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

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

is contained in N_L/K(L^×), where N_L/K is field norm map and ${\mathfrak {m}}_{K}$ is the maximal ideal of K.[1] Equivalently, n is the smallest integer such that the local Artin map is trivial on $U_{K}^{(n)}$ . Sometimes, the conductor is defined as ${\mathfrak {m}}_{K}^{n}$ 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 G_s is non-trivial, then ${\mathfrak {f}}(L/K)=\eta _{L/K}(s)+1$ , where η_L/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]

{\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), ${\mathfrak {f}}_{\chi }$ 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 L^ab/K, the maximal abelian extension of K in L, because of the "norm limitation theorem", which states that, in this situation,[8][9]

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.[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 : I^m → 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 ${\mathfrak {f}}(L/K)$ , to be the highest common factor of all moduli for which reciprocity holds; in fact reciprocity holds for ${\mathfrak {f}}(L/K)$ , 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 $\mathbf {Q} \left(\zeta _{n}\right)$ , where $\zeta _{n}$ 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 $n\infty$ otherwise.
Let L/K be $\mathbf {Q} \left({\sqrt {d}}\right)/\mathbf {Q}$ where d is a squarefree integer. Then,[16]
${\mathfrak {f}}\left(\mathbf {Q} \left({\sqrt {d}}\right)/\mathbf {Q} \right)={\begin{cases}\left|\Delta _{\mathbf {Q} \left({\sqrt {d}}\right)}\right|&{\text{for }}d>0\\\infty \left|\Delta _{\mathbf {Q} \left({\sqrt {d}}\right)}\right|&{\text{for }}d<0\end{cases}}$

where

\Delta _{\mathbf {Q} ({\sqrt {d}})}

is the discriminant of

\mathbf {Q} \left({\sqrt {d}}\right)/\mathbf {Q}

.

Relation to local conductors and ramification

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

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

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

Notes

Serre 1967, §4.2
As in Neukirch 1999, definition V.1.6
Neukirch 1999, proposition V.1.7
Milne 2008, I.1.9
Serre 1967, §4.2, proposition 1
Artin & Tate 2009, corollary to theorem XI.14, p. 100
As in Serre 1967, §4.2
Serre 1967, §2.5, proposition 4
Milne 2008, theorem III.3.5
As in Artin & Tate 2009, §XI.4. This is the situation in which the formalism of local class field theory works.
Cohen 2000, definition 3.4.1
Milne 2008, remark V.3.8
Janusz 1973, pp. 158,168–169
Some authors omit infinite places from the conductor, e.g. Neukirch 1999, §VI.6
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.
Milne 2008, example V.3.11
For the finite part Neukirch 1999, proposition VI.6.5, and for the infinite part Cohen 2000, definition 3.4.1
Neukirch 1999, corollary VI.6.6

References

Artin, Emil; Tate, John (2009) [1967], Class field theory, American Mathematical Society, ISBN 978-0-8218-4426-7, MR 2467155
Cohen, Henri (2000), Advanced topics in computational number theory, Graduate Texts in Mathematics, 193, Springer-Verlag, ISBN 978-0-387-98727-9
Janusz, Gerald (1973), Algebraic Number Fields, Pure and Applied Mathematics, 55, Academic Press, ISBN 0-12-380250-4, Zbl 0307.12001
Milne, James (2008), Class field theory (v4.0 ed.), retrieved 2010-02-22
Neukirch, Jürgen (1999). Algebraic Number Theory. Grundlehren der mathematischen Wissenschaften. 322. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021.
Serre, Jean-Pierre (1967), "Local class field theory", in Cassels, J. W. S.; Fröhlich, Albrecht (eds.), Algebraic Number Theory, Proceedings of an instructional conference at the University of Sussex, Brighton, 1965, London: Academic Press, ISBN 0-12-163251-2, MR 0220701

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[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