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 -functions are the Dirichlet -functions (cf. Dirichlet -function). The remaining -functions are more or less close analogues and generalizations of these -functions.
Nowadays -functions comprise a very large class of functions which are attached to representations of the Galois group . For example, choose a representation of the Galois group of an algebraic number field (cf. Representation of a group). For each prime , let be a Frobenius element in . Then the function
is the Artin -series corresponding to . In a similar way, the action of on the -torsion points of an elliptic curve , defined over , gives rise to the Hasse–Weil -function of . There exists a large body of fascinating conjectures about these -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).
|[a1]||S. Gelbart, "An elementary introduction to the Langlands program" Bull. Amer. Math. Soc. , 10 (1984) pp. 177–220|
|[a2]||M. Rapoport (ed.) N. Schappacher (ed.) P. Schneider (ed.) , Beilinson's conjectures on special values of -functions , Acad. Press (1988)|
L-function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=L-function&oldid=19281