a lot of special cases does not look very promising. Yes, there exists a theoretical possibility that the "algebraic" RH and RH for Maass forms are fundamentally different conjectures and should be treated separately. But for now I do not see any reason to suspect this.

I think it is an interesting question if one can make an elliptic curve flat by embedding it into the projective space of any dimension. Personally, I doubt it.

Yes, I meant the analytic topology, of course. Actually, I was pretty sure that these cohomology coincide, though did not know where to look for this. Thanks. But, what I am looking for are not some functorial properties but nontrivial results PUBLISHED somewhere. To read it!

It is true for $n=1$. I am sure it is not true for $n\gt 1$, but I do not know how to make a counterexample.

Thank you. This is not bad, though a bit too categorical to my taste. I am not satisfied with this (and other) sources for two reasons: 1) They ignore the nontorsion part in $H^2(X,\mathcal{O}^{\times})$. 2) They never go further, to $H^3(X,\mathcal{O}^{\times})$. I have some suspicion that for $H^3$ the things are completely different.

Perhaps it is better (less hard) to work with etale topology, but I am interested in transcendental methods. I think you are right that the etale cohomology is torsion, but the Cech one certainly is not.

And, apparently, all they care about is the torsion.

Indeen, after little googling I found some papers about this. Though, in the most of them the cohomology is etale.

To clarify the question: $H^n$ there is the Cech cohomology group (in the strong topology). Thank you for the idea, but if there exists an analog of the theorem of the cube, I would prefer to read about it. To get it for myself may be not so easy.

Thank you. But I would like to have some examples at hand. And, what about $n=3$?

I believe we cannot judge the value of 7th problem properly until the Schanuel's conjecture is resolved.

Thank you, Andreas. This is nice! Why didn't I see it before?

Lucas, If I got you right, you suggest to enumerable the statements which are equivalent to Con. I suspect that there are statements which are strictly weaker then Con but still undecidable.