DYNAMICAL ZETA FUNCTIONS

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0.1.3 Important types of zeta functions

For a general discussion of a zeta functions see article " Zeta functions " in the

Encyclopedic Dictionary of Mathematics [17]. The following is a classification

of the important types of zeta functions that are already known:

1) The zeta and //-functions of algebraic numbers fields: the Riemann zeta

function, Dirichlet L-functions, Dedekind zeta functions, Hecke //-functions,

Artin L-functions.

2) The p-adic L-functions of Leopoldt and Kubota.

3) The zeta functions of quadratic forms: Epstein zeta functions, Siegel

zeta functions.

4) The zeta functions associated with Hecke operators.

5)The zeta and L-functions attached to algebraic varieties defined over

finite fields: Artin zeta function, Hasse-Weil zeta functions.

6) The zeta functions attached to discontinuous groups : Selberg zeta

functions.

7) The dynamical zeta functions: Artin-Mazur zeta function, Lefschetz

zeta function, Ruelle zeta function for discrete dynamical systems, Ruelle

zeta function for flows.

0.1.4 Hasse-Weil zeta function

Let V be a nonsingular projective algebraic variety of dimension n over a

finite field k with q elements. The variety V is thus defined by homoge-

nous polynomial equations with coefficients in the field k for m + 1 variables

Xo,Xi,...,xm. These variables are in the algebraic closure k of the field fc,

and constitute the homogeneous coordinates of a point of V\The variety V

is invariant under the Frobenius map F : (x0,Xi, ...,xm) — »

(XQ,XI,

...,xqm).

Arithmetic considerations lead Hasse and Weil to introduce a zeta function

which counts the points of V with coordinates in the different finite exten-

sions of the field fc, or equivalently points of V which are fixed under Fn for

some n 1:

/ ~ #Fix (F") \

({z, V) := exp 2^

zn