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Modulus (algebraic number theory)
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<h2 id="definition">Definition</h2> <p>Let <i>K</i> be a global field with <a href="/facts/Ring_of_integers/KEzxxMbh">ring of integers</a> <i>R</i>. A modulus is a formal product<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p> m = ∏ p p ν ( p ) , ν ( p ) ≥ 0 {\displaystyle \mathbf {m} =\prod _{\mathbf {p} }\mathbf {p} ^{\nu (\mathbf {p} )},\,\,\nu (\mathbf {p} )\geq 0} <p>where p runs over all <a href="/facts/Place_(mathematics)/LPzDT5CA">places</a> of <i>K</i>, <a href="/facts/Finite_place/LPzDT5CA">finite</a> or <a href="/facts/Infinite_place/LPzDT5CA">infinite</a>, the exponents ν(p) are zero except for finitely many p. If <i>K</i> is a number field, ν(p) = 0 or 1 for real places and ν(p) = 0 for complex places. If <i>K</i> is a function field, ν(p) = 0 for all infinite places. </p><p>In the function field case, a modulus is the same thing as an <a href="/facts/Effective_divisor/V0QDypeo">effective divisor</a>,<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> and in the number field case, a modulus can be considered as special form of <a href="/facts/Arakelov_divisor/MlmGW0Gm">Arakelov divisor</a>.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> </p><p>The notion of <a href="/facts/Modular_arithmetic/0wtRa5dU">congruence</a> can be extended to the setting of moduli. If <i>a</i> and <i>b</i> are elements of <i>K</i>×, the definition of <i>a</i> ≡∗<i>b</i> (mod pν) depends on what type of prime p is:<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a><a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> </p> <ul><li>if it is finite, then</li></ul> a ≡ ∗ b ( m o d p ν ) ⇔ o r d p ( a b − 1 ) ≥ ν {\displaystyle a\equiv ^{\ast }\!b\,(\mathrm {mod} \,\mathbf {p} ^{\nu })\Leftrightarrow \mathrm {ord} _{\mathbf {p} }\left({\frac {a}{b}}-1\right)\geq \nu } where ordp is the <a href="/facts/Normalized_valuation/vqIgPIX3">normalized valuation</a> associated to p; <ul><li>if it is a real place (of a number field) and ν = 1, then</li></ul> a ≡ ∗ b ( m o d p ) ⇔ a b > 0 {\displaystyle a\equiv ^{\ast }\!b\,(\mathrm {mod} \,\mathbf {p} )\Leftrightarrow {\frac {a}{b}}>0} under the <a href="/facts/Real_embedding/FklJQo5j">real embedding</a> associated to p. <ul><li>if it is any other infinite place, there is no condition.</li></ul> <p>Then, given a modulus m, <i>a</i> ≡∗<i>b</i> (mod m) if <i>a</i> ≡∗<i>b</i> (mod pν(p)) for all p such that ν(p) > 0. </p> <h2 id="ray-class-group">Ray class group</h2> <p class="note">Main article: <a href="/facts/Ray_class_group/gGAMoSvE">Ray class group</a></p> <p>The ray modulo m is<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a><a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a><a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> </p> K m , 1 = { a ∈ K × : a ≡ ∗ 1 ( m o d m ) } . {\displaystyle K_{\mathbf {m} ,1}=\left\{a\in K^{\times }:a\equiv ^{\ast }\!1\,(\mathrm {mod} \,\mathbf {m} )\right\}.} <p>A modulus m can be split into two parts, mf and m∞, the product over the finite and infinite places, respectively. Let <i>I</i>m to be one of the following: </p> <ul><li>if <i>K</i> is a number field, the subgroup of the <a href="/facts/Group_of_fractional_ideals/I5rvJ6nt">group of fractional ideals</a> generated by ideals coprime to mf;<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a></li> <li>if <i>K</i> is a function field of an <a href="/facts/Algebraic_curve/LnWsPmDP">algebraic curve</a> over <i>k</i>, the group of divisors, rational over <i>k</i>, with support away from m.<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a></li></ul> <p>In both case, there is a <a href="/facts/Group_homomorphism/sDguaNYs">group homomorphism</a> <i>i</i> : <i>K</i>m,1 → <i>I</i>m obtained by sending <i>a</i> to the <a href="/facts/Principal_ideal/Cga66qgM">principal ideal</a> (resp. <a href="/facts/Principal_divisor/V0QDypeo">divisor</a>) (<i>a</i>). </p><p>The ray class group modulo m is the quotient <i>C</i>m = <i>I</i>m / i(<i>K</i>m,1).<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a><a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> A coset of i(<i>K</i>m,1) is called a ray class modulo m. </p><p><a href="/facts/Erich_Hecke/VUPBbT10">Erich Hecke</a>'s original definition of <a href="/facts/Hecke_character/hwA3cSkB">Hecke characters</a> may be interpreted in terms of <a href="/facts/Character_(mathematics)/M46SyYUp">characters</a> of the ray class group with respect to some modulus m.<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a> </p> <h3>Properties</h3> <p>When <i>K</i> is a number field, the following properties hold.<a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a> </p> <ul><li>When m = 1, the ray class group is just the <a href="/facts/Ideal_class_group/LLn5Pwj1">ideal class group</a>.</li> <li>The ray class group is finite. Its order is the ray class number.</li> <li>The ray class number is divisible by the <a href="/facts/Class_number_(number_theory)/LLn5Pwj1">class number</a> of <i>K</i>.</li></ul> <h2 id="notes">Notes</h2> <ul><li>Cohn, Harvey (1985), <i>Introduction to the construction of class fields</i>, Cambridge studies in advanced mathematics, vol. 6, <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-24762-7</li> <li>Janusz, Gerald J. (1996), <i>Algebraic number fields</i>, <a href="/facts/Graduate_Studies_in_Mathematics/APstozeU">Graduate Studies in Mathematics</a>, vol. 7, <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8218-0429-2</li> <li><a href="/facts/Serge_Lang/I7MW2rUu">Lang, Serge</a> (1994), <i>Algebraic number theory</i>, <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">Graduate Texts in Mathematics</a>, vol. 110 (2 ed.), New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-94225-4, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1282723">1282723</a></li> <li>Milne, James (2008), <a href="http://jmilne.org/math/CourseNotes/cft.html"><i>Class field theory</i></a> (v4.0 ed.), retrieved 2010-02-22</li> <li><a href="/facts/J%C3%BCrgen_Neukirch/nzVc4G3F">Neukirch, Jürgen</a> (1999). <i>Algebraische Zahlentheorie</i>. <i>Grundlehren der mathematischen Wissenschaften</i>. Vol. 322. Berlin: <a href="/facts/Springer_Science%2BBusiness_Media/nAesf6nT">Springer-Verlag</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-65399-8. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1697859">1697859</a>. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0956.11021">0956.11021</a>.</li> <li><a href="/facts/Jean-Pierre_Serre/1xSr7BCh">Serre, Jean-Pierre</a> (1988), <a href="https://archive.org/details/algebraicgroupsc0000serr"><i>Algebraic groups and class fields</i></a>, <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">Graduate Texts in Mathematics</a>, vol. 117, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-96648-9</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Lang 1994, §VI.1 - Lang, Serge (1994), Algebraic number theory, Graduate Texts in Mathematics, vol. 110 (2 ed.), New York: Springer-Verlag, ISBN 978-0-387-94225-4, MR 1282723 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1282723" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=1282723</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Cohn 1985, definition 7.2.1 - Cohn, Harvey (1985), Introduction to the construction of class fields, Cambridge studies in advanced mathematics, vol. 6, Cambridge University Press, ISBN 978-0-521-24762-7 <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Janusz 1996, §IV.1 - Janusz, Gerald J. (1996), Algebraic number fields, Graduate Studies in Mathematics, vol. 7, American Mathematical Society, ISBN 978-0-8218-0429-2 <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Serre 1988, §III.1 - Serre, Jean-Pierre (1988), Algebraic groups and class fields, Graduate Texts in Mathematics, vol. 117, New York: Springer-Verlag, ISBN 978-0-387-96648-9 <a href="https://archive.org/details/algebraicgroupsc0000serr" target="_blank">https://archive.org/details/algebraicgroupsc0000serr</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Serre 1988, §III.1 - Serre, Jean-Pierre (1988), Algebraic groups and class fields, Graduate Texts in Mathematics, vol. 117, New York: Springer-Verlag, ISBN 978-0-387-96648-9 <a href="https://archive.org/details/algebraicgroupsc0000serr" target="_blank">https://archive.org/details/algebraicgroupsc0000serr</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Neukirch 1999, §III.1 - Neukirch, Jürgen (1999). Algebraische Zahlentheorie. Grundlehren der mathematischen Wissenschaften. Vol. 322. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1697859" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=1697859</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Janusz 1996, §IV.1 - Janusz, Gerald J. (1996), Algebraic number fields, Graduate Studies in Mathematics, vol. 7, American Mathematical Society, ISBN 978-0-8218-0429-2 <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Serre 1988, §III.1 - Serre, Jean-Pierre (1988), Algebraic groups and class fields, Graduate Texts in Mathematics, vol. 117, New York: Springer-Verlag, ISBN 978-0-387-96648-9 <a href="https://archive.org/details/algebraicgroupsc0000serr" target="_blank">https://archive.org/details/algebraicgroupsc0000serr</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Milne 2008, §V.1 - Milne, James (2008), Class field theory (v4.0 ed.), retrieved 2010-02-22 <a href="http://jmilne.org/math/CourseNotes/cft.html" target="_blank">http://jmilne.org/math/CourseNotes/cft.html</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Janusz 1996, §IV.1 - Janusz, Gerald J. (1996), Algebraic number fields, Graduate Studies in Mathematics, vol. 7, American Mathematical Society, ISBN 978-0-8218-0429-2 <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Serre 1988, §VI.6 - Serre, Jean-Pierre (1988), Algebraic groups and class fields, Graduate Texts in Mathematics, vol. 117, New York: Springer-Verlag, ISBN 978-0-387-96648-9 <a href="https://archive.org/details/algebraicgroupsc0000serr" target="_blank">https://archive.org/details/algebraicgroupsc0000serr</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Janusz 1996, §IV.1 - Janusz, Gerald J. (1996), Algebraic number fields, Graduate Studies in Mathematics, vol. 7, American Mathematical Society, ISBN 978-0-8218-0429-2 <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Serre 1988, §V.1 - Serre, Jean-Pierre (1988), Algebraic groups and class fields, Graduate Texts in Mathematics, vol. 117, New York: Springer-Verlag, ISBN 978-0-387-96648-9 <a href="https://archive.org/details/algebraicgroupsc0000serr" target="_blank">https://archive.org/details/algebraicgroupsc0000serr</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Janusz 1996, §IV.1 - Janusz, Gerald J. (1996), Algebraic number fields, Graduate Studies in Mathematics, vol. 7, American Mathematical Society, ISBN 978-0-8218-0429-2 <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>Serre 1988, §VI.6 - Serre, Jean-Pierre (1988), Algebraic groups and class fields, Graduate Texts in Mathematics, vol. 117, New York: Springer-Verlag, ISBN 978-0-387-96648-9 <a href="https://archive.org/details/algebraicgroupsc0000serr" target="_blank">https://archive.org/details/algebraicgroupsc0000serr</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>Neukirch 1999, §VII.6 - Neukirch, Jürgen (1999). Algebraische Zahlentheorie. Grundlehren der mathematischen Wissenschaften. Vol. 322. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1697859" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=1697859</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li> <li id="fn:17"><p>Janusz 1996, §4.1 - Janusz, Gerald J. (1996), Algebraic number fields, Graduate Studies in Mathematics, vol. 7, American Mathematical Society, ISBN 978-0-8218-0429-2 <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li> </ol>