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 $𝔣(L/K)$, is the smallest non-negative integer $n$ such that the higher unit group

$U^{(n)}=1+{𝔪}_K^n=\{u\in {𝒪}^{\times }:u\equiv 1(\mathrm{mod}{𝔪}_K^n)\}$

is contained in $N_{L/K}(L^{\times })$, where $N_{L/K}$ is field norm map and ${𝔪}_K$ is the maximal ideal of $K$.Template:Sfnp Equivalently, $n$ is the smallest integer such that the local Artin map is trivial on $U_K^{(n)}$. Sometimes, the conductor is defined as ${𝔪}_K^n$ where $n$ is as above.Template:Sfnp

The conductor of an extension measures the ramification. Qualitatively, the extension is unramified if and only if the conductor is zero,Template:Sfnp and it is tamely ramified if and only if the conductor is 1.Template:Sfnp 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 G_s is non-trivial, then $𝔣(L/K)={\eta }_{L/K}(s)+1$, where ${\eta }_{L/K}$ is the function that translates from "lower numbering" to "upper numbering" of higher ramification groups.Template:Sfnp

The conductor of $L/K$ is also related to the Artin conductors of characters of the Galois group $\mathrm{Gal}(L/K)$. Specifically,Template:Sfnp

${𝔪}_K^{𝔣(L/K)}=\underset{\chi }{\mathrm{lcm}}{𝔪}_K^{{𝔣}_{\chi }}$

where $\chi$ varies over all multiplicative complex characters of $\mathrm{Gal}(L/K)$, ${𝔣}_{\chi }$ is the Artin conductor of $\chi$, 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.Template:Sfnp However, it only depends on $L^{\mathrm{a}\mathrm{b}}/K$, the maximal abelian extension of $K$ in $L$, because of the "norm limitation theorem", which states that, in this situation,Template:SfnpTemplate:Sfnp

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

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.

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.Template:Sfnp

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 $\theta :l^m\to \mathrm{Gal}(L/K)$ 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 $𝔣(L/K)$, to be the highest common factor of all moduli for which reciprocity holds; in fact reciprocity holds for $𝔣(L/K)$, so it is the smallest such modulus.Template:SfnpTemplate:Sfnp

Example

• Taking as base the field of rational numbers, the Kronecker–Weber theorem states that an algebraic number field $K$ is abelian over $\mathbb{Q}$ if and only if it is a subfield of a cyclotomic field $\mathbb{Q}({\zeta }_n)$, where ${\zeta }_n$ denotes a primitive $n$th root of unity.Template:Sfnp 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 $n\mathrm{\infty }$ otherwise.
• Let $L/K$ be $\mathbb{Q}(\sqrt{d})/\mathbb{Q}$ where $d$ is a squarefree integer. Then,Template:Sfnp

  $𝔣(\mathbb{Q}(\sqrt{d})/\mathbb{Q})=\{\begin{array}{cc}|{\mathrm{\Delta }}_{\mathbb{Q}(\sqrt{d})}|& \text{for }d>0\\ \mathrm{\infty }|{\mathrm{\Delta }}_{\mathbb{Q}(\sqrt{d})}|& \text{for }d<0\end{array}$

where ${\mathrm{\Delta }}_{\mathbb{Q}(\sqrt{d})}$ is the discriminant of $\mathbb{Q}(\sqrt{d})/\mathbb{Q}$.

Relation to local conductors and ramification

The global conductor is the product of local conductors:

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

As a consequence, a finite prime is ramified in $L/K$ if, and only if, it divides $𝔣(L/K)$.Template:Sfnp An infinite prime $v$ occurs in the conductor if, and only if, $v$ is real and becomes complex in $L$.

References

• Template:Cite book

• Template:Cite book

• Template:Cite book

• Template:Cite book

• Template:Cite book

• Template:Neukirch ANT

• Template:Citation