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}}_{K}^{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. Vol. 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, vol. 193, Springer-Verlag, ISBN 978-0-387-98727-9
Janusz, Gerald (1973), Algebraic Number Fields, Pure and Applied Mathematics, vol. 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). Algebraische Zahlentheorie. Grundlehren der mathematischen Wissenschaften. Vol. 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

[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. Vol. 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

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