Local zeta function

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In number theory, the local zeta function Z(Vs) (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 Fq with q elements and Nm is the number of points of V defined over the finite field extension Fqm of Fq.[1]

Making the variable transformation u = qs, 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(Vu) with coefficients in the finite field Fq is defined as a function whose logarithmic derivative generates the number Nm of solutions of the equation defining V in the degree m extension Fqm.


FormulationEdit

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

 .

ExamplesEdit

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 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]

MotivationsEdit

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/pZ as p runs over all prime numbers).

In these fields, the variable t is substituted by p−s, where s is the complex variable traditionally used in Dirichlet series. (For details see Hasse–Weil zeta function.)

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

Riemann hypothesis for curves over finite fieldsEdit

For projective curves C over F that are non-singular, it can be shown that

 

with P(t) a polynomial, of degree 2g, 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 q1/2. 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.)

General formulas for the zeta functionEdit

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 Frobq 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

 .

See alsoEdit

ReferencesEdit

  1. ^ 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
  2. ^ Daniel Bump, Algebraic Geometry (1998), p. 195.
  3. ^ Barry Mazur, Eigenvalues of Frobenius, p. 244 in Algebraic Geometry, Arcata 1974: Proceedings American Mathematical Society (1974).
  4. ^ Robin Hartshorne, Algebraic Geometry, p. 449 Springer 1977 APPENDIX C "The Weil Conjectures"