Difference between revisions of "L-function"
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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 | + | {{MSC|11Mxx}} |
+ | {{TEX|done}} | ||
+ | |||
+ | A generalization of the | ||
+ | [[Zeta-function|zeta-function]] at the cost of introducing characters (cf. | ||
+ | [[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 | ||
+ | [[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 $L$-functions are the Dirichlet $L$-functions (cf. | ||
+ | [[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 | + | Nowadays $L$-functions comprise a very large class of functions which are attached to representations of the |
− | + | [[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. | |
− | + | [[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 | + | $$L(p,s) = \prod_p \det(\textrm{Id} - \rho(F_p) p^{-s})^{-1}$$ |
+ | 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==== | ====References==== | ||
− | + | {| | |
+ | |- | ||
+ | |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}} | ||
+ | |- | ||
+ | |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}} | ||
+ | |- | ||
+ | |} |
Latest revision as of 23:23, 9 January 2015
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 |
L-function. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=L-function&oldid=19281