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In number theory, the **local zeta function** *Z*(*V*, *s*) (sometimes called the **congruent zeta function** or the Hasse–Weil zeta function) is defined as

where V is a non-singular n-dimensional projective algebraic variety over the field **F**_{q} with q elements and *N*_{m} is the number of points of *V* defined over the finite field extension **F**_{qm} of **F**_{q}.^{[1]}

Making the variable transformation *u* = *q*^{−s}, gives

as the formal power series in the variable .

Equivalently, the local zeta function is sometimes defined as follows:

In other words, the local zeta function *Z*(*V*, *u*) with coefficients in the finite field **F**_{q} is defined as a function whose logarithmic derivative generates the number *N*_{m} of solutions of the equation defining V in the degree m extension **F**_{qm}.

Given a finite field *F*, there is, up to isomorphism, only one field *F _{k}* 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 *F _{k}* 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.

equals the generating function

- .

For example, assume all the *N _{k}* are 1; this happens for example if we start with an equation like

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 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.^{[2]} 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.^{[3]}

For the definition and some examples, see also.^{[4]}

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.

The local *Z* zeta functions are multiplied to get global * * zeta functions. These generally involve different finite fields (for example the whole family of fields **Z**/*p***Z** as *p* runs over all prime numbers).

In these fields, the variable *t* is substituted by *p ^{−s}*, where

The global products of *Z* in the two cases used as examples in the previous section therefore come out as and after letting .

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.
(See étale cohomology for the basic formulae of the general theory.)

It is a consequence of the Lefschetz trace formula for the Frobenius morphism that

Here is a separated scheme of finite type over the finite field *F* with elements, and Frob_{q} is the geometric Frobenius acting on -adic étale cohomology with compact supports of , the lift of to the algebraic closure of the field *F*. This shows that the zeta function is a rational function of .

An infinite product formula for 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 , 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

- .

**^**Section V.2 of Silverman, Joseph H. (1992),*The arithmetic of elliptic curves*, Graduate Texts in Mathematics, vol. 106, New York: Springer-Verlag, ISBN 978-0-387-96203-0, MR 1329092**^**Daniel Bump,*Algebraic Geometry*(1998), p. 195.**^**Barry Mazur,*Eigenvalues of Frobenius*, p. 244 in*Algebraic Geometry, Arcata 1974: Proceedings American Mathematical Society*(1974).**^**Robin Hartshorne,*Algebraic Geometry*, p. 449 Springer 1977 APPENDIX C "The Weil Conjectures"