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A generalization of the [[Zeta-function|zeta-function]] at the cost of introducing characters (cf. [[Character of a group|Character of a group]]). The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057030/l0570302.png" />-functions form a complicated class of special functions of a complex variable, defined by a [[Dirichlet series|Dirichlet series]] or an [[Euler product|Euler product]] with characters. They are the basic instrument for studying by analytic methods the arithmetic of corresponding mathematical objects: the field of rational numbers, algebraic fields, algebraic varieties over finite fields, etc. The simplest representatives of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057030/l0570303.png" />-functions are the Dirichlet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057030/l0570304.png" />-functions (cf. [[Dirichlet-L-function|Dirichlet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057030/l0570305.png" />-function]]). The remaining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057030/l0570306.png" />-functions are more or less close analogues and generalizations of these <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057030/l0570307.png" />-functions.
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{{MSC|11Mxx}}
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{{TEX|done}}
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A generalization of the
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[[Zeta-function|zeta-function]] at the cost of introducing characters (cf.
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[[Character of a group|Character of a group]]). The $L$-functions form a complicated class of special functions of a complex variable, defined by a
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[[Dirichlet series|Dirichlet series]] or an
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[[Euler product|Euler product]] with characters. They are the basic instrument for studying by analytic methods the arithmetic of corresponding mathematical objects: the field of rational numbers, algebraic fields, algebraic varieties over finite fields, etc. The simplest representatives of $L$-functions are the Dirichlet $L$-functions (cf.
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[[Dirichlet-L-function|Dirichlet $L$-function]]). The remaining $L$-functions are more or less close analogues and generalizations of these $L$-functions.
  
  
  
 
====Comments====
 
====Comments====
Nowadays <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057030/l0570308.png" />-functions comprise a very large class of functions which are attached to representations of the [[Galois group|Galois group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057030/l0570309.png" />. For example, choose a representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057030/l05703010.png" /> of the Galois group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057030/l05703011.png" /> of an algebraic number field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057030/l05703012.png" /> (cf. [[Representation of a group|Representation of a group]]). For each prime <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057030/l05703013.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057030/l05703014.png" /> be a Frobenius element in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057030/l05703015.png" />. Then the function
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Nowadays $L$-functions comprise a very large class of functions which are attached to representations of the
 
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[[Galois group|Galois group]] $\def\Gal{\textrm{Gal}}\Gal(\overline{\Q}/\Q)$. For example, choose a representation $\rho : G \to {\textrm{GL}}(n,\C)$ of the Galois group $G$ of an algebraic number field $K$ (cf.
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057030/l05703016.png" /></td> </tr></table>
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[[Representation of a group|Representation of a group]]). For each prime $p$, let $F_p$ be a Frobenius element in $G$. Then the function
  
is the Artin <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057030/l05703018.png" />-series corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057030/l05703019.png" />. In a similar way, the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057030/l05703020.png" /> on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057030/l05703021.png" />-torsion points of an elliptic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057030/l05703022.png" />, defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057030/l05703023.png" />, gives rise to the Hasse–Weil <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057030/l05703025.png" />-function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057030/l05703026.png" />. There exists a large body of fascinating conjectures about these <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057030/l05703027.png" />-functions, which, on the one hand, relate them to automorphic forms (Langlands' conjectures) and, on the other hand, relate values at integral points to algebraic-geometric invariants (Beilinson's conjectures).
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$$L(p,s) = \prod_p \det(\textrm{Id} - \rho(F_p) p^{-s})^{-1}$$
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is the Artin $L$-series corresponding to $\rho$. In a similar way, the action of $\Gal(\overline{\Q}/\Q)$ on the $l^n$-torsion points of an elliptic curve $E$, defined over $\Q$, gives rise to the Hasse–Weil $L$-function of $E$. There exists a large body of fascinating conjectures about these $L$-functions, which, on the one hand, relate them to automorphic forms (Langlands' conjectures) and, on the other hand, relate values at integral points to algebraic-geometric invariants (Beilinson's conjectures).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> S. Gelbart,  "An elementary introduction to the Langlands program"  ''Bull. Amer. Math. Soc.'' , '''10'''  (1984)  pp. 177–220</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> M. Rapoport (ed.)  N. Schappacher (ed.)  P. Schneider (ed.) , ''Beilinson's conjectures on special values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057030/l05703028.png" />-functions'' , Acad. Press  (1988)</TD></TR></table>
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{|
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|valign="top"|{{Ref|Ge}}||valign="top"| S. Gelbart,  "An elementary introduction to the Langlands program"  ''Bull. Amer. Math. Soc.'', '''10'''  (1984)  pp. 177–220 {{MR|0733692}}  {{ZBL|0539.12008}}
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|-
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|valign="top"|{{Ref|RaScSc}}||valign="top"| M. Rapoport (ed.)  N. Schappacher (ed.)  P. Schneider (ed.), ''Beilinson's conjectures on special values of $L$-functions'', Acad. Press  (1988) {{MR|0944987}}  {{ZBL|0635.00005}}
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|-
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|}

Revision as of 18:09, 14 April 2012

2010 Mathematics Subject Classification: Primary: 11Mxx [MSN][ZBL]

A generalization of the zeta-function at the cost of introducing characters (cf. Character of a group). The $L$-functions form a complicated class of special functions of a complex variable, defined by a Dirichlet series or an Euler product with characters. They are the basic instrument for studying by analytic methods the arithmetic of corresponding mathematical objects: the field of rational numbers, algebraic fields, algebraic varieties over finite fields, etc. The simplest representatives of $L$-functions are the Dirichlet $L$-functions (cf. Dirichlet $L$-function). The remaining $L$-functions are more or less close analogues and generalizations of these $L$-functions.


Comments

Nowadays $L$-functions comprise a very large class of functions which are attached to representations of the Galois group $\def\Gal{\textrm{Gal}}\Gal(\overline{\Q}/\Q)$. For example, choose a representation $\rho : G \to {\textrm{GL}}(n,\C)$ of the Galois group $G$ of an algebraic number field $K$ (cf. Representation of a group). For each prime $p$, let $F_p$ be a Frobenius element in $G$. Then the function

$$L(p,s) = \prod_p \det(\textrm{Id} - \rho(F_p) p^{-s})^{-1}$$ is the Artin $L$-series corresponding to $\rho$. In a similar way, the action of $\Gal(\overline{\Q}/\Q)$ on the $l^n$-torsion points of an elliptic curve $E$, defined over $\Q$, gives rise to the Hasse–Weil $L$-function of $E$. There exists a large body of fascinating conjectures about these $L$-functions, which, on the one hand, relate them to automorphic forms (Langlands' conjectures) and, on the other hand, relate values at integral points to algebraic-geometric invariants (Beilinson's conjectures).

References

[Ge] S. Gelbart, "An elementary introduction to the Langlands program" Bull. Amer. Math. Soc., 10 (1984) pp. 177–220 MR0733692 Zbl 0539.12008
[RaScSc] M. Rapoport (ed.) N. Schappacher (ed.) P. Schneider (ed.), Beilinson's conjectures on special values of $L$-functions, Acad. Press (1988) MR0944987 Zbl 0635.00005
How to Cite This Entry:
L-function. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=L-function&oldid=19281
This article was adapted from an original article by A.F. Lavrik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article