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
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) »
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^
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