Local zeta function

In number theory, the local zeta function (sometimes called the congruent zeta function) is defined as
where is the number of points of defined over the finite field extension of and is a non-singular -dimensional projective algebraic variety over the field with elements. Making the variable transformation gives
as the formal power series in the variable formula_3.
Equivalently, the local zeta function is sometimes defined as follows: 
In other words, the local zeta function with coefficients in the finite field is defined as a function whose logarithmic derivative generates the number of solutions of the equation defining in the degree extension 
Formulation.
Given a finite field "F", there is, up to isomorphism, only one field "Fk" with
for "k" = 1, 2, ... . Given a set of polynomial equations — or an algebraic variety "V" — defined over "F", we can count the number
of solutions in "Fk" and create the generating function
The correct definition for "Z"("t") is to set log "Z" equal to "G", so
and "Z"(0) = 1, since "G"(0) = 0, and "Z"("t") is "a priori" a formal power series.
The logarithmic derivative
equals the generating function
Examples.
For example, assume all the "Nk" are 1; this happens for example if we start with an equation like "X" = 0, so that geometrically we are taking "V" to be a point. Then
is the expansion of a logarithm (for |"t"| < 1). In this case we have
To take something more interesting, let "V" be the projective line over "F". If "F" has "q" elements, then this has "q" + 1 points, including as we must the one point at infinity. Therefore, we have
and
for |"t"| small enough, and therefore
The first study of these functions was in the 1923 dissertation of Emil Artin. He obtained results for the case of a hyperelliptic curve, and conjectured the further main points of the theory as applied to curves. The theory was then developed by F. K. Schmidt and Helmut Hasse. The earliest known nontrivial cases of local zeta functions were implicit in Carl Friedrich Gauss's "Disquisitiones Arithmeticae", article 358. There, certain particular examples of elliptic curves over finite fields having complex multiplication have their points counted by means of cyclotomy.
For the definition and some examples, see also.
Motivations.
The relationship between the definitions of "G" and "Z" can be explained in a number of ways. (See for example the infinite product formula for "Z" below.) In practice it makes "Z" a rational function of "t", something that is interesting even in the case of "V" an elliptic curve over finite field.
It is the functions "Z" that are designed to multiply, to get global zeta functions. These involve different finite fields (for example the whole family of fields Z/"p"Z as "p" runs over all prime numbers). In that connection, the variable "t" undergoes substitution by "p−s", where "s" is the complex variable traditionally used in Dirichlet series. (For details see Hasse–Weil zeta function.)
With that understanding, the products of the "Z" in the two cases used as examples come out as formula_17 and formula_18.
Riemann hypothesis for curves over finite fields.
For projective curves "C" over "F" that are non-singular, it can be shown that
with "P"("t") a polynomial, of degree 2"g", where "g" is the genus of "C". Rewriting
the Riemann hypothesis for curves over finite fields states
For example, for the elliptic curve case there are two roots, and it is easy to show the absolute values of the roots are "q"1/2. Hasse's theorem is that they have the same absolute value; and this has immediate consequences for the number of points.
André Weil proved this for the general case, around 1940 ("Comptes Rendus" note, April 1940): he spent much time in the years after that writing up the algebraic geometry involved. This led him to the general Weil conjectures. Alexander Grothendieck developed scheme theory for the purpose of resolving these.
A generation later Pierre Deligne completed the proof. 
General formulas for the zeta function.
It is a consequence of the Lefschetz trace formula for the Frobenius morphism that
Here formula_23 is a separated scheme of finite type over the finite field "F" with formula_24 elements, and Frobq is the geometric Frobenius acting on formula_25-adic étale cohomology with compact supports of formula_26, the lift of formula_23 to the algebraic closure of the field "F". This shows that the zeta function is a rational function of formula_28.
An infinite product formula for formula_29 is
Here, the product ranges over all closed points "x" of "X" and deg("x") is the degree of "x".
The local zeta function "Z(X, t)" is viewed as a function of the complex variable "s" via the change of 
variables "q−s".
In the case where "X" is the variety "V" discussed above, the closed points 
are the equivalence classes "x=[P]" of points "P" on formula_31, where two points are equivalent if they are conjugates over "F". The degree of "x" is the degree of the field extension of "F"
generated by the coordinates of "P". The logarithmic derivative of the infinite product "Z(X, t)" is easily seen to be the generating function discussed above, namely