# Conductor (class field theory)

In algebraic number theory, the conductor of a finite abelian extension of local or global fields provides a quantitative measure of the ramification in the extension. The definition of the conductor is related to the Artin map.

## Local conductor

Let L/K be a finite abelian extension of non-archimedean local fields. The conductor of L/K, denoted ${\displaystyle {\mathfrak {f}}(L/K)}$, is the smallest non-negative integer n such that the higher unit group

${\displaystyle U^{(n)}=1+{\mathfrak {m}}^{n}=\left\{u\in {\mathcal {O}}^{\times }:u\equiv 1\,(\operatorname {mod} {\mathfrak {m}}_{K}^{n})\right\}}$

is contained in NL/K(L×), where NL/K is field norm map and ${\displaystyle {\mathfrak {m}}_{K}}$ is the maximal ideal of K.[1] Equivalently, n is the smallest integer such that the local Artin map is trivial on ${\displaystyle U_{K}^{(n)}}$. Sometimes, the conductor is defined as ${\displaystyle {\mathfrak {m}}_{K}^{n}}$ where n is as above.[2]

The conductor of an extension measures the ramification. Qualitatively, the extension is unramified if, and only if, the conductor is zero,[3] and it is tamely ramified if, and only if, the conductor is 1.[4] More precisely, the conductor computes the non-triviality of higher ramification groups: if s is the largest integer for which the "lower numbering" higher ramification group Gs is non-trivial, then ${\displaystyle {\mathfrak {f}}(L/K)=\eta _{L/K}(s)+1}$, where ηL/K is the function that translates from "lower numbering" to "upper numbering" of higher ramification groups.[5]

The conductor of L/K is also related to the Artin conductors of characters of the Galois group Gal(L/K). Specifically,[6]

${\displaystyle {\mathfrak {m}}_{K}^{{\mathfrak {f}}(L/K)}=\operatorname {lcm} \limits _{\chi }{\mathfrak {m}}_{K}^{{\mathfrak {f}}_{\chi }}}$

where χ varies over all multiplicative complex characters of Gal(L/K), ${\displaystyle {\mathfrak {f}}_{\chi }}$ is the Artin conductor of χ, and lcm is the least common multiple.

### More general fields

The conductor can be defined in the same way for L/K a not necessarily abelian finite Galois extension of local fields.[7] However, it only depends on Lab/K, the maximal abelian extension of K in L, because of the "norm limitation theorem", which states that, in this situation,[8][9]

${\displaystyle N_{L/K}(L^{\times })=N_{L^{\text{ab}}/K}\left((L^{\text{ab}})^{\times }\right).}$

Additionally, the conductor can be defined when L and K are allowed to be slightly more general than local, namely if they are complete valued fields with quasi-finite residue field.[10]

### Archimedean fields

Mostly for the sake of global conductors, the conductor of the trivial extension R/R is defined to be 0, and the conductor of the extension C/R is defined to be 1.[11]

## Global conductor

### Algebraic number fields

The conductor of an abelian extension L/K of number fields can be defined, similarly to the local case, using the Artin map. Specifically, let θ : Im → Gal(L/K) be the global Artin map where the modulus m is a defining modulus for L/K; we say that Artin reciprocity holds for m if θ factors through the ray class group modulo m. We define the conductor of L/K, denoted ${\displaystyle {\mathfrak {f}}(L/K)}$, to be the highest common factor of all moduli for which reciprocity holds; in fact reciprocity holds for ${\displaystyle {\mathfrak {f}}(L/K)}$, so it is the smallest such modulus.[12][13][14]

#### Example

• Taking as base the field of rational numbers, the Kronecker–Weber theorem states that an algebraic number field K is abelian over Q if and only if it is a subfield of a cyclotomic field ${\displaystyle \mathbf {Q} (\zeta _{n})}$, where ${\displaystyle \zeta _{n}}$ denotes a primitive nth root of unity.[15] If n is the smallest integer for which this holds, the conductor of K is then n if K is fixed by complex conjugation and ${\displaystyle n\infty }$ otherwise.
• Let L/K be ${\displaystyle \mathbf {Q} ({\sqrt {d}})/\mathbf {Q} }$ where d is a squarefree integer. Then,[16]
${\displaystyle {\mathfrak {f}}\left(\mathbf {Q} ({\sqrt {d}})/\mathbf {Q} \right)={\begin{cases}\left|\Delta _{\mathbf {Q} ({\sqrt {d}})}\right|&{\text{for }}d>0\\\infty \left|\Delta _{\mathbf {Q} ({\sqrt {d}})}\right|&{\text{for }}d<0\end{cases}}}$
where ${\displaystyle \Delta _{\mathbf {Q} ({\sqrt {d}})}}$ is the discriminant of ${\displaystyle \mathbf {Q} ({\sqrt {d}})/\mathbf {Q} }$.

#### Relation to local conductors and ramification

The global conductor is the product of local conductors:[17]

${\displaystyle \displaystyle {\mathfrak {f}}(L/K)=\prod _{\mathfrak {p}}{\mathfrak {p}}^{{\mathfrak {f}}(L_{\mathfrak {p}}/K_{\mathfrak {p}})}.}$

As a consequence, a finite prime is ramified in L/K if, and only if, it divides ${\displaystyle {\mathfrak {f}}(L/K)}$.[18] An infinite prime v occurs in the conductor if, and only if, v is real and becomes complex in L.

## Notes

1. ^ Serre 1967, §4.2
2. ^ As in Neukirch 1999, definition V.1.6
3. ^ Neukirch 1999, proposition V.1.7
4. ^ Milne 2008, I.1.9
5. ^ Serre 1967, §4.2, proposition 1
6. ^ Artin & Tate 2009, corollary to theorem XI.14, p. 100
7. ^ As in Serre 1967, §4.2
8. ^ Serre 1967, §2.5, proposition 4
9. ^ Milne 2008, theorem III.3.5
10. ^ As in Artin & Tate 2009, §XI.4. This is the situation in which the formalism of local class field theory works.
11. ^ Cohen 2000, definition 3.4.1
12. ^ Milne 2008, remark V.3.8
13. ^ Janusz 1973, pp. 158,168–169
14. ^ Some authors omit infinite places from the conductor, e.g. Neukirch 1999, §VI.6
15. ^ Manin, Yu. I.; Panchishkin, A. A. (2007). Introduction to Modern Number Theory. Encyclopaedia of Mathematical Sciences. 49 (Second ed.). pp. 155, 168. ISBN 978-3-540-20364-3. ISSN 0938-0396. Zbl 1079.11002.
16. ^ Milne 2008, example V.3.11
17. ^ For the finite part Neukirch 1999, proposition VI.6.5, and for the infinite part Cohen 2000, definition 3.4.1
18. ^ Neukirch 1999, corollary VI.6.6