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p-adic L-function
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<h2 id="dirichlet-l-functions">Dirichlet L-functions</h2> <p>The Dirichlet <i>L</i>-function is given by the analytic continuation of </p> L ( s , χ ) = ∑ n χ ( n ) n s = ∏ p prime 1 1 − χ ( p ) p − s {\displaystyle L(s,\chi )=\sum _{n}{\frac {\chi (n)}{n^{s}}}=\prod _{p{\text{ prime}}}{\frac {1}{1-\chi (p)p^{-s}}}} <p>The Dirichlet <i>L</i>-function at negative integers is given by </p> L ( 1 − n , χ ) = − B n , χ n {\displaystyle L(1-n,\chi )=-{\frac {B_{n,\chi }}{n}}} <p>where <i>B</i><i>n</i>,χ is a <a href="/facts/Generalized_Bernoulli_number/RH0mblhs">generalized Bernoulli number</a> defined by </p> ∑ n = 0 ∞ B n , χ t n n ! = ∑ a = 1 f χ ( a ) t e a t e f t − 1 {\displaystyle \displaystyle \sum _{n=0}^{\infty }B_{n,\chi }{\frac {t^{n}}{n!}}=\sum _{a=1}^{f}{\frac {\chi (a)te^{at}}{e^{ft}-1}}} <p>for χ a Dirichlet character with conductor <i>f</i>. </p> <h2 id="definition-using-interpolation">Definition using interpolation</h2> <p>The Kubota–Leopoldt <i>p</i>-adic <i>L</i>-function <i>L</i><i>p</i>(<i>s</i>, χ) interpolates the Dirichlet <i>L</i>-function with the Euler factor at <i>p</i> removed. More precisely, <i>L</i><i>p</i>(<i>s</i>, χ) is the unique continuous function of the <i>p</i>-adic number <i>s</i> such that </p> L p ( 1 − n , χ ) = ( 1 − χ ( p ) p n − 1 ) L ( 1 − n , χ ) {\displaystyle \displaystyle L_{p}(1-n,\chi )=(1-\chi (p)p^{n-1})L(1-n,\chi )} <p>for positive integers <i>n</i> divisible by <i>p</i> − 1. The right hand side is just the usual Dirichlet <i>L</i>-function, except that the Euler factor at <i>p</i> is removed, otherwise it would not be <i>p</i>-adically continuous. The continuity of the right hand side is closely related to the <a href="/facts/Kummer_congruence/6qmuz6IG">Kummer congruences</a>. </p><p>When <i>n</i> is not divisible by <i>p</i> − 1 this does not usually hold; instead </p> L p ( 1 − n , χ ) = ( 1 − χ ω − n ( p ) p n − 1 ) L ( 1 − n , χ ω − n ) {\displaystyle \displaystyle L_{p}(1-n,\chi )=(1-\chi \omega ^{-n}(p)p^{n-1})L(1-n,\chi \omega ^{-n})} <p>for positive integers <i>n</i>. Here χ is twisted by a power of the <a href="/facts/Teichm%25C3%25BCller_character/MtcdHbTo">Teichmüller character</a> ω. </p> <h2 id="viewed-as-a-p-adic-measure">Viewed as a <i>p</i>-adic measure</h2> <p><i>p</i>-adic <i>L</i>-functions can also be thought of as <a href="/facts/P-adic_measure/PJcN6H4x"><i>p</i>-adic measures</a> (or <a href="/facts/P-adic_distribution/PJcN6H4x"><i>p</i>-adic distributions</a>) on <i>p</i>-profinite Galois groups. The translation between this point of view and the original point of view of Kubota–Leopoldt (as Q<i>p</i>-valued functions on Z<i>p</i>) is via the Mazur–Mellin transform (and <a href="/facts/Class_field_theory/4cvKbn6a">class field theory</a>). </p> <h2 id="totally-real-fields">Totally real fields</h2> <p>Deligne & Ribet (1980), building upon previous work of Serre (1973), constructed analytic <i>p</i>-adic <i>L</i>-functions for totally real fields. Independently, Barsky (1978) and Cassou-Noguès (1979) did the same, but their approaches followed Takuro Shintani's approach to the study of the <i>L</i>-values. </p> <ul><li>Barsky, Daniel (1978), <a href="http://www.numdam.org/item?id=GAU_1977-1978__5__A9_0">"Fonctions zeta p-adiques d'une classe de rayon des corps de nombres totalement réels"</a>, in <a href="/facts/Yvette_Amice/YKc6Khuq">Amice, Y.</a>; Barsky, D.; Robba, P. (eds.), <i>Groupe d'Etude d'Analyse Ultramétrique (5e année: 1977/78)</i>, vol. 16, Paris: Secrétariat Math., <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-2-85926-266-2, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0525346">0525346</a></li> <li>Cassou-Noguès, Pierrette (1979), "Valeurs aux entiers négatifs des fonctions zêta et fonctions zêta p-adiques", <i><a href="/facts/Inventiones_Mathematicae/bARcL95Y">Inventiones Mathematicae</a></i>, 51 (1): 29–59, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1979InMat..51...29C">1979InMat..51...29C</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF01389911">10.1007/BF01389911</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0020-9910">0020-9910</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0524276">0524276</a></li> <li>Coates, John (1989), <a href="http://www.numdam.org/item?id=SB_1988-1989__31__33_0">"On p-adic L-functions"</a>, <i>Astérisque</i> (177): 33–59, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0303-1179">0303-1179</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1040567">1040567</a></li> <li>Colmez, Pierre (2004), <a href="http://www.math.jussieu.fr/~colmez/tsinghua.pdf"><i>Fontaine's rings and p-adic L-functions</i></a> (PDF)</li> <li><a href="/facts/Pierre_Deligne/wYC8fhfq">Deligne, Pierre</a>; Ribet, Kenneth A. (1980), "Values of abelian L-functions at negative integers over totally real fields", <i><a href="/facts/Inventiones_Mathematicae/bARcL95Y">Inventiones Mathematicae</a></i>, 59 (3): 227–286, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1980InMat..59..227D">1980InMat..59..227D</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF01453237">10.1007/BF01453237</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0020-9910">0020-9910</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0579702">0579702</a></li> <li>Iwasawa, Kenkichi (1969), "On p-adic L-functions", <i><a href="/facts/Annals_of_Mathematics/CUYktUqD">Annals of Mathematics</a></i>, Second Series, 89 (1), Annals of Mathematics: 198–205, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F1970817">10.2307/1970817</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0003-486X">0003-486X</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/1970817">1970817</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0269627">0269627</a></li> <li>Iwasawa, Kenkichi (1972), <i>Lectures on p-adic L-functions</i>, <a href="/facts/Princeton_University_Press/kwvFlKEj">Princeton University Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-691-08112-0, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0360526">0360526</a></li> <li>Katz, Nicholas M. (1975), "p-adic L-functions via moduli of elliptic curves", <i>Algebraic geometry</i>, Proc. Sympos. Pure Math., vol. 29, Providence, R.I.: <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>, pp. 479–506, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0432649">0432649</a></li> <li><a href="/facts/Neal_Koblitz/TebdlGqw">Koblitz, Neal</a> (1984), <i>p-adic Numbers, p-adic Analysis, and Zeta-Functions</i>, Graduate Texts in Mathematics, vol. 58, Berlin, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-96017-3, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0754003">0754003</a></li> <li><a href="/facts/Tomio_Kubota/Sn4Rpgfp">Kubota, Tomio</a>; <a href="/facts/Heinrich-Wolfgang_Leopoldt/xSbeLDXc">Leopoldt, Heinrich-Wolfgang</a> (1964), <a href="https://gdz.sub.uni-goettingen.de/id/PPN243919689_0214_0215?tify={%22pages%22:%5B334%5D,%22view%22:%22info%22}">"Eine p-adische Theorie der Zetawerte. I. Einführung der p-adischen Dirichletschen L-Funktionen"</a>, <i><a href="/facts/Journal_f%25C3%25BCr_die_reine_und_angewandte_Mathematik/YCcRSaAQ">Journal für die reine und angewandte Mathematik</a></i>, 214/215: 328–339, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1515%2Fcrll.1964.214-215.328">10.1515/crll.1964.214-215.328</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0075-4102">0075-4102</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0163900">0163900</a></li> <li><a href="/facts/Jean-Pierre_Serre/1xSr7BCh">Serre, Jean-Pierre</a> (1973), "Formes modulaires et fonctions zêta p-adiques", in Kuyk, Willem; <a href="/facts/Jean-Pierre_Serre/1xSr7BCh">Serre, Jean-Pierre</a> (eds.), <i>Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, 1972)</i>, Lecture Notes in Math, vol. 350, Berlin, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, pp. 191–268, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-540-37802-0_4">10.1007/978-3-540-37802-0_4</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-06483-1, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0404145">0404145</a></li></ul>