The **conductor** of an Artin representation is a positive integer that measures its ramification. It can be expressed as a product of local conductors.

Let $K/\Q$ be a Galois extension and $\rho:\Gal(K/\Q)\to\GL_n(\C)$ an Artin representation. Then the conductor of $\rho$ is $ \prod_p p^{f(\rho,p)} $ for non-negative integers $f(\rho,p)$, where the product is taken over prime numbers $p$.

To define the exponents $f(\rho,p)$, fix a prime $\mathfrak{p}$ of $K$ above $p$ and consider the corresponding extension of local fields $K_{\mathfrak{p}}/\Q_p$ with Galois group $G$. Then $G$ has a filtration of higher ramification groups in lower numbering $G_i$, as defined in Chapter IV of Serre's _Local Fields [MR:0554237, 10.1007/978-1-4757-5673-9]. In particular, $G_{-1}=G$, $G_0$ is the inertia group of $K_\mathfrak{p}/\Q_p$, and $G_1$ is the wild inertia group, which is a finite $p$-group.

Let $g_i = |G_i|$. Then \[ f(\rho, p) = \sum_{i\geq 0} \frac{g_i}{g_0} (\chi(1) - \chi(G_i))\] where $\chi$ is the character of the representation $\rho$ and $\chi(G_i)$ is the average value of $\chi$ on $G_i$. Since $\rho$ maps to $\GL_n(\C)$, $\chi(1)=n$.

Note that if $p$ is unramified in $K$, then $f(\rho,p)=0$ and conversely, if $\rho$ is faithful and $p$ is ramified in $K$, then $f(\rho,p)>0$.

**Authors:**

**Knowl status:**

- Review status: beta
- Last edited by Andrew Sutherland on 2019-05-10 11:11:55

**Referred to by:**

- artin.2.163.8t12.a.a.bottom
- artin.2.23.3t2.b.a.bottom
- artin.2.785.24t138.b.a.bottom
- artin.label
- cmf.galois_representation
- lmfdb/artin_representations/main.py (line 436)
- lmfdb/artin_representations/templates/artin-representation-galois-orbit.html (line 9)
- lmfdb/artin_representations/templates/artin-representation-index.html (line 20)
- lmfdb/artin_representations/templates/artin-representation-search.html (line 18)
- lmfdb/artin_representations/templates/artin-representation-show.html (line 9)
- lmfdb/number_fields/templates/nf-show-field.html (line 265)

**History:**(expand/hide all)

- 2019-05-10 11:11:55 by Andrew Sutherland
- 2019-05-10 10:29:45 by Andrew Sutherland
- 2019-05-10 10:29:08 by Andrew Sutherland
- 2019-05-09 14:50:26 by John Jones
- 2019-05-09 14:49:06 by John Jones
- 2019-05-08 19:39:47 by John Jones
- 2012-06-25 12:46:45 by John Jones

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