Congruent zeta functions. No.5
Yoshifumi Tsuchimoto
For any projecvite variety
over a field
,
we may define its congruent
zeta function
likewise for the affine varieties.
We quote the famous
It is a profound theorem, relating the number of rational points
of
over finite fields
and the topology of
.
For a further study we recommend [#!Ha!#, Appendix C],[#!Milne!#],
[#!milneLEC!#].
DEFINITION 5.2
Let

be a ring. A polynomial
![$f(X_0,X_1,\dots,X_n)\in R[X_0,X_1,\dots, X_n]$](img26.svg)
is said to be
homogenius of degree

if an equality
holds as a polynomial in

variables

.
DEFINITION 5.3
Let

be a field.
- We put
and call it (the set of
-valued points of) the projective space.
The class of an element
in
is
denoted by
.
- Let
be homogenious polynomials. Then we set
and call it (the set of
-valued point of) the projective variety
defined by
.
(Note that the condition

does not depend on the choice of the
representative

of
![$[x]\in \P ^n(k)$](img40.svg)
.)
LEMMA 5.4
We have the following picture of
.
That means,
is divided into two pieces
a
nd
.
That means,
is covered by three “open sets”
. Each of them is isomorphic to the
plane (that is, the affine space of dimension 2).