Arithmetic zeta function

If X is the spectrum of a finite field with q elements, then

${\displaystyle \zeta _{X}(s)={\frac {1}{1-q^{-s}}}.}$

For a variety X over a finite field, it is known by Grothendieck's trace formula that

${\displaystyle \zeta _{X}(s)=Z(X,q^{-s})}$

where ${\displaystyle Z(X,t)}$ is a rational function (i.e., a quotient of polynomials).

Given two varieties X and Y over a finite field, the zeta function of ${\displaystyle X\times Y}$ is given by

${\displaystyle Z(X,t)\star Z(Y,t)=Z(X\times Y,t),}$

where ${\displaystyle \star }$ denotes the multiplication in the ring ${\displaystyle W(\mathbf {Z} )}$ of Witt vectors of the integers.[1]

If X is the spectrum of the ring of integers, then ζ_X (s) is the Riemann zeta function. More generally, if X is the spectrum of the ring of integers of an algebraic number field, then ζ_X (s) is the Dedekind zeta function.

The zeta function of affine and projective spaces over a scheme X are given by

${\displaystyle {\begin{aligned}\zeta _{\mathbf {A} ^{n}(X)}(s)&=\zeta _{X}(s-n)\\\zeta _{\mathbf {P} ^{n}(X)}(s)&=\prod _{i=0}^{n}\zeta _{X}(s-i)\end{aligned}}}$

The latter equation can be deduced from the former using that, for any X that is the disjoint union of a closed and open subscheme U and V, respectively,

${\displaystyle \zeta _{X}(s)=\zeta _{U}(s)\zeta _{V}(s).}$

Even more generally, a similar formula holds for infinite disjoint unions. In particular, this shows that the zeta function of X is the product of the ones of the reduction of X modulo the primes p:

${\displaystyle \zeta _{X}(s)=\prod _{p}\zeta _{X_{p}}(s).}$

Such an expression ranging over each prime number is sometimes called Euler product and each factor is called Euler factor. In many cases of interest, the generic fiber X_Q is smooth. Then, only finitely many X_p are singular (bad reduction). For almost all primes, namely when X has good reduction, the Euler factor is known to agree with the corresponding factor of the Hasse-Weil zeta function of X_Q. Therefore, these two functions are closely related.

Hasse and Weil conjectured that ζ_X (s) has a meromorphic continuation to the complex plane and satisfies a functional equation with respect to s → n − s where n is the absolute dimension of X.

This is proven for n = 1 and some very special cases when n > 1 for flat schemes over Z and for all n in positive characteristic. It is a consequence of the Weil conjectures (more precisely, the Riemann hypothesis part thereof) that the zeta function has a meromorphic continuation up to ${\displaystyle \mathrm {Re} (s)>n-{\tfrac {1}{2}}}$.

According to the generalized Riemann Hypothesis the zeros of ζ_X (s) are conjectured to lie inside the critical strip 0 ≤ Re(s) ≤ n lie on the vertical lines Re(s) = 1/2, 3/2, ... and the poles of ζ_X (s) inside the critical strip 0 ≤ Re(s) ≤ n lie on the vertical lines Re(s) = 0, 1, 2, ....

This was proved (Emil Artin, Helmut Hasse, André Weil, Alexander Grothendieck, Pierre Deligne) in positive characteristic for all n. It is not proved for any scheme that is flat over Z. The Riemann hypothesis is a partial case of Conjecture 2.

Subject to the analytic continuation, the order of the zero or pole and the residue of ζ_X (s) at integer points inside the critical strip is conjectured to be expressible by important arithmetic invariants of X. An argument due to Serre based on the above elementary properties and Noether normalization shows that the zeta function of X has a pole at s = n whose order equals the number of irreducible components of X with maximal dimension.[2] Secondly, Tate conjectured[3]

${\displaystyle \mathrm {ord} _{s=n-1}\zeta _{X}(s)=rk{\mathcal {O}}_{X}^{\times }(X)-rk\mathrm {Pic} (X)}$

i.e., the pole order is expressible by the rank of the groups of invertible regular functions and the Picard group. The Birch and Swinnerton-Dyer conjecture is a partial case this conjecture. In fact, this conjecture of Tate's is equivalent to a generalization of Birch and Swinnerton-Dyer.

More generally, Soulé conjectured[4]

${\displaystyle \mathrm {ord} _{s=n-m}\zeta _{X}(s)=-\sum _{i}(-1)^{i}rkK_{i}(X)^{(m)}}$

The right hand side denotes the Adams eigenspaces of algebraic K-theory of X. These ranks are finite under the Bass conjecture.

These conjectures are known when n = 1, that is, the case of number rings and curves over finite fields. As for n > 1, partial cases of the Birch and Swinnerton-Dyer conjecture have been proven, but even in positive characteristic the conjecture remains open.