2 added 16 characters in body

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.

1

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 $\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.