Conductor (class field theory)

Let L/K be a finite abelian extension of non-archimedean local fields. The conductor of L/K, denoted $\backslash mathfrak\{f\}(L/K)$, is the smallest non-negative integer n such that the higher unit group

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

is contained in N_L/K(L^×), where N_L/K is field norm map and $\backslash mathfrak\{m\}\_K$ is the maximal ideal of K.{{harvnb|Serre|1967|loc=§4.2}} Equivalently, n is the smallest integer such that the local Artin map is trivial on $U\_K^\{(n)\}$. Sometimes, the conductor is defined as $\backslash mathfrak\{m\}\_K^n$ where n is as above.As in {{harvnb|Neukirch|1999|loc=definition V.1.6}}

The conductor of an extension measures the ramification. Qualitatively, the extension is unramified if, and only if, the conductor is zero,{{harvnb|Neukirch|1999|loc=proposition V.1.7}} and it is tamely ramified if, and only if, the conductor is 1.{{harvnb|Milne|2008|loc=I.1.9}} 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 $\backslash mathfrak\{f\}(L/K)\; =\; \backslash eta\_\{L/K\}(s)\; +\; 1$, where η_L/K is the function that translates from "lower numbering" to "upper numbering" of higher ramification groups.{{harvnb|Serre|1967|loc=§4.2, proposition 1}}

The conductor of L/K is also related to the Artin conductors of characters of the Galois group Gal(L/K). Specifically,{{harvnb|Artin|Tate|2009|loc=corollary to theorem XI.14, p. 100}}

:$\backslash mathfrak\{m\}\_K^\{\backslash mathfrak\{f\}(L/K)\}\; =\; \backslash operatorname\{lcm\}\backslash limits\_\backslash chi\; \backslash mathfrak\{m\}\_K^\{\backslash mathfrak\{f\}\_\backslash chi\}$

where χ varies over all multiplicative complex characters of Gal(L/K), $\backslash mathfrak\{f\}\_\backslash chi$ is the Artin conductor of χ, and lcm is the least common multiple.

The conductor can be defined in the same way for L/K a not necessarily abelian finite Galois extension of local fields.As in {{harvnb|Serre|1967|loc=§4.2}} 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,{{harvnb|Serre|1967|loc=§2.5, proposition 4}}{{harvnb|Milne|2008|loc=theorem III.3.5}}

: $N\_\{L/K\}\backslash left(L^\backslash times\backslash right)\; =\; N\_\{L^\{\backslash text\{ab\}\}/K\}\; \backslash left(\backslash left(L^\{\backslash text\{ab\}\}\backslash right)^\backslash times\; \backslash right).$

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.As in {{harvnb|Artin|Tate|2009|loc=§XI.4}}. This is the situation in which the formalism of local class field theory works.

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.{{harvnb|Cohen|2000|loc=definition 3.4.1}}

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 $\backslash mathfrak\{f\}(L/K)$, to be the highest common factor of all moduli for which reciprocity holds; in fact reciprocity holds for $\backslash mathfrak\{f\}(L/K)$, so it is the smallest such modulus.{{harvnb|Milne|2008|loc=remark V.3.8}}{{harvnb|Janusz|1973|pp=158,168–169}}Some authors omit infinite places from the conductor, e.g. {{harvnb|Neukirch|1999|loc=§VI.6}}

• 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 $\backslash mathbf\{Q\}\backslash left(\backslash zeta\_n\backslash right)$, where $\backslash zeta\_n$ denotes a primitive nth root of unity.{{cite book | first1=Yu. I. | last1=Manin | authorlink1=Yuri I. Manin | first2=A. A. | last2=Panchishkin | title=Introduction to Modern Number Theory | series=Encyclopaedia of Mathematical Sciences | volume=49 | edition=Second | year=2007 | isbn=978-3-540-20364-3 | issn=0938-0396 | zbl=1079.11002 | pages=155, 168 }} 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 $n\; \backslash infty$ otherwise.
• Let L/K be $\backslash mathbf\{Q\}\backslash left(\backslash sqrt\{d\}\backslash right)/\backslash mathbf\{Q\}$ where d is a squarefree integer. Then,{{harvnb|Milne|2008|loc=example V.3.11}}
• : $\backslash mathfrak\{f\}\backslash left(\backslash mathbf\{Q\}\backslash left(\backslash sqrt\{d\}\backslash right)/\backslash mathbf\{Q\}\backslash right)\; =\; \backslash begin\{cases\}$

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

The global conductor is the product of local conductors:For the finite part {{harvnb|Neukirch|1999|loc=proposition VI.6.5}}, and for the infinite part {{harvnb|Cohen|2000|loc=definition 3.4.1}}

:$\backslash mathfrak\{f\}(L/K)\; =\; \backslash prod\_\backslash mathfrak\{p\}\backslash mathfrak\{p\}^\{\backslash mathfrak\{f\}\backslash left(L\_\backslash mathfrak\{p\}/K\_\backslash mathfrak\{p\}\backslash right)\}.$

As a consequence, a finite prime is ramified in L/K if, and only if, it divides $\backslash mathfrak\{f\}(L/K)$.{{harvnb|Neukirch|1999|loc=corollary VI.6.6}} An infinite prime v occurs in the conductor if, and only if, v is real and becomes complex in L.

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Category:Class field theory