I have spent some time being confused by the nature of global methods in number theory. It seems that there are in some sense (for my purposes) three levels at which algebraic number theorists operate: local (at one prime), everywhere local (at all primes simultaneously, including the infinite ones) and global (actually playing with the number field). When we talk about things like local class field theory we mean the first one, and when we talk about global class field theory I guess we mean the last one (but the extent to which the ideles seem to get involved suggests to my naive mind a strong whiff of the second one also).


When we talk about local to global principles we most definitely mean the passage from the second to the third. I guess my question can therefore by crystallised in terms of elementary number theory as follows: what global techniques do we have for proving the non-existence of solutions to diophantine equations? In other words, given a failure of the local-global principle, what are the techniques one can use to demonstrate it independently of local information?


If I am given a diophantine equation and asked to show it has no solutions, I can think of very few methods that are not in some sense `local', certainly if we count working at the infinite primes also as local (which surely we should?). 


One possibility I have been considering is that something to do with heights/descent is perhaps a global method. However, the height of a point is still measurable by concatenating local data, and descent is normally via some trick involving congruences, but perhaps the `well-orderedness' of the process is a truly global trick.


Also, returning to a pessimistic analysis, in an important classical result that I have seen called a `measure of the failure of the local-global principle': the finiteness of the class number, it seems to me that the statement doesn't involve the infinite primes in any way, so to prove it we milk the infinite primes for all the have got and get the Minkowski bound by directly studying local behaviour above infinity. Is my opinion in this regard incorrect? If so, which of the arguments are truly global?


So to conclude, are there such things as "global methods", and if there are, what are they? Apologies for posing what is probably a naive, overly-simplistic and absurdly general question, but I am hoping several people may have thought about this and have interesting things to say.


Thanks,
Tom.



What are the truly 'global methods' in number theory?
The following example is perhaps expressed a bit loosely but it is morally correct, and it is fundamental.


Suppose that for each prime $p$ we have chosen a decomposition group $G\_p=\mathrm{Gal}(\bar{\mathbf{Q}}\_p|\mathbf{Q}\_p)\ $ of $G=\mathrm{Gal}(\bar{\mathbf{Q}}|\mathbf{Q})$. If we are given a global character $\chi:G\to\mathbf{C}^\times$, then we get by restriction a family of local characters $\chi\_p$ of $G\_p$ almost all of which are unramified in the sense of being trivial on the inertia subgroup $I\_p\subset G\_p$. Conversely, when does a family $(\chi\_p)\_p$ of local characters, almost all of which are unramified, come from a global character ?


The reciprocity isomorphism of local class field theory at the various primes allows us to attach a character of finite order $\xi\_p:\mathbf{Q}\_p^\times\to\mathbf{C}^\times\ $ to each $\chi\_p$. Since the $\chi\_p$ are almost all unramified, the $\xi\_p$ are almost all trivial on $\mathbf{Z}\_p^\times$, and hence give rise to a character $\xi:\mathbf{A}^\times\to\mathbf{C}^\times$ of the idèles of $\mathbf{Q}$. Now the condition for the $\chi\_p$ to come from a global $\chi:G\to\mathbf{C}^\times$ is that this $\xi$ should be trivial on the subgroup $\mathbf{Q}^\times\subset\mathbf{A}^\times$. In other words, $\xi$ should come from a character $\mathbf{A}^\times/\mathbf{Q}^\times\to\mathbf{C}^\times$. This is a truly global condition and cannot be expressed by any collection of local conditions. 


