L-function
2020 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 -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://encyclopediaofmath.org/index.php?title=L-function&oldid=36177