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Zeta functions of algebraic cycles over finite fields. (English) Zbl 0808.14016
From the text: In this paper, we generalize the concept of a zeta function from zero cycles to higher dimensional cycles and investigate its meromorphic continuation and order of pole at \(t=1\) by means of a Riemann-Roch approach.
Let \(X\) be a projective algebraic variety of dimension \(n\) over \(\mathbb{F}_ q\). For each integer \(r\) \((0 \leq r \leq n)\) we define the zeta function of \(r\)-cycles on \(X\) to be the following formal power series \(Z_ r (X,t) = \prod_ P (1 - t^{\deg P})^{-1}\), where \(P\) runs over all prime \(r\)- cycles of \(X\) defined over \(\mathbb F_ q\) and \(\deg P\) is the degree function on the group of \(r\)-cycles. When \(r=0\), \(Z_ r(X,t)\) is just the classical zeta function of a variety over a finite field. Let \(N_ r (d)\) be the number of prime \(r\)-cycles over \(\mathbb F_ q\) of degree \(d\) on \(X\). The set of effective \(r\)-cycles of degree \(d\) is parametrized (one- to-one) by an algebraic set (the Chow scheme) in a projective space. Thus, \(N_ r (d)\) is finite. Let \(W_ r (d)\) be the following weighted sum (each prime \(r\)-cycle of degree \(k\) is counted \(k\) times) \(W_ r(d) = \sum_{k | d} kN_ r (k)\). \(N_ r (d)\) can be obtained from \(W_ r (d)\) by Möbius inversion. Then, an alternative expression for \(Z_ r (X,t)\) is given by \[ Z_ r (X,t) = \prod^ \infty_{d = 1} (1 - t^ d)^{-N_ r (d)} = \exp \left( \sum^ \infty_{d = 1} {W_ r ( d) \over d} t^ d \right). \] This function incorporates complete information about the sequences \(N_ r (d)\) and \(W_ r (d)\) as \(d\) varies. Thus, our general question is reduced to the study of zeta functions of algebraic cycles. – We expect that these general zeta functions contain rich information about the arithmetic and geometry of \(X/\mathbb F_ q\), as in the case of zeta functions of zero cycles. Let \(A_ r (X)\) be the group of \(\mathbb F_ q\)-rational \(r\)-cycles modulo rational equivalence. Let \(E_ r (X)\) be the monoid of effective \(r\)- cycle classes in \(A_ r (X)\). In general, \(E_ r (X)\) may not be a finitely generated monoid.
I. On the arithmetic side, we conjecture that if \(E_ r (X)\) is a finitely generated monoid, then \(Z_ r (X,t)\) is a \(p\)-adic meromorphic function.
II. In connection with geometry, we conjecture that for if \(X/\mathbb F_ q\) is a smooth projective variety and if \(E_ r (X)\) is a finitely generated monoid, then the order of pole of \(Z_ r (X,t)\) at \(t=1\) equals the rank of the group \(A_ r (X)\), where the rank of an abelian group is defined to be the maximal number of \(\mathbb Z\)-linearly independent elements. One consequence of the conjecture is that for such \(X\), the mysterious Chow ring \(A(X/\mathbb F_ q) = \bigoplus^ n_{r = 0} A_ r (X)\) has finite rank equal to the order of pole at \(t=1\) of the complete zeta function \(Z(X,t) = \prod^ n_{r = 0} Z_ r (X,t)\). This conjecture is an analogue of a conjecture of Tate on algebraic cycles over finite fields.
We shall easily see that both of the conjectures in I and II are true for \(r=0\) (the case of the classical zeta functions) and for \(r=n\). Thus, the two conjectures hold for curves. The first new case to consider is the zeta function of curves (1-cycles, or divisors) on a projective surface. A corollary of our results is that both conjectures hold for sufficiently general surfaces which are complete intersections.
Our main object of study will be the zeta function of divisors. Our first main result states that if \(A_{n-1} (X)\) is finitely generated with rank one, then the zeta function of divisors is \(p\)-adic meromorphic with a simple pole at \(t=1\). Thus, both conjectures hold in this case.
Our second main result asserts that conjecture II is true for divisors. Namely, if the monoid \(E_{n-1} (X)\) of effective divisor classes in \(A_{n-1} (X)\) is a finitely generated monoid, then the zeta function of divisors is \(p\)-adic meromorphic on the closed unit disk and has a pole at \(t=1\) of order equal to the rank of \(A_{n - 1} (X)\).
In the last section, we attempt to generalize our results from divisors to cycles of higher codimension.

MSC:
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14C25 Algebraic cycles
14G15 Finite ground fields in algebraic geometry
11G25 Varieties over finite and local fields
11M41 Other Dirichlet series and zeta functions
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