# L-function

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.

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).