Zeta Functions Of Algebraic Cycles Over Finite Fields

Daqing Wan

Department of Mathematics University of Washington Seattle, WA 98195

1. Introduction.

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.

We begin with an example which motivated the present work. A classical problem about finite fields is to find a formula for the number N′_{q}(d, n) of irreducible polynomials over F_{q} of degree d in n variables as d varies. This question goes back to Gauss (see the notes of Chapter 3 in [LN]), who first found a formula in the case when there is only one variable and the finite field is a prime field F_{p}. Dickson generalized Gauss’s formula from a prime finite field to a general finite field F_{q}. Since then, attempts have been made to find a formula in the case of several variables. In 1936, Carlitz ([CA1]) found an analogous formula for the number of irreducible factorizable polynomials of degree d, where a polynomial over F_{q} is factorizable if it factors as a product of linear factors over the algebraic closure of F_{q}. This generalizes Gauss’s formula in some sense. Later, Carlitz ([CA2-3]) established the asymptotic behavior of N′_{q}(d, n) (as d → ∞). Some of Carlitz’s results were generalized by Cohen ([CO1-2]). Both Carlitz [CA2-3] and Cohen [CO1-2] said that no explicit formula for N′_{q}(d, n) is known in the several variable case. Using ideas from the theory of zeta functions, we found an explicit but complicated combinatorial formula for N′_{q}(d, n). The proof actually shows that in the several variable case, there is no combinatorial formula of the Gauss type; rather, there is a nice p-adic formula generalizing Gauss’s formula. In some sense, this answers the question of Carlitz and Cohen.

To understand this, we consider homogeneous polynomials. Let N_{q}(d, n) be the number of homogeneous irreducible polynomials over F_{q} of degree d in n variables, where two such polynomials are regarded as the same if they differ by a constant. Unique factorization for polynomials implies that

∑_{d=0}^{∞} \frac{(q^{\binom{n+d−1}{n−1}} − 1)}{(q − 1)}t^{d} = ∏_{P (x)}(1 − t^{degP (x)})^{−1}  (1.1)

= ∏_{d=1}^{∞}(1 − t^{d})^{−N_{q}(d,n)}  (1.2)

where the first product runs over all irreducible homogeneous polynomials in n variables. The Möbius inversion formula and (1.2) then give an explicit combinatorial formula for N_{q}(d, n). If n ≤ 2, the left side of (1.1) is a rational function, so one finds a simple formula in this case (of which Gauss’s formula is the affine version). If n > 2, the left side of (1.1) not only is not a rational function, but is even not convergent in any neighbourhood of the origin. This explains why for n > 2 there is no simple combinatorial formula for N_{q}(d, n). However, if we view the left side of (1.1) p-adically, we see that it is a p-adic meromorphic function. Thus, there is a nice p-adic formula for N_{q}(d, n) (see Theorem 2.1).

Note that an irreducible polynomial over F_{q} is simply a prime divisor over F_{q} in a projective space. Thus, the question of finding a formula for the number of irreducible polynomials over F_{q} of degree d has a natural generalization: that is, finding a formula for the number of prime divisors over F_{q} of degree d in a projective variety X/F_{q}. Why should one restrict to divisors? Thus, a general question is the following: Given a projective variety X/F_{q} of dimension n, for each integer 0 ≤ r ≤ n, find a formula for the number of prime r-cycles over F_{q} of degree d as d varies. In the case r = 0, this reduces (via Möbius inversion) to the well-known question of finding a formula for the number of F_{q^{d}}-rational points on X/F_{q} as d varies, which is an extremely fruitful question in number theory and algebraic geometry, answered in large part by Weil’s famous conjectures (Deligne’s theorem). We expect that the general question (for all r) will play a similar role.

Recall that the study of the general question for r = 0 is equivalent to the study of the zeta function of X. Thus, we should have a similar notion of the zeta function of r-cycles which plays the same role in the general question. This is indeed the case, as we shall see now.

Let X be a projective algebraic variety of dimension n over F_{q}. For each integer r (0 ≤ r ≤ n) we define the zeta function of r-cycles on X to be the following formal power series

Z_{r}(X, t) = ∏_{P}(1 − t^{degP})^{−1},  (1.3)

where P runs over all prime r-cycles of X defined over F_{q} and degP is the degree function on the group of r-cycles (for a more precise definition, see section 3). 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 F_{q} of degree d on X. By a theorem of Chow and van der Waerden (Satz 1 in [CW]), 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) = ∑_{k|d} kN_{r}(k).  (1.4)

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) = ∏_{d=1}^{∞}(1 − t^{d})^{−N_{r}(d)}

= exp(∑_{d=1}^{∞} \frac{W_{r}(d)}{d}t^{d}).  (1.5)

This function incorporates complete information about the sequences N_{r}(d) and W_{r}(d). 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/F_{q}, as in the case of zeta functions of zero cycles. More precisely, we should expect some similar results or conjectures corresponding to Weil’s conjectures ([WE]) on zeta functions of zero cycles and Tate’s conjectures ([TA1]) relating the pole of Z_{0}(X, t) to invariants of algebraic cycles. We now describe two such conjectures for zeta functions of algebraic cycles.

I. First, on the arithmetic side, we conjecture that all Z_{r}(X, t) are p-adic meromorphic functions. This would imply that there is a p-adic formula for the number of prime F_{q}-rational r-cycles of degree d, generalizing the well-known formula for the number of F_{q}-rational points on X, and thus answering our general question raised above. This conjecture generalizes Dwork’s theorem on p-adic meromorphicity of Z_{0}(X, t), which was at the heart of his proof on the rationality of Z_{0}(X, t) (see, e.g. Chapter 5 in [KO1]).

II. Second, in connection with geometry, we conjecture that for a smooth projective variety X/F_{q}, the order of pole of Z_{r}(X, t) at t = 1 equals the rank of the group A_{r}(X) of F_{q}-rational r-cycles modulo rational equivalence, where the rank of an abelian group is defined to be the maximal number of Z-linearly independent elements. In particular, this would imply that the rank of A_{r}(X) is finite. Another consequence of the conjecture is that the mysterious Chow ring A(X/F_{q}) = ⊕_{r=0}^{n}A_{r}(X) has finite rank equal to the order of pole at t = 1 of the complete zeta function Z(X, t) = ∏_{r=0}^{n} Z_{r}(X, t). This conjecture is an analogue of a conjecture of Tate on algebraic cycles over finite fields. Tate’s conjecture asserts that the order of pole of Z_{0}(X, t) at t = 1/q^{r} equals the rank of the group A_{r}^{hom}(X) of F_{q}-rational r-cycles modulo homological equivalence (see equation (12) in [TA1]).

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 (a more general result is given in Theorem 6.7). Thus, both conjectures hold in this case. Examples are: any nonsingular complete intersection of dimension at least three (Grothendieck’s theorem) and a sufficiently general surface which is a complete intersection (Deligne’s theorem). Our proof generalizes Weil’s rationality proof for the zeta function of zero cycles on a smooth curve and uses the Riemann-Roch theorem (the general form). This is one of the first arithmetic applications of the higher dimensional Riemann-Roch theorem (see the introduction in [BL]). In the special case when the homogeneous coordinate ring of X is a unique factorization domain, we shall give a much easier proof using the Hilbert-Serre theorem.

Our second main result is as follows: If the monoid of effective divisor classes in A_{n−1}(X) is a finitely generated monoid (by a theorem of Lang and Néron, A_{n−1}(X) itself is a finitely generated abelian group), 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). Thus, the second conjecture is true in this case.

A by-product of our proof is a formula for the special value of Z_{n−1}(t) at t = 1. The formula is of the form −\frac{1}{(q^{λ}−1)}hR(X), where h is the order of torsion in A_{n−1}(X), R(X) is a certain regulator, and q^{λ} − 1 is the number of roots of unity in the quotient field of the homogeneous coordinate ring S(X). This is a higher dimensional analogue in characteristic p of the residue formula for the Dedekind zeta function or the Birch-Swinnerton-Dyer conjecture for the Hasse-Weil zeta function of an elliptic curve.

This result suggests that one should look at the special value of Z_{r}(X, t) at t = 1. Hopefully, one can formulate a conjectural formula for the special value. Unfortunately, this has not even been done for the classical zeta function Z_{0}(X, t), although Artin and Tate did have a conjecture for the special value of the middle dimensional part of Z_{0}(X, t) at t = 1/q in the case when X is a surface over F_{q} (see Conjecture (C) in [TA2]). The conjectural Artin-Tate formula involves the Brauer group of X and the order of torsion in the Néron-Severi group (the group of algebraic cycles modulo algebraic equivalence). Our results suggest that the special value of Z_{r}(X, t) at t = 1 should at least involve the order of torsion of A_{r}(X).

In the last section, we attempt to generalize our results from divisors to cycles of higher codimension. For this to work, it is necessary to generalize the Riemann-Roch problem to higher codimensions. This suggests another interesting problem in algebraic geometry. For an effective r-cycle C, we define L(C) to be the set of effective r-cycles rationally equivalent to C. This is a connected algebraic set, which is close to being linear. The generalized Riemann-Roch problem is then to study the behaviour of the sequence of varieties L(dC) for d large. In case C = D is a divisor, we reduce to the ordinary Riemann-Roch problem, that is, the behaviour of the dimensions l(dD) for d large. For our purpose over a finite field, a satisfactory answer to the generalized Riemann-Roch problem would give information about the number of F_{q}-rational points on the sequence of varieties L(dC), which is what we need in higher codimensions. Assuming that the monoid of effective r-cycle classes in A_{r}(X) is finitely generated and that certain properties hold for the higher codimensional Riemann-Roch problem (properties which hold in the case of divisors), we can then prove that Z_{r}(X, t) is meromorphic on the closed unit disk and has a pole of order ρ_{r} = rank(A_{r}(X)) at t = 1 with special value c_{r}h_{r}R_{r}(X), where h_{r} is the order of torsion of A_{r}(X), R_{r}(X) is a regulator, and c_{r} is a rational number under our assumptions.

If X/F_{q} is a quasi-projective variety in P^{m}, but is not projective, one can define the zeta function Z_{r}(X, t) of algebraic cycles in a similar way. We may suppose that X = Y − Z, where Z ⊂ Y are projective. For a prime r-cycle C on X, define its degree to be the degree of its closure in Y. The zeta function Z_{r}(X, t) is defined in a way similar to (1.3). One has the factorization Z_{r}(X, t) = Z_{r}(Y, t)/Z_{r}(Z, t). Thus it suffices to study Z_{r}(X, t) for X projective.

The above is mostly concerned with the similarities between the classical zeta function Z_{0}(X, t) and the generalized zeta function Z_{r}(X, t). We now say a few words about the differences between them. First, the duality and the Riemann hypothesis in Weil’s conjectures do not seem to generalize to zeta functions of higher cycles. The best duality one can hope for would be that the order of pole of Z_{r}(X, t) at t = 1 equals the order of pole of Z_{n−r}(X, t) at t = 1. Assuming Conjecture II, this is equivalent to equality between the rank of A_{r}(X) and the rank of A_{n−r}(X). This is indeed true for r = 0 and r = n. However, Katz’s version of Griffith’s famous example (Exposé XX in [DK]) implies that for a certain three-fold X over a finite field F_{q}, the rank of A_{1}(X) is greater than the rank of A_{2}(X). In this sense, we cannot expect duality for the Z_{r}(X, t). The Riemann hypothesis does not generalize, because Z_{r}(X, t) is expected to be a p-adic meromorphic function, and we shall show that Z_{r}(X, t) is never rational for 0 < r < n. In fact, these functions are never complex analytic in any neighborhood of the origin. A p-adic question would be to determine the Newton polygon of the reciprocal zeros and poles of Z_{r}(X, t), for example, giving an analogue of the Katz conjecture (Mazur’s theorem, see [MA]).

Concerning the various methods to study zeta functions over finite fields, we recall that there are three quite independent approaches to proving rationality of the zeta function of zero cycles on a curve. The first is Weil’s Riemann-Roch approach (see [MO]). The second is the p-adic method (Dwork). The third is the l-adic approach. Both the p-adic approach and the l-adic approach work for the zeta function of zero cycles on higher dimensional X. Each has its own advantages. It has been somewhat mysterious why the Riemann-Roch approach does not generalize. The present work provides some explanations and shows that there is some hope that the Riemann-Roch approach will work for all zeta functions Z_{r}(X, t), at least in certain important cases. An advantage of the Riemann-Roch approach is that it gives not only meromorphic continuation but also the special value at t = 1. However, we expect more difficulties for Z_{r}(X, t) (0 < r < n) than for Z_{0}(X, t).

The idea of defining zeta functions of algebraic cycles can be adapted to diophantine problems over other fields, not just the finite field F_{q}. For example, for a projective variety X over a number field K, one can consider the sequence of numbers a_{h} of K-rational points on X of height h, where the degree and dimension of the points are fixed and the height function is a suitable generalization (via Chow varieties) to algebraic cycles of the ordinary height function. One can define the zeta function to be the Dirichlet series associated with the sequence a_{h}. Fundamental questions are then meromorphic continuation, the order of pole and the special value. Results in analytic number theory, such as Hardy-Littlewood’s the circle method, can be applied to study these questions. One can also define zeta functions of algebraic cycles over a p-adic field, generalizing Igusa’s local zeta function ([IG]) and Serre’s zeta function on the reductions of a p-adic analytic space (see Section 3 in [SE]). These zeta functions are expected to be rational functions. We hope to examine these questions in more detail in the future.

Acknowledgment. I would like to thank Neal Koblitz for his constant help over the years and for his many valuable comments and suggestions.

2. The Number of Irreducible Polynomials

Let F_{q} be a finite field of q elements. Fix a positive integer n. For each nonnegative integer d, let M_{d} be the number of homogeneous polynomials over F_{q} of degree d in n variables (M_{0} = 1). Let N_{d} be the number of irreducible ones (N_{0} = 0). Two polynomials are identified if they differ by a constant. The purpose of this section is to prove a p-adic formula and a combinatorial formula for N_{d}. The special case treated in this section also serves to motivate the later more general results. It is clear that N_{1} = (q^{n} − 1)/(q − 1). We need only consider N_{d} for d > 1.

Theorem 2.1 (p-adic formula). If n > 2, there is a sequence of p-adic integers α_{i} approaching zero such that for all positive integers d > 1,

N_{d} = −\frac{1}{d} ∑_{k|d} μ(\frac{d}{k}) ∑_{i=1}^{∞} α_{i}^{k}.  (2.1)

If n = 1, N_{d} = 0 for d > 1. If n = 2, then for all positive integers d > 1,

N_{d} = \frac{1}{d} ∑_{k|d} μ(\frac{d}{k})q^{k}.  (2.2)

Proof. It is easy to see that

M_{d} = \frac{q^{\binom{n+d−1}{n−1}} − 1}{q − 1}.  (2.3)

Define

Z(t) = ∏_{P (x)}(1 − t^{degP (x)})^{−1},

where the product is over all irreducible polynomials in n variables over F_{q}. Because any homogeneous polynomial over F_{q} can be uniquely factored as a product of irreducible homogeneous polynomials over F_{q}, one checks that

Z(t) = ∏_{d=1}^{∞}(1 − t^{d})^{−N_{d}}  (2.4)

= exp(∑_{d=1}^{∞} \frac{W_{d}}{d} t^{d})  (2.5)

= ∑_{d=0}^{∞} M_{d}t^{d},

where W_{d} is the following weighted sum (each irreducible polynomial of degree k is counted k times)

W_{d} = ∑_{k|d} kN_{k}.  (2.6)

According to (2.3), one computes that

Z(t) = \frac{1}{q − 1}(−\frac{1}{1 − t} + ∑_{d=0}^{∞} q^{\binom{n+d−1}{n−1}}t^{d})

= \frac{1}{1 − t} ∑_{d=0}^{∞} \frac{q^{\binom{n+d−1}{n−1}} − q^{\binom{n+d−2}{n−1}}}{q − 1} t^{d}  (2.7)

= \frac{Q(t)}{1 − t},

where Q(t) is defined by (2.7). If n > 2, (2.7) shows that Q(t) is a p-adic entire function with integral coefficients and constant term 1. The p-adic Weierstrass factorization theorem shows that there is a sequence of p-adic integers α_{i} approaching zero such that

Z(t) = \frac{∏_{i=1}^{∞}(1 − α_{i}t)}{1 − t}.  (2.8)

This and (2.5) give the following p-adic formula:

W_{d} = 1 − ∑_{i=1}^{∞} α_{i}^{d}.  (2.9)

To obtain the formula for N_{d}, we apply Möbius inversion and deduce the theorem for n > 2. The case n ≤ 2 is similar. One computes that for n = 2,

Z(t) = \frac{1}{q − 1}(\frac{q}{1 − qt} − \frac{1}{1 − t}) = \frac{1}{(1 − t)(1 − qt)}.  (2.10)

If n = 1, then Z(t) = 1/(1 − t). The theorem is proved.

Theorem 2.2 (p-adic estimate). Let n > 2. In Theorem 2.1, we order the α_{i} so that ord_{q}α_{i} is increasing, where ord_{q} is the q-adic valuation such that ord_{q}q = 1. Then for all d ≥ 1,

ord_{q}α_{d} = \binom{n + d − 2}{n − 1} − \binom{n + d − 3}{n − 1}.  (2.11)

Proof. By (2.7), the Newton polygon of the entire function Q(t) is the convex closure in R^{2} of the following vertices

(d, \binom{n + d − 2}{n − 1}), d ≥ 0.  (2.12)

Theorem 2 follows from this.

Next, we derive an explicit combinatorial formula for N_{d}. This type of formula will be used in a future article [WA] to prove the rationality of a different type of zeta functions of algebraic cycles (roughly speaking, we will fix the degree and let the field F_{q} varies).

Theorem 2.3 (Combinatorial Formula). For each positive integer m, we have

N_{m} = ∑_{d|m} \frac{−d}{m}μ(\frac{m}{d}) ∑_{k_{1}+2k_{2}+···+dk_{d}=d,k_{i}≥0} \frac{(−1)^{k_{1}+k_{2}+···+k_{d}}}{k_{1} + k_{2} + ⋯ + k_{d}} \binom{k_{1} + k_{2} + ⋯ + k_{d}}{k_{1} k_{2} ⋯ k_{d}}M_{1}^{k_{1}} ⋯ M_{d}^{k_{d}},

where M_{d} is given by (2.3) (the inner summation is similar to Waring’s formula for power sums in terms of elementary symmetric functions).

Proof. Taking the logarithm of (2.5), one finds that

∑_{d=1}^{∞} \frac{W_{d}}{d} t^{d} = log(1 + ∑_{d=1}^{∞} M_{d}t^{d})

= − ∑_{k=1}^{∞} \frac{(−1)^{k}}{k} (∑_{d=1}^{∞} M_{d}t^{d})^{k}.  (2.13)

Comparing the cofficients of (2.13), we obtain

W_{d} = −d ∑_{k_{1}+2k_{2}+···+dk_{d}=d,k_{i}≥0} \frac{(−1)^{k_{1}+k_{2}+···+k_{d}}}{k_{1} + k_{2} + ⋯ + k_{d}} \binom{k_{1} + k_{2} + ⋯ + k_{d}}{k_{1} k_{2} ⋯ k_{d}}M_{1}^{k_{1}} ⋯ M_{d}^{k_{d}}.

Möbius inversion then gives Theorem 2.3.

Remark. If we let M_{d} be the number of non-associated polynomials of total degree d in n variables over F_{q} (not necessarily homogeneous), and let N_{d} be the number of irreducible ones, then the same method shows that Theorem 2.3 remains true with M_{d} given by

M_{d} = \frac{q^{\binom{n+d}{n}} − q^{\binom{n+d−1}{n}}}{q − 1} .  (2.14)

If n > 1, the p-adic formula for N_{d} is of the form (for all d > 0)

N_{d} = −\frac{1}{d} ∑_{k|d} μ(\frac{d}{k}) ∑_{i=1}^{∞} α_{i}^{k},  (2.15)

where the α_{d} are p-adic numbers whose q-adic valuations are given by

ord_{q}α_{d} = \binom{n + d − 1}{n − 1} − \binom{n + d − 2}{n − 1}.  (2.16)

Thus, we have an explicit formula and a p-adic formula for the number of irreducible polynomials of degree d in n variables. In some sense, this answers the question of Carlitz and Cohen, which is our first motivation in studying zeta functions of algebraic cycles.

3. Zeta Functions of Algebraic Cycles

In this section, we introduce the zeta functions of algebraic cycles of a projective variety in a given projective space, and we formulate two general conjectures associated with such zeta functions. Some simple properties of the zeta functions will be proved.

We first recall the definition of the degree of a projective variety. Let n and m be positive integers with n ≤ m. Let X be a closed n dimensional subscheme of the m dimensional projective space P^{m} over F_{q} (the embedding will be fixed). We shall always work over the ground field F_{q} unless otherwise stated. Let S(X) = ⊕_{d=0}^{∞}S_{d}(X) be the homogeneous coordinate ring of X, where S_{d}(X) consists of the homogeneous elements of degree d and S(X) is finitely generated by S_{1}(X) as an F_{q} algebra (S_{0}(X) = F_{q}). Let

l(d) = dim_{F_{q}}S_{d}(X).

For large d, l(d) is a polynomial of degree n, called the Hilbert polynomial of X. The degree of X, denoted degX, is defined to be the coefficient of d^{n}/n! in this polynomial (see I.7 in [HA2]). If X is irreducible, an alternative definition of degX is the intersection number X · H^{n}, where H is a hyperplane in P^{m} (see Example 2.5.2 in [FU]). This means that the degree of X is the number of intersection points cutout by the intersection of n sufficiently general hyperplanes.

Fix a closed n dimensional subscheme X of P^{m} over F_{q}. A subvariety (or a prime cycle) over F_{q} is a closed integral subscheme of X (i.e., a reduced and irreducible closed subscheme). For each integer 0 ≤ r ≤ n, we define the zeta function of r-cycles on X to be the following formal power series

Z_{r}(X, t) = ∏_{P}(1 − t^{degP})^{−1},  (3.1)

where P runs over all subvarieties of dimension r on X and degP is the degree of P viewed as a closed subscheme of P^{m}. The complete zeta function of X is then defined to be the product of all Z_{r}(X, t) (0 ≤ r ≤ n). Let N_{r}(d) (resp. M_{r}(d)) be the number of prime r-cycles (resp. effective r cycles) of degree d on X. By a theorem of Chow and van der Waerden, 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, M_{r}(d) and N_{r}(d) are all finite. All zeta functions are well defined. They are easily seen to be p-adic analytic in the open unit disk.

Alternative expressions for Z_{r}(X, t) are given by

Z_{r}(X, t) = ∏_{d=1}^{∞}(1 − t^{d})^{−N_{r}(d)}

= ∑_{d=0}^{∞} M_{r}(d)t^{d}

= exp(∑_{d=1}^{∞} \frac{W_{r}(d)}{d}t^{d}),  (3.2)

where W_{r}(d) is the following weighted sum (each prime r-cycle of degree k is counted k times)

W_{r}(d) = ∑_{k|d} kN_{r}(k).  (3.3)

Geometrically, a point counted in N_{r}(d) consists of d conjugate “non closed points” of dimension r. Thus, W_{r}(d) can be interpreted as the number of “non closed points” of degree d and dimension r. In particular, W_{0}(d) is just the number of F_{q^{d}}-rational points on X and Z_{0}(X, t) is the classical zeta function of a projective scheme over a finite field. It is well known that Z_{0}(X, t) is a rational function (Dwork’s theorem [DW]). The example in section 2 shows that in general we cannot expect that the Z_{r}(X, t) are rational functions; in fact, we shall show that Z_{r}(X, t) is never rational for 0 < r < n. However, the example does suggest that Z_{r}(X) is a p-adic meromorphic function. If Z_{r}(X, t) is p-adic meromorphic, this would imply that there are two sequences of p-adic integers α_{i} and β_{i} approaching zero such that for all positive integers d

W_{r}(d) = α_{1}^{d} + α_{2}^{d} + ⋯ − β_{1}^{d} − β_{2}^{d} − · · · .

This would generalize the formula for W_{0}(d), in which there are only finitely many terms, because Z_{0}(X, t) is rational.

Let X_{1}, X_{2}, ⋯, X_{c} be the distinct closed integral components of X. The zeta function of n cycles

Z_{n}(X, t) = ∏_{i=1}^{c}(1 − t^{degX_{i}})^{−1}  (3.4)

is a rational function. Thus, the zeta function of top dimension is very simple and well understood. Since a scheme X and its underlying reduced subscheme X_{red} have the same subvarieties,

Z_{r}(X, t) = Z_{r}(X_{red}, t),  (3.5)

for all r. If X_{1} and X_{2} are closed subschemes of X such that X = X_{1} ∪ X_{2}, then one checks that

Z_{r}(X, t) = \frac{Z_{r}(X_{1}, t)Z_{r}(X_{2}, t)}{Z_{r}(X_{1} ∩ X_{2}, t)},  (3.4)

where X_{1} ∩ X_{2} is the scheme theoretic intersection of X_{1} and X_{2}. These observations show that in studying zeta functions of algebraic cycles we may suppose that X is a variety (closed integral subscheme of P^{m}). Thus, the homogeneous coordinate ring S(X) of X is an integral domain.

Let H be a hyperplane in P^{m}. Let P be an irreducible subvariety in X of dimension r. As we know, the degree of P is the intersection number P · H^{r}. If r = 0, the number P · H^{0} is clearly independent of the projective embedding of X. If r > 0, however, the degree is dependent on the projective embedding of X. It would be interesting to find the general behaviour of zeta functions as the embedding varies.

Conjecture I (meromorphic continuation). Let X be an n dimensional projective variety in P^{m} over F_{q}. Then for all integers 0 ≤ r ≤ n, Z_{r}(X, t) is a p-adic meromorphic function.

Even for projective space X = P^{n} (n ≥ 3), we do not have a complete proof of this conjecture. The following simple result shows that the zeta function of divisors is p-adic meromorphic if the homogeneous coordinate ring S(X) is a unique factorization domain. This result is a direct generalization of the polynomial example treated in section 1. In the next section, we shall establish more evidence for the above conjecture in the case of divisors.

Theorem 3.1. Let X be a factorial projective variety in P^{m} (i.e., the homogeneous coordinate ring S(X) of X is a unique factorization domain). Then, the zeta function of divisors on X is a p-adic meromorphic function. It is rational if and only if X is a curve.

Proof. Since S(X) is a unique factorization domain, all prime divisors are principal. Thus, all effective divisors are principal. If f ∈ S_{d}(X), then intersection theory shows that

deg(div(f)) = d × degX,  (3.6)

where div(f) is the divisor defined by the hypersurface section f = 0. Let

Z^{∗}(X, t) = ∏_{f}(1 − t^{deg(f)})^{−1} = ∑_{d=0}^{∞} M_{d}t^{d},  (3.7)

where f runs through all irreducible homogeneous elements of S(X) and M_{d} is the number of homogeneous elements of degree d in S(X) (two such elements are considered the same if they differ by a constant). Then (3.6) shows that

Z_{n−1}(X, t) = Z^{∗}(X, t^{deg(X)}).  (3.8)

Since S(X) is a unique factorization domain, it is integrally closed. Thus, two non-zero elements of S_{d}(X) define the same divisor if and only if they differ by an element of F_{q}. Let l(d) be the dimension of S_{d}(X) as an F_{q} vector space; then

M_{d} = \frac{q^{l(d)} − 1}{q − 1}.  (3.9)

The Hilbert-Serre theorem states that l(d) is a polynomial of degree n for large d. Substituting (3.9) into (3.7), we deduce that Z^{∗}(X, t) is a p-adic meromorphic function. Because the degree of l(d) is n, it follows that Z_{n−1}(X, t) is rational if and only if n = 1. The theorem is proved.

Remark 3.2. X is factorial if and only if X is projectively normal (S(X) is integrally closed) and the group A^{1}(X) of Weil divisor classes is generated by a hyperplane section.

Theorem 3.3. Let X/F_{q} be an n-dimensional projective variety in P^{m}. Then Z_{r}(X, t) is never rational for 0 < r < n.

Proof. The simple reason is that for 0 < r < n there are too many r-cycles of a given degree, and this prevents Z_{r}(X, t) from being rational. To prove the theorem, we may assume n > 1. By (3.2),

Z_{r}(X, t) = ∑_{d=0}^{∞} M_{r}(d)t^{d}.

If Z_{r}(X, t) were rational, we would have the following upper bound

| M_{r}(d) |≤ q^{cd},  (3.10)

for some constant c.

We first treat the case r = n − 1. Observe that the integral closure of F_{q} in the quotient field of S(X) is a finite extension F_{q^{λ}} of F_{q}. Each homogeneous element f in S_{d}(X) determines an effective divisor of degree d × degX. Because X is projective, two different such elements determine the same divisor if and only if they differ by an element in the quotient field of S(X) which is integral over F_{q}. Such an element is then in F_{q^{λ}}. Thus, the following lower bound holds

M_{n−1}(d × degX) ≥ \frac{q^{l(d)} − 1}{q^{λ} − 1},  (3.11)

where l(d) is the dimension of S_{d}(X) as an F_{q} vector space. By the Hilbert-Serre theorem, l(d) is a polynomial in d of degree n ≥ 2. This and (3.11) show that M_{n−1}(d) grows faster than q^{cd} for any constant c. Thus, Z_{n−1}(X, t) cannot be rational.

For 1 ≤ r ≤ n − 2, we choose a subvariety Y of X of dimension r + 1 and consider only those effective r-cycles contained in Y. The above argument for divisors implies that M_{r}(d) grows faster than q^{cd} for any constant c. The theorem is proved.

The proof shows that for 0 < r < n, Z_{r}(X, t) is not complex analytic in any neighborhood of the origin.

Next, we formulate a conjecture relating the order of pole of the zeta function at t = 1 to the rank of the group of algebraic cycles modulo rational equivalence.

Let X be an n dimensional projective variety in P^{m} over F_{q}. Let r be an integer between 0 and n. A prime r-cycle V is a closed integral subvariety of dimension r on X which is defined over F_{q}. An r-cycle over F_{q} is an element of the free abelian group C_{r}(X) generated by the prime r-cycles. We write cycles in the form C = ∑_{i} n_{i}V_{i}, where the V_{i} are prime cycles, the n_{i} are integers, and only finitely many n_{i} are different from zero. Define the degree degC of C by

degC = ∑_{i} n_{i}degV_{i},  (3.12)

where degV_{i} is the degree of V_{i} viewed as a projective subvariety of P^{m}. If all n_{i} ≥ 0, C is called effective. An r-cycle is also called an (n − r)-cocycle. An (n − 1)-cycle is also called a divisor.

We now recall the definition of various equivalences on an algebraic scheme X (separated and of finite type) over an arbitary field F. All varieties here are assumed to be defined over F.

For any r+1-dimensional subvariety V of X, and any non-zero rational function f on V (i.e., f ∈ R(V)^{∗}), define an r-cycle div(f) on X by

div(f) = ∑ ord_{W} (f)W,

where the sum is over all codimension one subvarieties W of V and ord_{W} is the order function on R(V)^{∗} determined by the one-dimensional local ring O_{W,V} (see Chapter 1 in [FU]). A cycle C in C_{r}(X) is rationally equivalent to zero if and only if there are a finite number of (r + 1)-dimensional subvarieties V_{i} of X and f_{i} ∈ R(V_{i})^{∗} such that

C = ∑ div(f_{i}).

In the case r = n − 1, rational equivalence coincides with linear equivalence.

An alternate definition is as follows: Let X × P^{1} be the Cartesian product of X with the projective line P^{1}. A cycle C in C_{r}(X) is rationally equivalent to zero if and only if there are (r + 1)-dimensional subvarieties V_{1}, ⋯, V_{t} of X × P^{1} such that the projections from V_{i} to P^{1} are dominant and

C = ∑_{i=1}^{t} V_{i}(0) − V_{i}(∞).

A cycle C in C_{r}(X) is algebraically equivalent to zero if and only if there are nonsingular projective curves C(i) defined over the gound field and (r + 1)-dimensional subvarieties V_{i} of X × C(i) (i = 1, ⋯, t) such that the projections from V_{i} to C(i) are dominant and

C = ∑_{i=1}^{t} V_{i}(u_{i}) − V_{i}(v_{i}),

where u_{i} and v_{i} are two rational points on C(i).

A cycle C in C_{r}(X) is torsion equivalent to zero if some integral multiple of C is algebraically equivalent to zero.

A cycle C in C_{r}(X) is l-adic homologically equivalent to zero if its l-adic cohomology class in H_{2r}(X, Q_{l}) is zero, where l is a fixed prime number different from the characteristic of the ground field F.

A cycle C in C_{r}(X) is numerically equivalent to zero if for all subvarieties D of codimension r on X, the intersection number C · D is zero.

All of the above definitions give equivalence relations. Let

C_{r}^{num} ⊃ C_{r}^{hom} ⊃ C_{r}^{τ} ⊃ C_{r}^{alg} ⊃ C_{r}^{rat}

denote the groups of r-cycles which are, respectively, numerically, l-adic homologically, torsion, algebraically and rationally equivalent to zero. Let

A_{r}^{num} ⊂ A_{r}^{hom} ⊂ A_{r}^{τ} ⊂ A_{r}^{alg} ⊂ A_{r}^{rat} = A_{r}(X)

be the corresponding quotient groups of C_{r}(X). The groups A_{r}(X) are the finest among all cycle groups. In general, they are very difficult to compute. The direct sum A(X) = A^{rat}(X) = ⊕_{r=0}^{n}A_{r}(X) is the Chow ring with multiplication given by intersection product. This is a very mysterious ring; only in a few cases has its structure been determined. Similarly, one can define the Chow rings A^{num}(X), A^{hom}(X), A^{τ} (X) and A^{alg}(X) of numerical equivalence, homological equivalence, torsion equivalence and algebraic equivalence. One often uses the alternate notation A^{r}(X) = A_{n−r}(X). Similarly, instead of A^{num}(X), one often writes A_{num}(X), etc. For an abelian group A, not necessarily finitely generated, we define the rank r(A) of A to be the maximal number of Z-linearly independent elements in A. The rank r(A) is the dimension of the Q-vector space A ⊗ Q. Any torsion group has rank 0. The group of rational numbers (a divisible group) has rank 1. In general, r(A) can be infinite.

Conjecture II (rank and pole). Let X/F_{q} be a smooth projective variety of dimension n in P^{m}. Then, for all r, the rank of A_{r}(X) equals the order of pole of Z_{r}(X, t) at t = 1.

This conjecture means that Z_{r}(X, t) is p-adic meromorphic on the closed unit disk and its order of pole at t = 1 equals the rank of A_{r}(X). In particular, the conjecture implies that the rank is finite. The last property has the possibility of being true because we are working over a finite field. It would not hold over the complex numbers C. For example, over the complex numbers, the group C_{n−1}^{alg}(X)/C_{n−1}^{rat}(X) is an abelian variety (torus). If its dimension is non-zero, then the torus contains a copy of C. Hence, the Z-rank of A_{n−1}(X) is infinite. However, it is known that A_{r}^{num}(X) and A_{r}^{hom}(X) are finitely generated over any field. This follows from the finite dimensionality of l-adic cohomology, see [HA1]. Quillen’s theorem on higher K-theory shows that A^{r}(X) is isomorphic to the cohomology group H^{r}(X, K_{r}), where K_{r} is the sheaf of higher K groups on X. Thus, Conjecture II implies that over a finite field, the cohomology groups H^{r}(X, K_{r}) have finite rank.

Proposition 3.4. Let X/F_{q} be a projective variety and X̄ be the base extension of X to the algebraic closure F̄_{q}. Then a cycle C in A_{r}(X) is a torsion element if and only if it is a torsion element of A_{r}(X̄).

Proof. Let C be a cycle in A_{r}(X). It suffices to prove that if C is zero in A_{r}(X̄), then C is a torsion element in A_{r}(X). Thus, let C be rationally equivalent to zero over F̄_{q}. By definition, there are a finite number of (r+1)-dimensional subvarieties V̄_{i} of X̄ ×P^{1} such that the projections from V_{i} to P^{1} are dominant and

C = ∑_{i=1}^{t} V_{i}(0) − V_{i}(∞).

Let σ be the Frobenius automorphism a → a^{q} of F̄_{q}. σ acts naturally on each V_{i} and has finite order. Thus, for any integer a,

C = ∑_{i=1}^{t} σ^{a}(V_{i}(0) − V_{i}(∞)).

Choose a positive integer k such that

∑_{j=0}^{k−1} σ^{j}(V_{i}) = k_{i}W_{i}

for all i, where the W_{i} are subvarieties of X̄ × P^{1} which can be defined over F_{q} and the k_{i} are positive integers. Then

kC = ∑_{i=1}^{t} k_{i}(W_{i}(0) − W_{i}(∞)).

This shows that kC is zero in A_{r}(X). The proposition is proved.

In the case that X is nonsingular in codimension 1, Weil proved a stronger result for divisors, which says that a divisor D over a field is linearly equivalent to zero if and only if it is linearly equivalent to zero after base extension (see Section VI.5 in [LA]).

Corollary 3.5. Let X/F_{q} be a projective variety. Then C_{r}^{τ} (X)/C_{r}^{rat}(X) is a torsion group. In particular, A_{r}(X) has finite rank if and only if A_{r}^{τ}(X) does.

Proof. By the definition and Proposition 3.4, we only need to prove that C_{r}^{alg}(X̄)/C_{r}^{rat}(X̄) is a torsion group, where X̄ is the base extension of X from F_{q} to its algebraic closure. Now, it is known (see Example 19.1.2 in [FU]) that C_{r}^{alg}(X̄)/C_{r}^{rat}(X̄) is generated by classes in A_{r}^{rat}(X̄) of the form f_{∗}g^{∗}(β) for V a subvariety of X̄ × C, where C is a projective nonsingular curve over F̄_{q}, β is a zero cycle of degree zero on C (hence a point on the Jacobian variety J(C)), f is the projection from V to X and g is the projection from V to C. Since an abelian variety over F̄_{q} is a torsion group, the corollary is proved.

It would be interesting to know whether C_{r}^{τ} (X)/C_{r}^{rat}(X) is finite. This is true for divisors, at least under certain non-singularity assumptions.

Corollary 3.6. Let X/F_{q} be a projective variety. Then A_{0}(X) has rank 1.

Proof. Let X̄ = X ⊗ F̄_{q} be the base extension of X from F_{q} to its algebraic closure. First assume that X is absolutely irreducible, i.e., X̄ is irreducible. Let B_{0}(X̄) be the kernel of the degree map A_{0}(X̄) → Z. B_{0}(X̄) is generated by the images f_{∗}(J(C)), where f : C → X, C is a non-singular projective curve over F̄_{q} and J(C) is the Jacobian variety of C (see Example 1.6.6 in [FU]). Since an abelian variety over F̄_{q} is a torsion group, A_{0}(X̄) has rank 1. The corollary then follows from Proposition 3.4. If X is not absolutely irreducible, one uses the above proof together with a Galois argument.

We now prove

Theorem 3.7. Conjectures I-II hold for r = 0 and r = n.

Proof. It remains to prove Conjecture II for r = 0. Let X/F_{q} be smooth and projective. Weil’s conjectures show that there are integral polynomials P_{i}(t) such that

Z_{0}(X, t) = ∏_{i=0}^{2n} P_{i}(t)^{(−1)^{i+1}},

where P_{0}(t) = 1 − t and each reciprocal root of P_{i}(t) has absolute value q^{i/2}. Thus, t = 1 is a simple pole of Z_{0}(X, t). The theorem is proved.

4. Zeta Functions of Divisors: Rank One Case

In this section, we prove Conjectures I and II for the zeta function of divisors in the case when A^{1}(X) is finitely generated with rank 1.

Let X be an n-dimensional projective variety in P^{m} with homogeneous coordinate ring S(X). Let K(X) be the function field of X (i.e., the quotient field of S(X)). Let C^{1}(X/F_{q}) be the group of divisors. Each nonzero element f ∈ K(X) determines a principal divisor, denoted div(f). Intersection theory shows that a principal divisor has degree 0 (see I.7 in [HA2]). Two divisors are linearly equivalent if their difference is a principal divisor. Let A^{1}(X/F_{q}) be the group of linear equivalence classes of divisors in C^{1}(X/F_{q}). Then the degree function extends to A^{1}(X/F_{q}). An alternative definition of the degree of a divisor D in C^{1}(X/F_{q}) is given by the intersection number D · H^{n−1}, where H is a hyperplane in P^{m}. The two definitions coincide, see (Example 2.5.2 in [FU]). Assume that A^{1}(X/F_{q}) is a finitely generated abelian group (by a theorem of Lang-Néron [LN], this is true if X̄ is nonsingular in codimension 1). By analogy with the Dedekind zeta function of a number field or the Birch-Swinnerton-Dyer conjecture for elliptic curves over a number field, a fundamental question in characteristic p is then to relate the rank ρ of A^{1}(X) and the order h of the torsion in A^{1}(X) to the order of pole of Z_{n−1}(X, t) at t = 1 and its special value (for p-adic analogues of this question, see Chapter 4 in [KO2]). We use the letter h to denote the torsion order because it is the higher dimensional analogue of the class number. The following result shows that we have a complete understanding of the above question in the case when A^{1}(X) is finitely generated of rank 1. In particular, Conjectures I and II hold in this case.

Theorem 4.1. Let X be an n-dimensional projective variety in P^{m}, n ≥ 2. Assume that A^{1}(X/F_{q}) is finitely generated of rank 1 and torsion order h. Then the zeta function Z_{n−1}(X, t) of divisors is a p-adic meromorphic function and has a simple pole at t = 1 with residue \frac{h}{μ(q^{λ}−1)}, where μ is the smallest positive degree of an element of A^{1}(X) and λ is the positive integer such that F_{q^{λ}} is the integral closure of F_{q} in the quotient field of S(X).

We first give some preliminaries. Let F_{q^{λ}} be the integral closure of F_{q} in the quotient field K(X) of S(X). If X is projectively normal, then λ = 1. More generally, if X is normal, then λ = 1. Becasue in this case, certain d-uple embedding of X will be projectively normal and the function field K(X) is independent of embeddings.

If D is a divisor on X, define an F_{q}-vector space as follows:

L(D) = {f ∈ K(X) | f = 0 or div(f) + D > 0}.  (4.1)

Set l(D) = dim_{F_{q}}L(D). L(D) can also be viewed as an F_{q^{λ}}-vector space, whose dimension is denoted by l^{∗}(D) = l(D)/λ.

Lemma 4.2. Let X/F_{q} be a projective variety. Let D be a divisor on X. Then L(D) is a finite dimensional vector space over F_{q} and the number of effective divisors on X linearly equivalent to D is (q^{l(D)} − 1)/(q^{λ} − 1).

Proof. Let {f_{i}} be a basis of the F_{q}-vector space L(D). Every effective divisor D_{1} linearly equivalent to D has the form div(∑_{i} a_{i}f_{i}) + D with a_{i} ∈ F_{q} not all zero. Now div(f) = div(g) if and only if div(f/g) = 0.

Since X is a projective variety, this can happen if and only if f/g is integral over F_{q}. That is, f/g is an element of F_{q^{λ}}. Thus, {a_{i}} and {b_{i}} determine the same effective divisor D_{1} if and only if b_{i} = ca_{i} for some nonzero element c in F_{q^{λ}}. Since the number of effective divisors of a given degree is finite, it follows that l(D) is finite and the number of effective divisors on X linearly equivalent to D is (q^{l(D)} − 1)/(q^{λ} − 1). The lemma is proved.

Lemma 4.3. Let D be a Cartier divisor (i.e., locally principal) on an n-dimensional projective variety X over F_{q}. Let C be an ample Cartier divisor on X over F_{q}. There is an integer d_{0} such that for all d ≥ d_{0}, l(D + dC) is a polynomial in d of degree n with leading coefficient C^{n}/n!, where C^{n} is the self intersection number.

Proof. On an integral scheme X, there is a one-to-one correpondence between Cartier divisor classes and isomorphy classes of invertible sheaves. Lemma 4.3 then follows from the general Riemann-Roch theorem (Example 18.3.6 in [FU]) and the vanishing theorem for cohomology (see III. 5.3 in [HA2]).

Proof of Theorem 4.1. Let M_{d} and N_{d} be the number of effective divisors and prime divisors of degree d, respectively. Then,

Z_{n−1}(X, t) = ∑_{d=0}^{∞} M_{d}t^{d}.  (4.2)

Since A^{1}(X) has rank 1, an element is torsion if and only if its degree is zero. Let D_{1}, ⋯, D_{h} be h mutually inequivalent divisors of degree 0. Let μ be the smallest positive integer such that there is a divisor D^{∗} of degree μ. If μ does not divide d, then M_{d} = 0. If μ divides d, then any divisor of degree d is equivalent to D_{i} + (d/μ)D^{∗} for some i. In this case,

M_{d} = ∑_{i=1}^{h} \frac{q^{l(D_{i}+ \frac{d}{μ} D^{∗})} − 1}{q^{λ} − 1}.  (4.3)

Thus, we can write

Z_{n−1}(X, t) = \frac{1}{q^{λ} − 1} ∑_{d=0}^{∞} ∑_{i=1}^{h}(q^{l(D_{i}+dD^{∗})} − 1)t^{dμ}

= \frac{1}{q^{λ} − 1}(∑_{d=0}^{∞} ∑_{i=1}^{h} q^{l(D_{i}+dD^{∗})}t^{dμ} − \frac{h}{(1 − t^{μ})}).  (4.4)

Let H_{X} be a hyperplane section on X over F_{q}. H_{X} is ample with positive degree d(X). By the choice of μ, we have μ | d(X). Thus, H_{X} is linearly equivalent to D_{i} + \frac{d(X)}{μ}D^{∗} for some i. Since hD_{i} = 0, it follows

that hH_{X} is linearly equivalent to \frac{d(X)h}{μ}D^{∗}. This proves that aD^{∗} is an ample effective Cartier divisor for some positive integer a.

Fix such an integer a and write d = j + ak, where 0 ≤ j < a. For each pair (i, j), l(D_{i} + jD^{∗} + kaD^{∗}) is unbounded as k varies, because aD^{∗} is a multiple of the hyperplane section and all components of D_{i} + jD^{∗} are components of some hypersurface section. Thus, we may assume that l(D_{i} + jD^{∗} + baD^{∗}) > λ for some positive integer b and for all pairs (i, j). This means that there is a non-constant function f_{ij} in the vector space L(D_{i} + jD^{∗} + baD^{∗}). Then, because aD^{∗} is effective,

l(D_{i} + jD^{∗} + (d + b)aD^{∗}) = l(D_{i} + jD^{∗} + baD^{∗} + daD^{∗})
≥ l(daD^{∗}).  (4.5)

Since aD^{∗} is an ample Cartier divisor, Lemma 4.3 shows that for large d, l(daD^{∗}) is a polynomial in d of degree n. This and (4.4) show that Z_{n−1}(X, t) is a p-adic meromorphic function. That Z_{n−1}(X, t) has a simple pole at t = 1 with the given residue follows from (4.4). The theorem is proved.

Example 4.4. There are many varieties X such that A^{1}(X) = Z (cf. [LI]). Results of this type are called Noether theorems or Noether-Lefschetz theorems. Examples are Grassmann varieties (Igusa’s analogue of the Severi theorem), nonsingular complete intersections of dimension ≥ 3 (Grothendieck’s analogue of the Lefschetz theorem [GR]) and sufficiently general complete intersections of dimension 2 (Deligne’s analogue of the Noether theorem, see Exposé XIX in [DK]).

Remark 4.5. It is clear that Theorem 4.1 remains true if X is a non-singular curve. Let Pic(X/F_{q}) be the group of isomorphy classes of invertible sheaves on X/F_{q}. This is also the group of Cartier divisor classes because X is integral. One may define the zeta function of Cartier divisors (or invertible sheaves) as an Euler product over all effective prime Cartier divisors. This is a factor of Z_{n−1}(X, t). Then Theorem 4.1 remains true for the zeta function of Cartier divisors with A^{1}(X) replaced by Pic(X). The proof is the same (in fact, even easier). If X is locally factorial, then Pic(X) is isomorphic to A^{1}(X/F_{q}). It follows from Kleiman’s generalization of the Lang-Néron theorem that Pic(X/F_{q}) always has finite rank (Proposition IV.4 in [KL]).

From (4.4), one sees that the Newton polygon of the numerator of Z_{n−1}(X, t) is related to the Riemann-Roch problem.

5. Effective Cone and the Riemann-Roch Problem

In this section, we establish some properties of interior effective divisors, which will be used later on. Let D be an effective divisor on an n-dimensional projective variety X defined over any field F. The

Riemann-Roch problem is to determine the behaviour of l(dD) as d goes to infinity. This is well understood in the case when D is ample. In that case, the general Riemann-Roch theorem and the cohomology vanishing theorem show that l(dD) is a polynomial in d of degree n with leading coefficient equal to D^{n}/n! > 0, where D^{n} is the self-intersection number.

Zariski ([ZA2]) proved that if X is a nonsingular surface over an algebraically closed field and D is an effective divisor on X, then there are finitely many polynomials p_{1}(x), ⋯, p_{l}(x) such that for all large d, l(dD) = p_{ω(d)}(d), where ω(d) is a function of d with values in the set {1, ⋯, l}. These polynomials differ at most in their constant term. It is an open question whether or not ω(d) is always periodic. In higher dimensions, no such result is known.

Here, we are interested in giving a lower bound for the Riemann-Roch problem. Let X/F be a projective variety and D an effective divisor on X. Following Zariski [ZA2], we define two rings as follows:

R[D] = ∪_{d=0}^{∞}L(dD), R^{∗}[D] = ⊕_{d=0}^{∞}L(dD).

R^{∗}[D] is a graded ring. If R^{∗}[D] is finitely generated over F, then l(dD) (the dimension of the degree d part) is a polynomial for large d. This fails in general, even in the case of surfaces, as Zariski’s counter example shows. The question of whether or not R[D] is finitely generated is a generalization of Hilbert’s 14th problem. Finite generation holds for surfaces (Zariski’s theorem [ZA1]), but fails for three-folds (counter examples were given by Rees and later by Zariski [ZA2]).

Define the transcendence degree t(D) of D to be the transcendence degree of R[D]. t(D) takes values between 0 and n. The significance of t(D) is the following simple result, whose proof is immediate.

Proposition 5.1. There is a positive constant c such that for all large d, l(dD) ≥ cd^{t(D)}.

We now assume that A^{1}(X) has finite rank ρ. Then, A^{1}(X)/A^{1}(X)_{tor} is a lattice L of full rank in the ρ-dimensional Euclidean space A^{1}(X) ⊗ R. Let Δ(X) be the effective cone generated by the effective divisor classes. This cone is ρ-dimensional. If σ is a face of Δ of any dimension, we say that an effective divisor D is an interior point of σ if its image in A^{1}(X) ⊗ R is an interior point of σ, and we say that D is in σ if this image is in σ.

Proposition 5.2. (i). Let σ be an open face of Δ(X). Then all effective divisors in σ have the same transcendence degree. (ii). The interior face of Δ has transcendence degree n. (iii). The hyperplane section H_{X} is an interior point of Δ.

Proof. Let D_{1} and D_{2} be two effective divisors in σ and let L_{1} and L_{2} be their lattice points in σ. Since σ is open, for all large integers a, aL_{1} − L_{2} = L(a) is also a lattice point in σ. Thus, for a certain positive integer d we have adD_{1} = dD_{2} + D(a), where D(a) is an effective divisor. This shows that t(D_{1}) ≥ t(D_{2}). By symmetry, we see that t(D_{1}) = t(D_{2}), which proves (i).

Given an effective divisor D, aH_{X} − D is an effective divisor class for all large integers a, because H_{X} is very ample. This proves that H_{X} is an interior point, and (iii) follows. Clearly, t(H_{X}) = n. Thus (ii) follows from (i). The proposition is proved.

By part (i) of this proposition, we can define the transcendental degree t(σ) of any open face σ in Δ to be the transcendence degree of any of its lattice points. An interesting question is to determine t(σ) for each open face σ of Δ(X). This is trivial if ρ(X) = 1. In general, it might be a difficult problem. For our later application, we would like to know the answers to the following much weaker questions: When is the minimum value t(D) on Δ(X) positive? When is the minimum of t(D) greater than 1?

Example 5.3. Let a_{1}, ⋯, a_{ρ} be ρ positive integers. Let X be the product of the projective spaces P^{a_{i}}. Let D_{i} be the inverse image in X of a hyperplane in P^{a_{i}}. D_{i} is an effective divisor on X. It is well-known that A^{1}(X) is the free abelian group of rank ρ generated by the D_{i}. Furthermore, it is not difficult to prove that the effective cone Δ(X) is also generated by the D_{i}. Now, it is clear that t(D_{i}) is at least a_{i} (simply choose a_{i} algebraically independent functions on P^{a_{i}}). Thus, the minimum transcendence degree on Δ(X) is at least the minimum of the a_{i}.

Corollary 5.4. Let D be an effective divisor on a projective variety X of dimension n. Suppose that D is an interior point of Δ(X). Then there are positive numbers c_{1} < c_{2} such that for large d we have

c_{1}d^{n} ≤ l(dD) ≤ c_{2}d^{n}.

Proof. The lower bound follows from the above two propositions. The upper bound is true because D < aH_{X} for some hyperplane section H_{X} and some positive integer a.

Corollary 5.5. Let X/F_{q} be a projective variety. Let Δ_{a} be the cone generated by the ample divisors, and let Δ^{∘} be the interior cone of Δ. Then Δ^{∘} ⊃ Δ_{a}.

Proposition 5.2 shows that the very ample divisors are interior points of Δ. If D is ample, then some multiple of D is very ample. The corollary follows.

A related result is a theorem of Kleiman [KL] which says that over an algebraically closed field, Δ_{a} is the interior of the closed pseudo ample cone in A_{num}^{1}(X).

Remark 5.6. The inclusion in Corollary 5.5 can be strict. For example, let f : X′ → X be the blowing up at a point of a nonsingular surface X. Let E be the exceptional curve on X′. Then, A^{1}(X′) = f^{∗}A^{1}(X) ⊕ Z, where Z is generated by E. For any effective divisor D on X which is an interior point of Δ(X), the effective divisor f^{∗}D + E on X′ is an interior point of Δ(X′). But f^{∗}D + E is not ample because (f^{∗}D + E) · E = E^{2} = −1. If we consider the quadric surface defined by xy = zw, then it is easy to check that the interior of the effective cone coincides with the ample cone. Both of them are two dimensional open cones.

If Δ is open, Corollary 5.4 gives a good estimate for l(dD) for high multiples of any effective divisor. If ρ > 1, l(dD) does not tend to have a good uniform lower bound. This suggests that Δ should not be open. It would be interesting to know whether or not Δ(X) is closed. More precisely,

Question 5.7. Let X/F be a non-singular projective variety. Is it true that the monoid E^{1}(X) of effective divisor classes in Num(X) is finitely generated?

An affirmative answer to this question would have important applications. For example, in the next section we shall prove that if the monoid of effective divisors in A^{1}(X/F_{q}) is finitely generated, then Conjecture II is true for the zeta function of divisors. For another application, let us consider a projective nonsingular surface X over an algebraically closed field. Nakai’s criterion says that a divisor D on X is ample if and only if D^{2} > 0 and D · C > 0 for all irreducible curves C in X. If, now, the monoid of effective divisor classes is finitely generated, then it is generated by a finite number of prime divisors C_{i}. Applying Nakai’s criterion, one concludes that a divisor D is ample if and only if D^{2} > 0 and D · C_{i} > 0. This would prove that the determination of whether or not a divisor is ample can be achieved in a finite amount of time.

6. Zeta Functions of Divisors: The General Case

Our main purpose in this section is to prove that Conjecture II is true for divisors if the monoid of effective divisor classes is finitely generated. In fact, we also obtain information about the torsion part of A^{1}(X).

Let X/F_{q} be a projective variety. Let E^{1}(X) be the monoid generated by all effective prime divisors modulo linear equivalence. Equivalently, E^{1}(X) is the monoid of effective divisor classes. E^{1}(X) is a torsionfree monoid, because the degree of an effective divisor is positive. Let D_{1}, ⋯, D_{r} be effective divisors. They are said to be Z-linearly independent if there are no non-negative integers a_{i} and b_{j}, not all zero, such that ∑_{i} a_{i}D_{i} = ∑_{j} b_{j}D_{j}, where i ≠ j. The rank of E^{1}(X) is defined to be the maximum number of Z-linearly independent elements in E^{1}(X). It is easy to check that the rank of E^{1}(X) is the same as that of A^{1}(X). Even if E^{1}(X) is finitely generated, it may not have a basis. Let A^{1}(X) be finitely generated with rank ρ. Then A^{1}(X) ⊗ R is a ρ dimensional Euclidean space. Let Δ(X) be the cone generated by the images of E^{1}(X). Δ is a ρ dimensional cone in A^{1}(X) ⊗ R. A^{1}(X)/A^{1}(X)_{tor} (the free part) is a lattice L of full rank. That E^{1}(X) is finitely generated means that the cone Δ(X) is closed with finitely many faces. In this case, A^{1}(X) is a finitely generated abelian group.

To describe the special value lim_{t→1^{−}}(1−t)^{ρ}Z_{n−1}(X, t), we need to define the regulator R(X). For each minimal parallelepiped B_{i} of the lattice L lying in the cone Δ with the origin as vertex, let D_{i,j} (1 ≤ j ≤ ρ) be the remaining vertices which generate the parallelepiped. Let d_{i,j} be the degree of D_{i,j}. These are positive integers because they are in the cone Δ. A set B of such B_{i} is called a fundamental cover of Δ if the B_{i} are distinct and the cones generated by the B_{i} cover Δ. Since Δ is closed with finitely many faces, such a fundamental cover contains only finitely many B_{i}. In general, there are infinitely many fundamental covers of Δ. One can give a partial ordering on the set of all fundamental covers. Then there is an initial element with respect to this partial ordering. Choose a fundamental cover B = {B_{i}} of Δ and define the regulator by

R(X) = ∑_{i} \frac{1}{∏_{j=1}^{ρ} d_{i,j}} .

It can be proved directly that R(X) is independent of the choice of the fundamental cover B. However, we shall not do this, because independence will automatically follow from the proof of Theorem 6.1. In particular, the regulator is well defined.

If Δ(X) is a simplicial cone (this is the case if ρ ≤ 2), there are ρ uniquely determined lattice points P_{j} generating Δ(X) such that each P_{j} is minimal. The regulator R(X) then admits the decomposition R^{∗}/ ∏ d_{j}, where d_{j} is the degree of P_{j} and R^{∗} is the number of lattice points in the parallelepiped generated by the P_{j}. Thus, R^{∗} is intrinsic and independent of the embedding of X. The geometric invariant R^{∗} can be interpreted as the volume of a fundamental domain in Δ (suitably normalized). Thus, R^{∗} is the analogue of the classical regulator of a number field. The other factor ∏ d_{j} in R can be viewed as an analogue of the discriminant of a number field. This is an algebraic invariant, which depends on the projective embedding of X. Theorem 6.1 below shows that if Δ(X) is simplicial, we have a complete analogue of the Dedekind residue formula. On the other hand, if Δ(X) is not simplicial, it seems that the regulator R does not admit such a decomposition into a product of a geometric quantity and an algebraic quantity (they are somehow mixed together).

Now we can state our second main result.

Theorem 6.1. Let X/F_{q} be a projective variety. Let the monoid E^{1}(X) be finitely generated over Z_{≥0}. Let A^{1}(X) have rank ρ and torsion order h. Then Z_{n−1}(X, t) is a p-adic meromorphic function on the closed unit disk | t |≤ 1 with a pole of order ρ at t = 1. Furthermore, the special value is given by

lim_{t→1^{−}}(1 − t)^{ρ}Z_{n−1}(X, t) = −\frac{1}{q^{λ} − 1}hR(X),

where λ as before is the integer such that F_{q^{λ}} is the algebraic closure of F_{q} in the quotient field K(X) of S(X) (thus, q^{λ} − 1 is the number of roots of unity in K(X)).

Lemma 6.2. Let l(a_{1}, ⋯, a_{s}) be an increasing function from Z_{≥0}^{s} to Z_{≥0}, i.e., for all i,
l(a_{1}, ⋯, a_{i}, ⋯, a_{s}) ≤ l(a_{1}, ⋯, a_{i} + 1, ⋯, a_{s}).  (6.1)

Define the following formal power series

f(t) = ∑_{a_{i}=0}^{∞} q^{l(a_{1},···,a_{s})}t^{a_{1}d_{1}+···+a_{s}d_{s}}.  (6.2)

Then f(t) is p-adic meromorphic on the closed unit disk with a pole of order at most s at t = 1. The order equals s if and only if the function l(a) is bounded, in which case f(t) is a rational function.

Proof. We first assume that l(a) is bounded with maximum value k. Let l(b_{1}, ⋯, b_{s}) = k. Because l(a) is increasing, we have

f(t) = q^{k} \frac{t^{b_{1}d_{1}} ⋯ t^{b_{s}d_{s}}}{(1 − t^{d_{1}}) ⋯ (1 − t^{d_{s}})} + ∑′q^{l(a_{1},···,a_{s})}t^{∑ a_{i}d_{i}},  (6.3)

where the summation ∑′ runs over all nonnegative integers a_{i} with at least one of the a_{i} smaller than b_{i}. This reduces us to a finite number of summations over s − 1 copies of Z_{≥0}. By induction, the second term of (6.3 ) is a rational function with a pole of order at most s − 1. Thus, f(t) is a rational function with a pole of order s at t = 1.

Next, we assume that l(a) is unbounded. It suffices to prove that for any high power q^{h} of q, the reduction of f(t) modulo q^{h} is a meromorphic function on the closed unit disk with a pole of order at most s−1 at t = 1. Fix a positive integer h and let l(b_{1}, ⋯, b_{r}) ≥ h for some integers b_{i}. Since l(a) is an increasing function in the variables a_{i}, it follows that

f(t) ≡ ∑′q^{l(a_{1},···,a_{s})}t^{∑_{i} a_{i}d_{i}} (modq^{h}),  (6.4)

where the summation ∑′ runs over all nonnegative integers a_{i} with at least one of the a_{i} smaller than b_{i}. This reduces f(t) to a finite number of summations over s − 1 copies of Z_{≥0}. By induction, each of the summations is meromorphic on the closed unit disk with a pole of order at most s − 1. The proposition is proved.

In certain cases, a better meromorphic continuation of f(t) can be obtained. We need the following simple lemma, whose proof is obvious.

Lemma 6.3. In the notation of Lemma 6.2, assume l(ke_{i}) ≥ ck for some constant c and all large k, where e_{i} is the vector a with a_{i} = 1 and a_{j} = 0 for j ≠ i. Then f(t) is p-adic analytic in a disk | t |< 1 + ϵ. If we assume that lim_{k→∞} l(ke_{i})/k = ∞, then f(t) is a p-adic entire function.

Proof of Theorem 6.1. Fix a decomposition of the abelian group A^{1}(X) = L ⊕ T, where L is a free abelian group of rank ρ containing the hyperplane section H_{X} and T is the torsion group of order h with elements T_{1} = 0, ⋯, T_{h}. The degree of T_{i} is zero. Consider the ρ-dimensional vector space L ⊗ R = R^{ρ}, in which L is a lattice of full rank. A lattice point is effective if the divisor class representing it is effective. The set of all linear combinations of effective lattice points with nonnegative real coefficients forms a ρdimensional closed cone Δ with vertex at the origin. Because E^{1}(X) is a finitely generated monoid, Δ has finitely many faces. Note that there may be lattice points in Δ which are not effective. However, such lattice points can be written as linear combinations of effective lattice points with non-negative rational coefficients; thus, some multiple of such a lattice point will be an effective lattice point. Any effective divisor is linearly equivalent to P + T_{j} for some j and a lattice point P in Δ.

Let H_{X} be the lattice point corresponding to the hyperplane section on X. Let L_{X} be the half line emanating from the origin and passing through H_{X}. We fix a simplicial decomposition of Δ. This means that we decompose Δ into a finite union of closed ρ-dimensional simplicial cones Δ_{i} such that Δ_{i} ∩Δ_{j} (i ≠ j) has dimension less than ρ. By the definition of the zeta function, Lemma 4.3 and the inclusion-exclusion principle, we have

Z_{n−1}(X, t) = ∑_{j=1}^{h} ∑_{P ∈Δ} \frac{q^{l(T_{j}+P)} − 1}{q^{λ} − 1}t^{deg(P)},

= ∑_{j=1}^{h} ∑_{i} ∑_{P ∈Δ_{i}} \frac{q^{l(T_{j}+P)} − 1}{q^{λ} − 1}t^{deg(P)} + R(t),  (6.5)

= ∑_{j=1}^{h} ∑_{i} G_{j,i}(t) + R(t),

where P denotes a lattice point in Δ and R(t) is a finite alternating sum of power series similar to the G_{j,i}(t) with summation over cones of smaller dimension.

We claim that each G_{j,i}(t) is p-adic meromorphic on the closed unit disk with a pole of order ρ at t = 1 such that the value of (1 − t)^{ρ}G_{i}(t) at t = 1 is a positive rational number. The same proof will also show that R(t) is p-adic meromorphic on the closed unit disk with a pole of order at most ρ − 1 at t = 1. This is sufficient to prove the first part of the theorem.

Without loss of generality, we consider G_{1,1}(t). Let M_{1} be the monoid of lattice points in Δ_{1}. Let {D_{1}, ⋯, Dρ} be ρ effective lattice points of Δ_{1} which span the cone Δ_{1}. Thus, the D_{i} are actually effective divisors (if this is not true of the D_{i}, then replace them by the multiples hD_{i}). Let M′_{1} be the monoid generated by the D_{i}. Then the quotient monoid M_{1}/M′_{1} is finite. Choose representatives E_{k} such that each lattice point of Δ_{1} can be uniquely expressed as E_{k} + P for some k and P ∈ M′_{1}. Let d_{i} = degD_{i}. Then to prove the claim it is sufficient to prove it for the following power series:

(q^{λ} − 1)G(t) = ∑_{a_{i}=0}^{∞}(q^{l(T_{1}+E+∑_{i} a_{i}D_{i})} − 1)t^{e+∑_{i} a_{i}d_{i}}

= ∑_{a_{i}=0}^{∞} q^{l(T_{1}+E+∑_{i} a_{i}D_{i})}t^{e+∑_{i} a_{i}d_{i}} − ∑_{a_{i}=0}^{∞} t^{e+∑_{i} a_{i}d_{i}}

= F_{1}(t) − F_{2}(t),  (6.6)

where E is one of the E_{k} and e = deg(E).

One computes that

F_{2}(t) = \frac{t^{e}}{(1 − t^{d_{1}}) ⋯ (1 − t^{d_{ρ}})}.  (6.7)

This is a rational function with a pole of order ρ at t = 1 such that the value of (1 − t)^{ρ}F_{2}(t) at t = 1 equals 1/ ∏_{i} d_{i}, which is a positive rational number. To prove the claim for G(t), it suffices to prove that F_{1}(t) is meromorphic on the closed unit disk with a pole of order less than ρ at t = 1.

Because the D_{i} are effective, l(T_{1} + E + ∑_{i} a_{i}D_{i}) is an increasing function in the a_{i}. Since ∑_{i} D_{i} is an interior point of Δ, Proposition 5.2 shows that there is a positive integer b such that T_{1} + E + b ∑_{i} D_{i} is an effective divisor class. Corollary 5.4 then implies that l(T_{1} + E + ∑_{i} a_{i}D_{i}) is unbounded. The desired conclusion about F_{1}(t) follows from Lemma 6.2. This proves that Z_{n−1}(X, t) is p-adic meromorphic on the closed unit disk with a pole of order ρ at t = 1.

From (6.5)-(6.7) and the above proof, one sees that the special value is given by −\frac{1}{q^{λ}−1}hR′(X), where

R′(X) = ∑_{i} \frac{| M_{i}/M′_{i} |}{d_{i,1} ⋯ d_{i,ρ}} ,

M_{i} is the monoid of lattice points in Δ_{i} and M′_{i} is a submonoid of M_{i} generated by ρ effective lattice points of degree d_{i,j} such that | M_{i}/M′_{i} | is finite. The above proof also shows that R′(X) is the special value of the following rational function

Z(Δ, t) = ∑_{P ∈Δ} t^{degP} .

Thus, R′(X) is of a combinatorial nature, and it is independent of the choice of simplicial decomposition of Δ. Using a finer simplicial decomposition of Δ, if necessary, we may assume that we can choose M′_{i} = M_{i}. Such a fine simplicial decomposition corresponds uniquely to a fundamental cover B of Δ. Thus, R′(X) = R(X).

Because the special value is independent of the choice of simplicial decomposition, it follows that the regulator is well defined. The theorem is proved.

Example 6.4. Let π : X ⟶ C be a ruled surface over a nonsingular curve C with invariant e ≥ 0 (see Proposition V.2.20 in [HA2]). Let C_{0} ⊂ X be a section and let f be a fibre. Then A^{1}(X) = Z ⊕ π^{∗}A^{1}(C) is finitely generated of rank 2. Furthermore, the effective cone Δ(X) is the two dimensional closed simplicial cone spanned by C_{0} and f, because the intersection class of any irreducible curve on X is contained in the cone spanned by C_{0} and f. Thus Theorem 4.1 holds for a ruled surface with invariant e ≥ 0. Note that the ruled surface C × P^{1} has invariant e = 0.

Assuming further conditions on the Riemann-Roch problem, we have the following stronger meromorphic continuation.

Theorem 6.5. Let X/F_{q} be a projective variety. Let E^{1}(X) be finitely generated over Z_{≥0} with a set of generators D_{i}. Suppose that for each i, l(bD_{i}) is unbounded. Then Z_{n−1}(X, t) is p-adic meromorphic in a disk | t |< 1 + ϵ.

Proof. Since l(bD_{i}) is unbounded, there are nonconstant functions f_{i} ∈ L(D_{i}). Using the powers of the f_{i}, we see that l(bD_{i}) ≥ ck for some positive number c. This shows that l(kD) ≥ ck for large k, where D is any effective divisor. The theorem follows from Lemma 6.3 and the proof of Theorem 6.1.

Similarly, using Lemma 6.3, one can prove

Theorem 6.6. Let X/F_{q} be a projective variety. Assume that E^{1}(X) is finitely generated over Z_{≥0} with a set of generators D_{i}. Suppose that for each i, l(kD_{i})/k is unbounded. Then Z_{n−1}(X, t) is p-adic meromorphic.

Examples 6.7. Let X be the nonsingular quadric surface defined by xy = zw in P^{3}. Then it is well known that A^{1}(X) = Z ⊕ Z is generated by D_{xw} and D_{yw}, where D_{xw} is the prime divisor defined by x = w = 0. Furthermore, all effective divisors are of the form aD_{xw} + bD_{yw} with nonnegative integral coefficients. Thus, E^{1}(X) is generated by D_{xw} and D_{yw} of rank 2. One computes that

div(\frac{z}{x}) = D_{zy} − D_{xw} > −D_{xw}
div(\frac{z}{y}) = D_{zx} − D_{yw} > −D_{yw}.

Thus, the conditions of Theorem 6.5 are satisfied with ρ = 2. Another example of Theorem 6.5 is any abelian variety X such that E^{1}(X) is finitely generated. Over an abelian variety X, l(kD) is always unbounded for any effective divisor D, see [MU, Chapter II].

The quadric surface X is isomorphic to P^{1} × P^{1}. More generally, let a_{1}, ⋯, a_{ρ} be ρ positive integers. Let X be the product of the projective spaces P^{a_{i}}. By Example 5.4, the monoid E^{1}(X) of effective divisor classes is finitely generated of rank ρ. The minimum transcendence degree of the effective cone Δ(X) is at least the minimum of the a_{i}, which is positive. Thus X satisies the conditions of Theorem 6.5. If all of the a_{i} are greater than 1, then X satisfies the conditions of Theorem 6.6. If some of the a_{i} is 1, the conclusion of Theorem 6.6 also holds (by a direct computation similar to section 2). Another example of Theorem 6.6 is the product X = ∏_{i=1}^{ρ} X_{i}, where each X_{i} is a non-singular complete intersection of dimension at least 3. Grothendieck’s theorem implies that E^{1}(X) is the free monoid of rank ρ.

If X is n-dimensional with n ≥ 2 and A^{1}(X) is finitely generated of rank 1, then the minimum weight of Δ(X) is n. Thus X satisfies the conditions of Theorem 6.6. This shows that Theorem 6.6 is a generalization of Theorem 4.1.

7. Connections with the Higher Codimensional Riemann-Roch Problem

In this last section, we discuss possible approaches which might lead to a proof (at least in certain cases) of the conjectures formulated in this paper. The Riemann-Roch approach used in sections 4 and 5 naturally suggests a higher codimensional Riemann-Roch problem. We show how information about the higher codimensional Riemann-Roch problem can be used to prove Conjecture II.

We start with the various rationality proofs for the zeta function of zero cycles on a curve. Recall that there are three quite independent rationality proofs for the classical zeta function of a smooth curve over a finite field. The first proof, due to Weil, is to use the Riemann-Roch theorem for curves. The second type of proof is p-adic in nature (Dwork, Washnitzer-Monsky, Lubkin, Grothendieck-Berthelot). The third is the l-adic proof (Grothendieck). The last two approaches are cohomological in nature.

For zeta functions of zero cycles in higher dimensions, both the p-adic and the l-adic approaches generalize and work very well. Each approach has its advantages. For example, the l-adic approach has been successfully used to prove the Riemann hypothesis. However, to study p-adic analytic variation one has to use p-adic methods. In the general case, because Z_{r}(X, t) is expected to be a non-rational p-adic meromorphic function for 0 < r < n, it is almost certain that the l-adic approach would not work for these more general zeta functions. However, we do not know any consideration that would exclude the p-adic approach. Thus, one might hope that Z_{r}(X, t) can be expressed in terms of the Fredholm determinants of certain nuclear operators (generalizing the Frobenius operators) on an infinite dimensional p-adic Banach space. For this to work, one needs a trace formula of Dwork type. This would generalize Dwork’s proof and establish the p-adic meromorphic continuation of Z_{r}(X, t). One might also hope that there is a p-adic cohomology theory (of Dwork type or some other type) and a fixed point theorem which completely describe all the zeta functions, as in the case of zero cycles. Such a theory, if it exists, is necessarily infinite dimensional. This suggests that in studying the meromorphic continuation of zeta functions of r-cycles for 0 < r < n, one would expect much more complexity than in the case of zero cycles (because of the difficulties of extension from rational functions to meromorphic functions or from finite dimensional vector spaces to infinite dimensional vector spaces).

It is mysterious why the Riemann-Roch approach does not generalize to higher dimensions, even in the case of zero cycles. In section 4, we showed how the general Riemann-Roch theorem can be successfully applied to study zeta functions of divisors in certain cases. Motivated by this, we now discuss a generalization of the Riemann-Roch approach to zeta functions of all cycles, not just zero cycles and divisors. This involves a generalization of the Riemann-Roch problem from divisors to arbitrary cycles.

Let X/F_{q} be an n-dimensional projective variety in a fixed projective space. For any 0 ≤ r ≤ n, write

Z_{r}(X, t) = ∑_{d=0}^{∞} M_{d}t^{d},  (7.1)

where M_{d} is the number of effective r-cycles of degree d on X. Let E_{r}(X) be the monoid of effective r-cycles modulo rational equivalence.

First, we assume that the monoid E_{r}(X) is finitely generated. Then one can construct the cone Δ_{r}(X) with vertex at the origin that is generated by the effective r-cycle classes. Δ_{r} is closed and has finitely many faces. A lattice is given by A_{r}(X) modulo torsion elements. By a simplicial decomposition and a finite quotient argument, as in the proof of Theorem 6.1, we are reduced to the power series

G_{r}(X, t) = ∑_{a_{i}=0}^{∞} M(a_{1}, ⋯, a_{s})t^{a_{1}d_{1}+···+a_{s}d_{s}},  (7.2)

where M(a_{1}, ⋯, a_{s}) is the number of effective r-cycles rationally equivalent to C_{0} + a_{1}C_{1} + ⋯ + a_{s}C_{s}, C_{0} is an r-cycle, the C_{i} (i ≥ 1) are effective r-cycles which are independent (as lattice points) and d_{i} is the degree of C_{i}.

Let X be a projective variety over a field F. Let CH_{r}(d, X) be the Chow scheme, which is an algebraic set parametrizing effective r-cycles over F of degree d on X. Given an r-cycle C of degree d on X, let L(C) be the set of effective r-cycles on X rationally equivalent to C. L(C) is either empty or a connected algebraic subset of CH_{r}(d, X) with the property that any two points of L(C) can be joined by a chain of rational curves. Thus, L(C) is in some sense almost a linear variety, as in the case of divisors. The simple geometric nature of L(C) suggests that over a finite field F = F_{q} it should be possible to compute or p-adically estimate the number of F_{q}-rational points N(C) on L(C), and, in particular, determine the behaviour of the function N(C_{0} + d_{1}C_{1} + ⋯ + d_{s}C_{s}), where C_{0} is an r-cycle and the C_{i} (i ≥ 1) are rationally independent effective r-cycles.

In the case when C = D is a divisor, L(D) is a vector space of dimension l(D); and if F = F_{q}, then N(D) = (q^{l(D)} − 1)/(q^{λ} − 1), where λ is a positive integer independent of D. The general Riemann-Roch problem is to determine the function l(D_{0} + d_{1}D_{1} + ⋯ + d_{s}D_{s}), where D_{0} is a divisor and the D_{i} (i ≥ 1) are linearly independent effective divisors. This is equivalent to determining the behaviour of the varieties L(D_{0} + d_{1}D_{1} + ⋯ + d_{s}D_{s}) as the d_{i} vary.

In higher codimension, the general Riemann-Roch problem is then to determine the behaviour of the algebraic sets L(C_{0} + d_{1}C_{1} + ⋯ + d_{s}C_{s}) as the d_{i} vary, where C_{0} is an r-cycle and the C_{i} (i ≥ 1) are rationally independent effective r-cycles. Over a finite field, this means determining the behaviour of the function N(C_{0} + d_{1}C_{1} + ⋯ + d_{s}D_{s}). The higher codimensional Riemann-Roch problem can be expected to be a fruitful problem in algebraic geometry.

Returning to zeta functions of algebraic cycles, we say that X has the property (P_{r}) if for all rationally independent effective r-cycles C_{i} and any r-cycle C_{0}, there is a constant c_{r} such that N(C_{0} + ∑ d_{i}C_{i}) either is constant or else is congruent to c_{r} modulo any power q^{h} as long as all the d_{i} are sufficiently large depending on h. Note that N(∑ d_{i}C_{i}) is an increasing function of non-negative integers. In the case of divisors, this is always true, where one simply takes c_{n−1} = −1/(q^{λ} − 1).

Imitating the proof of Theorem 6.1, we can prove the following generalization for zeta functions of r-cycles.

Theorem 7.1. Let X/F_{q} be an n-dimensional projective variety. Let r be an integer with 0 ≤ r ≤ n. Assume that X/F_{q} has property (P_{r}) and E_{r}(X) is a finitely generated monoid. Let ρ_{r} and h_{r} be the rank and torsion order of A^{r}(X). Then the zeta function Z_{r}(X, t) of r-cycles is p-adic meromorphic on the closed unit disk with a pole of order ρ_{r} at t = 1. Its special value at t = 1 is given by the formula c_{r}h_{r}R_{r}(X), where R_{r}(X) is the regulator defined in a manner similar to case of divisors.

Thus, the two major problems left are the finiteness of the number of generators for the monoid E_{r}(X) and a determination (or good p-adic estimate) of the function N(C_{0} + d_{1}C_{1} + ⋯ + d_{s}C_{s}). This approach, if it works, would give more information about the zeta function of zero cycles than the p-adic approach or l-adic approach because it also gives the special value at t = 1, which involves the torsion order and the regulator.

To see an example, let us consider the zeta function of zero cycles on a projective space X = P^{n}. In this case,

Z_{0}(P^{n}, t) = \frac{1}{(1 − t)(1 − qt) ⋯ (1 − q^{n}t)}.

E_{0}(X) is generated by a point. Thus, A_{0}(X) has rank 1 and torsion order 1. The regulator R_{0}(X) is easily seen to be 1. The Riemann-Roch problem for zero cycles is then to compute the number M_{d} of effective zero cycles of degree d, which is simply the coefficient of t^{d} in Z_{0}(P^{n}, t). One checks that for large d, M_{d} is congruent to c_{0} = 1/(1 − q) ⋯ (1 − q^{n}). Thus, both the condition and the conclusion of Theorem 7.1 hold.

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