Conductor (class field theory)

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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[edit]

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\equiv1\, (\mathrm{mod}\,\mathfrak{m}_K^n)\right\}

is contained in NL/K(L×), where NL/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 Gs is non-trivial, then \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]

\mathfrak{m}_K^{\mathfrak{f}(L/K)}=\underset{\chi}{\mathrm{lcm}}\,\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[edit]

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]

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[edit]

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[edit]

Algebraic number fields[edit]

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 \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[edit]

  • 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}(\zeta_n).[15] The conductor of K is then the smallest such n.
  • Let L/K be \mathbf{Q}(\sqrt{d})/\mathbf{Q} where d is a squarefree integer. Then,[16]
\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 \Delta_{\mathbf{Q}(\sqrt{d})} is the discriminant of \mathbf{Q}(\sqrt{d})/\mathbf{Q}.

Relation to local conductors and ramification[edit]

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

\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 \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[edit]

  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[edit]