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Concept
Local zeta function
Summary
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 Fq with q elements and Nm is the number of points of V defined over the finite field extension Fqm of Fq.
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 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.
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
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. The earliest known nontrivial cases of local zeta functions were implicit in Carl Friedrich Gauss's Disquisitiones Arithmeticae, article 358.
Official source
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