Timeline for Reference request ( Conductor of Galois representation associated to Dirichlet character)
Current License: CC BY-SA 4.0
2 events
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| Jul 8, 2019 at 14:09 | comment | added | reuns | If $N = p$ the character is ramified only at $p$ and $\Phi_p(x)$ is irreducible in $\Bbb{Q}_p$ and $v_p(\xi_p-1) = \frac{1}{\deg(\Phi_p)}v_p(\Phi_p(1)) = 1/(p-1)$ is totally ramified, $Gal(\Bbb{Q}_p(\xi_p)/\Bbb{Q}_p)=Gal(\Bbb{Q}(\xi_p)/\Bbb{Q})$ so the local Artin conductor is $f(\chi,p)=\sum_{i\ge 0}\frac{g_i}{g_0}(\chi(1)-\chi(G_i)) = \frac{g_0}{g_0}(1-0) = 1$ and the global conductor is $\prod_p p^{f(\chi,p)}=p$ | |
| Jul 8, 2019 at 7:19 | history | asked | ililiil | CC BY-SA 4.0 |