L-function
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.
Comments
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
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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).
References
[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 ![]() |
L-function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=L-function&oldid=19281