Modulus (algebraic number theory)

{{For|the operation that gives a number's remainder|Modulo operation}}

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 places of a global field (i.e. an algebraic number field or a global function field). It is used to encode ramification data for abelian extensions of a global field.

==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 places of ''K'', finite or infinite, the exponents ν('''p''') are zero except for finitely many '''p'''. If ''K'' is a number field, ν('''p''') = 0 or 1 for real places and ν('''p''') = 0 for complex places. If ''K'' is a function field, ν('''p''') = 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 congruence can be extended to the setting of moduli. If ''a'' and ''b'' are elements of ''K''×, the definition of ''a'' ≡∗''b'' (mod '''p'''ν) 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'''p''' 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'' ≡∗''b'' (mod '''m''') if ''a'' ≡∗''b'' (mod '''p'''ν('''p''')) for all '''p''' such that ν('''p''') > 0.

==Ray class group==

{{main|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'''f and '''m'''∞, the product over the finite and infinite places, respectively. Let ''I'''''m''' 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'''f;<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 over ''k'', with support away from '''m'''.<ref>{{harvnb|Serre|1988|loc=§V.1}}</ref> In both case, there is a group homomorphism ''i'' : ''K'''''m''',1 → ''I'''''m''' obtained by sending ''a'' to the principal ideal (resp. divisor) (''a'').

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

Erich Hecke's original definition of Hecke characters may be interpreted in terms of characters 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|loc=§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 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 | url-access=registration | url=https://archive.org/details/algebraicgroupsc0000serr }}

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