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

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

is contained in N_L/K(L^×), where N_L/K is field norm map and ${\displaystyle {\mathfrak {m}}_{K}}$ 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 G_s is non-trivial, then ${\displaystyle {\mathfrak {f}}(L/K)=\eta _{L/K}(s)+1}$, 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]

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

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 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]}

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

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 ${\displaystyle {\mathfrak {f}}(L/K)}$, 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 ${\displaystyle \mathbf {Q} \left(\zeta _{n}\right)}$, 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 ${\displaystyle \mathbf {Q} \left({\sqrt {d}}\right)/\mathbf {Q} }$ where d is a squarefree integer. Then, ^[16]

  ${\displaystyle {\mathfrak {f}}\left(\mathbf {Q} \left({\sqrt {d}}\right)/\mathbf {Q} \right)={\begin{cases}\left|\Delta _{\mathbf {Q} \left({\sqrt {d}}\right)}\right|&{\text{for }}d>0\\\infty \left|\Delta _{\mathbf {Q} \left({\sqrt {d}}\right)}\right|&{\text{for }}d<0\end{cases}}}$

where ${\displaystyle \Delta _{\mathbf {Q} ({\sqrt {d}})}}$ is the discriminant of ${\displaystyle \mathbf {Q} \left({\sqrt {d}}\right)/\mathbf {Q} }$.

Relation to local conductors and ramification

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

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

As a consequence, a finite prime is ramified in L/K if, and only if, it divides ${\displaystyle {\mathfrak {f}}(L/K)}$. ^[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

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). Algebraische Zahlentheorie. 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