Local zeta-function

In number theory, a local zeta-function is a generating function

Related Topics:
Number theory - Generating function

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:Z(t)

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for the number of solutions of a set of equations defined over a finite field F, in extension fields Fk of F. The analogy with the Riemann zeta function

Related Topics:
Finite field - Riemann zeta function

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:zeta(s)

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comes via consideration of the logarithmic derivative

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:zeta'(s)/zeta(s).

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Given F, there is, up to isomorphism, just one field Fk with

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: = k,

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for k = 1,2, ... . Given a set of polynomial equations — or an algebraic variety V — defined over F, we can count the number

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:Nk

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of solutions in Fk; and create the generating function

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:G(t) = N1.t + N2.t2/2 + ... .

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The correct definition for Z(t) is to make log Z equal to G, and so

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:Z = exp(G);

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we will have Z(0) = 1 since G(0) = 0, and Z(t) is a priori a formal power series.

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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 a point. Then

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:G(t) = log(1 - t)

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is the expansion of a logarithm (for |t| < 1). In this case we have

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:Z(t) = 1/(1 − t).

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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 shall have

Related Topics:
Projective line - Point at infinity

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:Nk = qk + 1

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and

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:G(t) = log(1 − t) + log(1 − qt),

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for |t| small enough.

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In this case we have

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:Z(t) = 1/{(1 − t)(1 − qt)}.

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The relationship between the definitions of G and Z can be explained in a number of ways. 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.

Related Topics:
Rational function - Elliptic curve

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It is the functions Z that are designed to multiply, to get global zeta functions. Those involve different finite fields (for example the whole family of fields Z/p.Z as p runs over all prime numbers. In that relationship, 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). This explains too why the logarithmic derivative with respect to s is used.

Related Topics:
Prime number - Dirichlet series - Hasse-Weil zeta function

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With that understanding, the products of the Z in the two cases come out as zeta(s) and zeta(s)zeta(s-1).

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