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In [[mathematics]], in the field of [[algebraic number theory]], a '''modulus''' (plural '''moduli''') (or '''cycle''',<ref>{{harvnb|Lang|1994|loc=§VI.1}}</ref> or '''extended ideal'''<ref>{{harvnb|Cohn|1985|loc=definition 7.2.1}}</ref>) is a formal product of [[Place (mathematics)|place]]s of a [[global field]] (i.e. an [[algebraic number field]] or a [[global function field]]).  It is used to encode [[ramification]] data for [[abelian extension]]s of a global field. See [[Modulo operation]] for a definition most people will be seeking.

==Definition==

Let ''K'' be a global field with ring of integers ''R''.  A '''modulus''' is a formal product<ref>{{harvnb|Janusz|1996|loc=§IV.1}}</ref><ref>{{harvnb|Serre|1988|loc=§III.1}}</ref>

:<math>\mathbf{m} = \prod_{\mathbf{p}} \mathbf{p}^{\nu(\mathbf{p})},\,\,\nu(\mathbf{p})\geq0 </math>

where '''p''' runs over all [[place (mathematics)|places]] of ''K'', [[finite place|finite]] or [[infinite place|infinite]], the exponents ν('''p''') are zero except for finitely many '''p'''. If ''K'' is a number field, ν('''p''')&nbsp;=&nbsp;0 or 1 for real places and ν('''p''')&nbsp;=&nbsp;0 for complex places. If ''K'' is a function field, ν('''p''')&nbsp;=&nbsp;0 for all infinite places.

In the function field case, a modulus is the same thing as an [[effective divisor]],<ref>{{harvnb|Serre|1988|loc=§III.1}}</ref> and in the number field case, a modulus can be considered as special form of [[Arakelov divisor]].<ref>{{harvnb|Neukirch|1999|loc=§III.1}}</ref>

The notion of [[modular arithmetic|congruence]] can be extended to the setting of moduli. If ''a'' and ''b'' are elements of ''K''<sup>×</sup>, the definition of ''a''&nbsp;≡<sup>∗</sup>''b''&nbsp;(mod&nbsp;'''p'''<sup>ν</sup>) depends on what type of prime '''p''' is:<ref>{{harvnb|Janusz|1996|loc=§IV.1}}</ref><ref>{{harvnb|Serre|1988|loc=§III.1}}</ref>
*if it is finite, then
::<math>a\equiv^\ast\!b\,(\mathrm{mod}\,\mathbf{p}^\nu)\Leftrightarrow \mathrm{ord}_\mathbf{p}\left(\frac{a}{b}-1\right)\geq\nu</math>
:where ord<sub>'''p'''</sub> is the [[normalized valuation]] associated to '''p''';
*if it is a real place (of a number field) and ν = 1, then
::<math>a\equiv^\ast\!b\,(\mathrm{mod}\,\mathbf{p})\Leftrightarrow \frac{a}{b}>0</math>
:under the [[real embedding]] associated to '''p'''.
*if it is any other infinite place, there is no condition.
Then, given a modulus '''m''', ''a''&nbsp;≡<sup>∗</sup>''b''&nbsp;(mod&nbsp;'''m''') if ''a''&nbsp;≡<sup>∗</sup>''b''&nbsp;(mod&nbsp;'''p'''<sup>ν('''p''')</sup>) for all '''p''' such that ν('''p''')&nbsp;&gt;&nbsp;0.

==Ray class group==
The '''ray modulo m''' is<ref>{{harvnb|Milne|2008|loc=§V.1}}</ref><ref>{{harvnb|Janusz|1996|loc=§IV.1}}</ref><ref>{{harvnb|Serre|1988|loc=§VI.6}}</ref>
:<math>K_{\mathbf{m},1}=\left\{ a\in K^\times : a\equiv^\ast\!1\,(\mathrm{mod}\,\mathbf{m})\right\}.</math>

A modulus '''m''' can be split into two parts, '''m'''<sub>f</sub> and '''m'''<sub>∞</sub>, the product over the finite and infinite places, respectively. Let ''I''<sup>'''m'''</sup> to be one of the following:
*if ''K'' is a number field, the subgroup of the [[group of fractional ideals]] generated by ideals coprime to '''m'''<sub>f</sub>;<ref>{{harvnb|Janusz|1996|loc=§IV.1}}</ref>
*if ''K'' is a function field of an [[algebraic curve]] over ''k'', the group of divisors, [[rational divisor|rational]] over ''k'', with [[support of a divisor|support]] away from '''m'''.<ref>{{harvnb|Serre|1988|loc=§V.1}}</ref>
In both case, there is a [[group homomorphism]] ''i'' : ''K''<sub>'''m''',1</sub> → ''I''<sup>'''m'''</sup> obtained by sending ''a'' to the [[principal ideal]] (resp. [[principal divisor|divisor]]) (''a'').

The '''ray class group modulo m''' is the quotient ''C''<sub>'''m'''</sub> = ''I''<sup>'''m'''</sup> / i(''K''<sub>'''m''',1</sub>).<ref>{{harvnb|Janusz|1996|loc=§IV.1}}</ref><ref>{{harvnb|Serre|1988|loc=§VI.6}}</ref> A coset of i(''K''<sub>'''m''',1</sub>) is called a '''ray class modulo m'''.

[[Erich Hecke]]'s original definition of [[Hecke character]]s may be interpreted in terms of [[Character (mathematics)|character]]s of the ray class group with respect to some modulus '''m'''.<ref>{{harvnb|Neukirch|1999|loc=§VII.6}}</ref>

===Properties===
When ''K'' is a number field, the following properties hold.<ref>{{harvnb|Janusz|1996|§4.1}}</ref>
* When '''m''' = 1, the ray class group is just the [[ideal class group]].
* The ray class group is finite.  Its order is the '''ray class number'''.
* The ray class number is divisible by the [[Class number (number theory)|class number]] of ''K''.

==Notes==
{{reflist|2}}

==References==
*{{Citation
| last=Cohn
| first=Harvey
| title=Introduction to the construction of class fields
| series=Cambridge studies in advanced mathematics
| volume=6
| publisher=[[Cambridge University Press]]
| year=1985
| isbn=978-0-521-24762-7
}}
*{{Citation
| last=Janusz
| first=Gerald J.
| title=Algebraic number fields
| publisher=[[American Mathematical Society]]
| series=Graduate Studies in Mathematics
| volume=7
| year=1996
| isbn=978-0-8218-0429-2
}}
*{{Citation
| last=Lang
| first=Serge
| author-link=Serge Lang
| title=Algebraic number theory
| edition=2
| publisher=[[Springer-Verlag]]
| year=1994
| series=[[Graduate Texts in Mathematics]]
| volume=110
| place=New York
| isbn=978-0-387-94225-4
| mr=1282723
}}
*{{Citation
| last=Milne
| first=James
| title=Class field theory
| url=http://jmilne.org/math/CourseNotes/cft.html
| edition=v4.0
| year=2008
| accessdate=2010-02-22
}}
*{{Neukirch ANT}}
*{{Citation
| last=Serre
| first=Jean-Pierre
| author-link=Jean-Pierre Serre
| title=Algebraic groups and class fields
| year=1988
| isbn=978-0-387-96648-9
| publisher=[[Springer-Verlag]]
| location=New York
| series=[[Graduate Texts in Mathematics]]
| volume=117
}}

{{DEFAULTSORT:Modulus (Algebraic Number Theory)}}
[[Category:Algebraic number theory]]