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I think there must be an upper bound for A somewhere. The step troubling you should be the triangle inequality applied, but to conclude that the term you discard is at most 1/*p* you do need something.
Yes, I do mean the Talk page on Wikipedia (please do not call it "Wiki"), since the point you raise is expository, and if the article needs to be improved, that is what should happen. It is also typical of Wikipedia that heuristic remarks get taken out because one person doesn't find them helpful, when maybe ten others with a different background would find them helpful. The root of unity is of form σ(F)/F where F is in the function field and σ is in the Galois group. That is exactly how roots of unity occur in Kummer theory.
A π-adic expansion works just like a Taylor expansion, and here the sin and tan terms are of order 1, while the square root of p is of order (p - 1)/2.
The root of w(P,Q) is defined as the ratio of two functions. What you actually need to know here is the existence theorem for a function with a given divisor on the curve: the Abel-Jacobi map in this case, but the basic result is in any book on classical elliptic functions, i.e. the divisors of functions have the same number of zeroes and poles, and when you "sum up" the divisor on the curve you get 0. Write down a divisor determined by P and Q, note that it is the same when you translate it by P, and so the function changes by a constant multiple. But please write on the Talk page!
Some contemporary problems with origins in the Jugendtraum is easy to find online. Langlands says that Hilbert's 12th problem has been neglected, whether justly or not, and then goes on to talk about other stuff.
Shrug. You can treat my comment as a straw man if you insist. It doesn't mention "provability" at all. If you want a paraphrase of the whole thought, it would be that the hierarchy of results that matters to the mainstream tradition of number theory is no kind of logical hierarchy.
This link uses it in that kind of fashion: mathworld.wolfram.com/GoedelsFirstIncompletenessTheorem.html Of course the link there is to number theory as "the higher arithmetic" instead. While I can't prove that a logician wrote that text, it's what I meant.
