Special values of L-functions

<h2 id="conjectures">Conjectures</h2>
<p>There are two families of conjectures, formulated for general classes of <a href="/facts/L-function/REVCKvU7">L-functions</a> (the very general setting being for L-functions associated to <a href="/facts/Chow_motive/o0rNsaPo">Chow motives</a> over <a href="/facts/Number_field/e0K0TXSh">number fields</a>), the division into two reflecting the questions of:
</p>
<ol>
<li> how to replace 
  
    
      
        π
      
    
    {\displaystyle \pi }
  
 in the <a href="/facts/Leibniz_formula_for_pi/xkF1WgLD">Leibniz formula</a> by some other "transcendental" number (regardless of whether it is currently possible for <a href="/facts/Transcendental_number_theory/tjyLHm5G">transcendental number theory</a> to provide a proof of the transcendence); and</li>
<li> how to generalise the rational factor in the formula (class number divided by number of roots of unity) by some algebraic construction of a rational number that will represent the ratio of the L-function value to the "transcendental" factor.</li>
</ol>
<p>Subsidiary explanations are given for the integer values of 
  
    
      
        n
      
    
    {\displaystyle n}
  
 for which a formulae of this sort involving 
  
    
      
        L
        (
        n
        )
      
    
    {\displaystyle L(n)}
  
 can be expected to hold.
</p><p>The conjectures for (a) are called <i>Beilinson's conjectures</i>, for <a href="/facts/Alexander_Beilinson/dDfhNJzs">Alexander Beilinson</a>.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a><a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> The idea is to abstract from the <a href="/facts/Regulator_of_a_number_field/IQDk3QAR">regulator of a number field</a> to some "higher regulator" (the <a href="/facts/Beilinson_regulator/M2zhEDep">Beilinson regulator</a>), a determinant constructed on a real vector space that comes from <a href="/facts/Algebraic_K-theory/0RY5M49W">algebraic K-theory</a>.
</p><p>The conjectures for (b) are called the <i>Bloch–Kato conjectures for special values </i>(for <a href="/facts/Spencer_Bloch/xywM6bIw">Spencer Bloch</a> and <a href="/facts/Kazuya_Kato/F05g52S9">Kazuya Kato</a>; this circle of ideas is distinct from the <a href="/facts/Bloch%E2%80%93Kato_conjecture/0myRUsDm">Bloch–Kato conjecture</a> of K-theory, extending the <a href="/facts/Milnor_conjecture_(K-theory)/5vXHU1AM">Milnor conjecture</a>, a proof of which was announced in 2009). They are also called the <i>Tamagawa number conjecture</i>, a name arising via the <a href="/facts/Birch%E2%80%93Swinnerton-Dyer_conjecture/Issh7Yfr">Birch–Swinnerton-Dyer conjecture</a> and its formulation as an <a href="/facts/Elliptic_curve/60jvaHSA">elliptic curve</a> analogue of the <a href="/facts/Tamagawa_number/NRYVaI7S">Tamagawa number</a> problem for <a href="/facts/Linear_algebraic_group/eMLtu5Nh">linear algebraic groups</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> In a further extension, the equivariant Tamagawa number conjecture (ETNC) has been formulated, to consolidate the connection of these ideas with <a href="/facts/Iwasawa_theory/7t25cwJX">Iwasawa theory</a>, and its so-called <a href="/facts/Main_conjecture_of_Iwasawa_theory/WfPLvMTT">Main Conjecture</a>.
</p>
<h3>Current status</h3>
<p>All of these conjectures are known to be true only in special cases.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Brumer%E2%80%93Stark_conjecture/18tbmCIf">Brumer–Stark conjecture</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li>Kings, Guido (2003), <a href="http://jtnb.cedram.org/item?id=JTNB_2003__15_1_179_0">"The Bloch–Kato conjecture on special values of <i>L</i>-functions. A survey of known results"</a>, <i>Journal de théorie des nombres de Bordeaux</i>, 15 (1): 179–198, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.5802%2Fjtnb.396">10.5802/jtnb.396</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/1246-7405">1246-7405</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2019010">2019010</a></li>
<li><a href="https://www.encyclopediaofmath.org/index.php?title=Beilinson_conjectures">"Beilinson conjectures"</a>, <i><a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a></i>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a>, 2001 [1994]</li>
<li><a href="https://www.encyclopediaofmath.org/index.php?title=K-functor_in_algebraic_geometry">"K-functor in algebraic geometry"</a>, <i><a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a></i>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a>, 2001 [1994]</li>
<li>Mathar, Richard J. (2010), "Table of Dirichlet L-Series and Prime Zeta Modulo Functions for small moduli", <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1008.2547">1008.2547</a> [<a href="https://arxiv.org/archive/math.NT">math.NT</a>]</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="http://www.mpg.de/459553/L-Funktionen_gesamt.pdf">L-funktionen und die Vermutingen von Deligne und Beilinson (L-functions and the conjectures of Deligne and Beilsnson)</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Peter Schneider, Introduction to the Beilinson Conjectures (PDF) <a href="http://wwwmath.uni-muenster.de/u/pschnei/publ/beilinson-volume/Schneider.pdf" target="_blank">http://wwwmath.uni-muenster.de/u/pschnei/publ/beilinson-volume/Schneider.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Jan Nekovář, Beilinson's Conjectures (PDF) <a href="http://webusers.imj-prg.fr/~jan.nekovar/pu/mot.pdf" target="_blank">http://webusers.imj-prg.fr/~jan.nekovar/pu/mot.pdf</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Matthias Flach, The Tamagawa Number Conjecture (PDF) <a href="http://www.dpmms.cam.ac.uk/Seminars/Kuwait/abstracts/L56.pdf" target="_blank">http://www.dpmms.cam.ac.uk/Seminars/Kuwait/abstracts/L56.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
</ol>

Special values of L-functions open-in-new

Special values of L-functions