Poles of Zeta Functions of Complete Intersections

Daqing Wan Department of Mathematics University of California Irvine, CA 92697-3875 dwan@math.uci.edu

Abstract. A vanishing theorem is proved for ℓ-adic cohomology with compact support on a singular affine complete intersection. As an application, it is shown that for an affine complete intersection defined over a finite field of q elements, the reciprocal “poles” of the zeta function are always divisible by q as algebraic integers. A p-adic proof is also given, which leads to further q-divisibility of the poles or equivalently an improvement of the polar part of the Ax-Katz theorem for an affine complete intersection. Similar results hold for a projective complete intersection.

1. Introduction

Let F_{q} be a finite field of q elements with characteristic p. Let X be an n-dimensional algebraic set defined over F_{q}. The zeta function of X/F_{q} is defined by

\begin{aligned}Z(X, T) & = ∏_{x∈X_{0}} \frac{1}{1 − T^{deg(x)}} \\ & = exp(∑_{d=1}^{∞} \frac{T^{d}}{d} #X(F_{q^{d}})), \end{aligned}

where X_{0} denotes the set of closed points of X/F_{q} and #X(F_{q^{d}}) denotes the number of F_{q^{d}}-rational points on X. It is easy to see that Z(X, T) is a power series with integer coefficients. Dwork’s rationality theorem [Dw] shows that the zeta function Z(X, T) is rational in T. Thus, there are algebraic integers ρ_{1}, ⋯, ρ_{r}, β_{1}, ⋯, β_{s} such that

Z(X, T)^{(−1)^{n−1}} = \frac{∏_{i=1}^{r}(1 − ρ_{i}T)}{∏_{j=1}^{s}(1 − β_{j}T)}.

There is a good reason that we modified the above zeta function by the power (−1)^{n−1}. The ρ_{i} are called the reciprocal zeroes. The β_{j} are called the reciprocal poles. Deligne’s theorem on Riemann hypothesis [De2] shows that the absolute values of the ρ_{i} and the β_{j} are integral powers of \sqrt{q}.

In this article, we are interested in entireness properties of Z(X, T)^{(−1)^{n−1}}. Namely, we hope to say some “vanishing” information about the reciprocal poles β_{j}. Ideally, one would like Z(X, T)^{(−1)^{n−1}} to be essentially a polynomial except for possible trivial poles. This is indeed the case if X is sufficiently nice. The following theorem is well known and follows from the weak Lefschetz theorem and the Poincare duality in ℓ-adic cohomology developed in mid-sixties.

Theorem 1.1a (projective strong entireness). If X is an n-dimensional smooth projective complete intersection defined over F_{q}, then

Z(X, T)^{(−1)^{n−1}} = \frac{P (T)}{∏_{i=0}^{n}(1 − q^{i}T)^{(−1)^{n−1}}} ,

where P (T) is a polynomial with integer coefficients.

The hypersurface case of this theorem was first proved by Dwork in early sixties using p-adic methods. The general case can also be proved using Dwork’s methods, as shown recently by Adolphson-Sperber [AS2]. This theorem and the excision sequence of ℓ-adic cohomology immediately imply the following affine version which can be traced back to mid-sixties, see also [AS2] for a recent p-adic proof.

Theorem 1.1b (affine strong entireness). If X is an n-dimensional affine complete intersection such that both its projective closure and its infinity part are a projective smooth complete intersection over F_{q}, then

Z(X, T)^{(−1)^{n−1}} = \frac{P (T)}{(1 − q^{n}T)^{(−1)^{n−1}}} ,

where P (T) is a polynomial with integer coefficients.

Unfortunately, the above strong entireness does not hold in singular cases. Motivated by Bombieri’s observation [Bo, Theorem 1.1] and Dwork’s question [DGS, Remark 7.5 in Chapter II], it would be interesting to say some non-trivial information about the reciprocal poles β_{j} even if X is a singular complete intersection. In this direction, we shall prove the following result here. Theorem 1.2a (projective weak entireness). If X is an n-dimensional projective set theoretic complete intersection over F_{q}, then

Z(X, T)^{(−1)^{n−1}} = \frac{P (T)}{(1 − T)^{(−1)^{n−1}} ∏_{j=1}^{s}(1 − qγ_{j}T)},

where P (T) ∈ Z[T] and the γ_{j} are algebraic integers. Namely, the non-trivial reciprocal poles β_{j} (≠ 1) are divisible by q as algebraic integers.

If X^{aff} denotes the affine cone of a projective variety X, then it is easy to see that

Z(X^{aff}, T) = \frac{Z(X, qT)}{(1 − T)Z(X, T)}.

Thus, Theorem 1.2a is an immediate consequence of the following affine version. Theorem 1.2b (affine weak entireness). If X is an n-dimensional affine set theoretic complete intersection over F_{q}, then

Z(X, T)^{(−1)^{n−1}} = \frac{P (T)}{∏_{j=1}^{s}(1 − qγ_{j}T)},

where P (T) ∈ Z[T] and the γ_{j} are algebraic integers. Namely, the reciprocal poles β_{j} are divisible by q as algebraic integers.

Note that the weak entireness is false if we drop the complete intersection condition. For example, consider the affine variety

X = Spec F_{q}[X_{1}, X_{2}, X_{3}, X_{4}]/(X_{1}X_{3}, X_{1}X_{4}, X_{2}X_{3}, X_{2}X_{4}).  (1.1)

It is easy to see that X is just the union in A^{4} of the two planes X_{1} = X_{2} = 0 and X_{3} = X_{4} = 0 intersecting at the origin. One computes that Z(X, T) = (1 − T)/(1 − q^{2}T)^{2} and Z(X, T)^{(−1)^{2−1}} = (1 − q^{2}T)^{2}/(1 − T). Thus, the above weak entireness fails for this connected affine surface X.

Remark. For L-functions of exponential sums on the affine (or toric) n-space, the analogue of Theorem 1.2b is an immediate consequence of Dwork’s trace formula. Although the zeta function Z(X, T) can be expressed in terms of L-functions of exponential sums on an affine space, Theorem 1.2b cannot be derived directly from the exponential sum version. Theorem 1.2b is more subtle and involves cancellation or divisibility of functions, see section 3.

We shall give two proofs of Theorem 1.2b. Both are reduced to proving a vanishing theorem on cohomology. The first approach is conceptually very simple, uses Deligne’s integrality theorem and some standard properties of ℓ-adic cohomology. It works in greater generality, see section 2 for an extension of the above weak entireness theorem to certain L-functions over more general varieties. The second approach, which is how we first proved Theorem 1.2b, uses Dwork’s p-adic method. As Ogus pointed out, one could also use the rigid cohomology developed by Berthelot, which is a significant generalization of the Dwork-Monsky theory. The argument, entirely cohomological, would be similar to the proof for ℓ-adic case. We shall, however, contend with the much simpler (although somewhat less general) setting as studied by Dwork. This simpler p-adic approach is more explicit. It has the advantage to be able to improve Theorem 1.2b further and also to improve the polar part of the Ax-Katz theorem in the complete intersection case. The author does not know how to obtain this improvement by using ℓ-adic approach.

To recall the Ax-Katz theorem [Ax][Ka1], let X be an affine algebraic set defined by the vanishing of r polynomials in N variables of degrees d_{1}, ⋯, d_{r}. Let μ be the smallest non-negative integer which is greater than or equal to

\frac{N − ∑_{i=1}^{r} d_{i}}{max_{i} d_{i}} .

The Ax-Katz theorem says that all reciprocal zeroes ρ_{i} and all reciprocal poles β_{j} are divisible by q^{μ} as algebraic integers. Since this theorem is best possible in general for each N and each multi-degree {d_{1}, ⋯, d_{r}}, it would be unreasonable to expect uniform improvements of q-divisibility for all the α_{i} and β_{j} in general. However, in various special cases, improvements have been made by Adolphson-Sperber [AS1] if one takes into account the terms actually appearing in the f_{i} and by Moreno-Moreno [MM] if one takes into account the p-digits of the exponents of the terms actually appearing in the f_{i}. In the latter case, the characteristic p is necessarily small, say p < max d_{i}, in order for an improvement to occur.

Now, if X is an affine complete intersection, the above weak entireness theorem shows that all the reciprocal poles β_{j} are already divisible by q as algebraic integers. This suggests that it might be reasonable to expect an improvement of the polar part of the Ax-Katz theorem for complete intersections. We shall show that this is indeed the case in some sense. Recall that if X is a complete intersection of co-dimension r in A^{N}, Macaulay’s unmixedness theorem [Ku, p187] shows that one can choose a regular sequence of polynomials f_{1}, ⋯, f_{r} of F_{q}[x_{1}, ⋯, x_{N}] such that X is defined by the vanishing of the set {f_{1}, ⋯, f_{r}}. The sequence {f_{1}, ⋯, f_{r}} being a regular sequence means that, for each 1 ≤ i ≤ r, the multiplication map f_{i} on the ring F_{q}[x_{1}, ⋯, x_{N}]/(f_{1}, ⋯, f_{i−1}) is injective. Theorem 1.3. Let X be a complete intersection in A^{N} defined by the vanishing of a regular sequence of polynomials {f_{1}, ⋯, f_{r}} over F_{q} in N variables of degrees d_{1}, ⋯, d_{r}. Let ν be the smallest non-negative integer which is greater than or equal to

\frac{(N − 1 − ∑_{i=1}^{r} d_{i})}{max_{i} d_{i}}.

Then, all the reciprocal poles β_{j} of Z(X, T)^{(−1)^{N−r−1}} are divisible by q^{1+ν} as algebraic integers.

Since 1 + ν ≥ 1, Theorem 1.3 is an improvement of Theorem 1.2b. It is easy to see that

1 + ν = \{ \begin{matrix}μ, & if (N − 1 − ∑_{i} d_{i})/ max d_{i} is a non-negative integer, \\ 1 + μ, & otherwise. \end{matrix}

Thus, Theorem 1.3 is also an improvement of the polar part of the Ax-Katz theorem in the case of affine complete intersections. We shall show that Theorem 1.3 is best possible for some hypersurfaces of degree d in A^{N} provided that d is greater than 2 and d does not divide (N − 1)(N − 2). Presumably, Theorem 1.3 has a direct Hodge-theoretic analogue, improving earlier results of Deligne-Dimca etc [DD] in this direction. Of course, Theorem 1.3 gives new information only for singular complete intersections. For a sufficiently smooth complete intersection, much more precise information about the poles is already given in Theorem 1.1b.

Acknowledgement. The author wishes to thank N. Katz for fruitful discussions on the ℓ-adic proof of Theorem 1.2b.

2. Poles of L-functions: ℓ-adic methods

In this section, we use ℓ-adic cohomology and Deligne’s integrality theorem to prove a more general weak entireness result. The main idea is to use the excision sequence to prove a vanishing theorem on affine complete intersections. The vanishing theorem also implies a weak Lefschetz theorem for singular projective complete intersections. A quick summary of the basic properties for ℓ-adic cohomology is given in Katz [Ka2].

Let X be a separated scheme of finite type over a field K. Let ℓ be a prime number different from the characteristic of K. Let E be a constructible ℓ-adic sheaf on X. We shall use H_{c}^{i}(X, E) to denote the i-th ℓ-adic cohomology group with compact support. The restriction E|U of a sheaf E to a subscheme U will also be denoted by E. Lemma 2.1. Let X be an n-dimensional set theoretic complete intersection in some affine, smooth and equi-dimensional Y of finite type over K. Suppose that E is an lisse ℓ-adic sheaf on Y. Then for all i < n, we have

H_{c}^{i}(X, E) = 0.

Proof. We may assume that X is an ideal theoretic complete intersection defined by the vanishing of r regular functions on Y with dim(X) = dim(Y)−r. We prove the lemma using induction on r.

If r = 0, then X = Y is smooth, affine and equi-dimensional. The Poincare duality (valid for smooth, geometrically connected X and lisse E) shows that as ℓ-adic vector spaces,

dimH_{c}^{i}(X, E) = dimH^{2n−i}(X, E),

where H^{j} denotes j-th ℓ-adic cohomology without support. On the other hand, the Lefschetz affine theorem (valid for affine X and any constructible E) shows that

H^{j}(X, E) = 0 (j > n),

Thus, the lemma is true for r = 0 (this case is well known).

Suppose now that the lemma is true for all complete intersections which can be defined by less than r(> 0) equations in a smooth affine equi-dimensional scheme of finite type. Assume now that X is a complete intersection defined by the vanishing of r equations {f_{1}, ⋯, f_{r}} on some smooth affine equi-dimensional Y of finite type. Since Y is Cohen-Macaulay, by re-choosing the generators if necessary, we may assume that the sequence {f_{1}, ⋯, f_{r}} forms a regular sequence of the ring of regular functions on Y. In particular, for each 1 ≤ i ≤ r, the subsequence {f_{1}, ⋯, f_{i}} also defines a complete intersection in Y.

Let Z be the affine complete intersection in Y defined by the vanishing of the first r − 1 functions {f_{1}, ⋯, f_{r−1}}. Let U be the compliment of X in Z. The variety U is just the localization of Z at f_{r} and thus U is the complete intersection defined by the vanishing of the r − 1 functions {f_{1}, ⋯, f_{r−1}} in the smooth affine equi-dimensional Y_{r}, where Y_{r} is the localization of Y at f_{r}. The sheaf E is clearly lisse on both Z and U as it is lisse on the ambient space Y. By induction, we have

H_{c}^{i}(Z, E) = 0 (i < dim(Z)), H_{c}^{j}(U, E) = 0 (j < dim(Z)).  (2.1)

Since U is open in Z with closed complement X, we have the excision long exact sequence for ℓ-adic cohomology with compact support:

H_{c}^{i}(Z, E) ⟶ H_{c}^{i}(X, E) ⟶ H_{c}^{i+1}(U, E) ⟶ H_{c}^{i+1}(Z, E).

This exact sequence and (2.1) together show that

H_{c}^{i}(X, E) = 0

for all i < dim(Z) − 1 = dim(X). The proof is complete. Lemma 2.2. Let X be an n-dimensional projective set theoretic complete intersection in some projective, smooth and equi-dimensional Y over K. Let E be an lisse ℓ-adic sheaf on Y. Then the natural map

H^{i}(Y, E) ⟶ H^{i}(X, E)

is an isomorphism for all i < n and injective for i = n.

Proof. This is a consequence of Lemma 2.1. We may assume that X is defined by the vanishing of a regular sequence {f_{1}, ⋯, f_{r}} of elements in the homogeneous coordinate ring of Y with the same degree. For 0 ≤ j ≤ r, let Y_{j} be the projective complete intersection in Y defined by the vanishing of {f_{1}, ⋯, f_{j}}. Thus, we have

X = Y_{r} ⊂ Y_{r−1} ⊂ ⋯ ⊂ Y_{0} = Y.

Let U_{j} be the compliment Y_{j} − Y_{j+1} of Y_{j+1} in Y_{j}. It is easy to see that U_{j} is the complete intersection in the smooth affine and equi-dimensional Y − {f_{j+1} = 0} defined by the vanishing of the regular functions {f_{1}/f_{j+1}, ⋯, f_{j}/f_{j+1}} on Y − {f_{j+1} = 0}. Lemma 2.1 shows that

H_{c}^{i}(U_{j}, E) = 0, for i < dim(Y_{j}).

This and the long exact excision sequence

H_{c}^{i}(U_{j}, E) ⟶ H^{i}(Y_{j}, E) ⟶ H^{i}(Y_{j+1}, E) ⟶ H_{c}^{i+1}(U_{j}, E)

show that the natural map

H^{i}(Y_{j}, E) ⟶ H^{i}(Y_{j+1}, E)

is an isomorphism for i < dim(Y_{j+1}) and injective for i = dim(Y_{j+1}). The proof is complete.

Remark. The condition in Lemma 2.2 can be weakened somewhat. For example, the above proof shows that the condition that E is lisse on Y can be weaken to the condition that E is lisse on the compliment Y − S, where S is the hypersurface in Y defined by the vanishing of the product f_{1} ⋯ f_{r}. Lemma 2.2 can be viewed as a weak Lefschetz theorem for complete intersections. It is well known if X is a non-singular complete intersection in Y. The general case does not seem to have been noted before.

We now return to L-functions over finite fields and prove a result more general than Theorem 1.2b. Let F_{q} be the finite field of q elements with characteristic p. Let X be an affine scheme of finite type over F_{q}. Let E be a constructible ℓ-adic sheaf on X. Following Deligne [De1], an ℓ-adic sheaf E is called integral if for each closed point x ∈ X_{0}, the eigenvalues of the geometric Frobenius Frob_{x} acting on the vector space E_{x} are algebraic integers in a fixed number field. Assume that E is integral. The L-function of the sheaf E on X is defined by

L(E/X, T) = ∏_{x∈X_{0}} \frac{1}{det(I − Frob_{x}T^{deg(x)})} ∈ 1 + T O[[T]],

where O denotes the ring of integers in the given number field. Grothendieck’s rationality theorem [Gr] shows that L(E/X, T) is a rational function in T.

Theorem 2.3 (weak entireness). Let X be a set theoretic complete intersection in some affine, smooth and equi-dimensional Y of finite type over F_{q}. Suppose that E is an integral and lisse ℓ-adic sheaf on Y. Then

L(E/X, T)^{(−1)^{dim(X)−1}} = \frac{P (T)}{∏_{j=1}^{s}(1 − qγ_{j}T)},

where P (T) is a polynomial in O[T] and the γ_{j} are algebraic integers.

Proof. Let n = dim(X). Let F̄_{q} be a fixed algebraic closure of F_{q}. The ℓ-adic cohomological formula [Gr] gives

L(E/X, T) = ∏_{i=0}^{2n} det(I − F T |H_{c}^{i}(X ⊗_{F_{q}} F̄_{q}, E))^{(−1)^{i−1}},

where F is the geometric Frobenius F acting on H_{c}^{i}(X ⊗_{F_{q}} F̄_{q}, E). Deligne’s integrality theorem [De1, Theorem 5.2.2] shows that for i > n, q^{i−n} divides the eigenvalues of F on H_{c}^{i}(X ⊗_{F_{q}} F̄_{q}, E) as algebraic integers. On the other hand, Lemma 2.1 shows that H_{c}^{i}(X ⊗_{F_{q}} F̄_{q}, E) = 0 for i < n. The proof is complete.

Taking E to be the constant sheaf Q_{ℓ} on Y, we obtain the following corollary which generalizes Theorem 1.2b. Corollary 2.4. Let X be a set theoretic complete intersection in some affine, smooth and equi-dimensional Y of finite type over F_{q}. Then

Z(X, T)^{(−1)^{dim(X)−1}} = \frac{P (T)}{∏_{j=1}^{s}(1 − qγ_{j}T)},

where P (T) is a polynomial in Z[T] and the γ_{j} are algebraic integers.

For another example, we consider the L-function attached to the exponential sum of the type

∑_{f(x)=0, g(x)≠0} Ψ(tr(\frac{h(x)}{g(x)})),

where Ψ is a non-trivial additive character of F_{q} and the f, g, h are pairwise prime polynomials in N variables over F_{q}. The space X is just the hypersurface {f = 0} contained in the smooth connected affine Y = A^{N} − {g ≠ 0}. The associated ℓ-adic sheaf on X is clearly the restriction to X of an lisse ℓ-adic sheaf on Y. Thus, Theorem 2.3 applies to this example.

Remark. If X is the n-dimensional torus G_{m}^{n}, one can say even more about the reciprocal poles qγ_{j} if E gives rise a continuous complex (or more general overconvergent p-adic) representation of the arithmetic fundamental group π_{1}^{arith}(X/F_{q}). In fact, Dwork trace formula shows that each γ_{j} is of the form q^{i}ρ_{j} for some integer i ≥ 0, where ρ_{j} is a reciprocal zero of P (T), the numerator of L(E/X, T)^{(−1)^{dim(X)−1}}. See Bombieri [Bo, Theorem 1.1] for more details in the case of L-functions of exponential sums over the torus G_{m}^{n}.

It would be interesting to know if the assumption in Theorem 2.3 that E is lisse on Y can be weakened to the condition that E is lisse on X. A positive answer would imply that the L-function of any complex continuous representation on any equi-dimensional affine curve over F_{q} (automatically a set theoretic complete intersection by Cowsik-Nori [CN]) is always a polynomial up to some trivial reciprocal poles of the form qγ_{j}, where the γ_{j} are roots of unity. This consequence is of course well known if X is smooth, but seems open if X is singular. As a partial evidence, a characteristic p version of our question was raised in [TW1] and answered positively in [TW2]. See the survey in [Wa] for related questions.

3. Poles of zeta functions: p-adic methods

In this section, we study the poles of the zeta function of an affine complete intersection using p-adic methods. This is the way we first proved Theorem 1.2b.

It is well known that the zeta function can be expressed as an alternating product of the Fredholm determinant of certain completely continuous Frobenius operator acting on a certain complex of p-adic Banach spaces. By induction, we show that the eigenvalues of the Frobenius operator acting on each piece of the complex are algebraic integers. To obtain strong entireness result, one needs to prove that the complex is acyclic in positive dimensions. Unfortunately, the complex is not acyclic in general, unless X is a sufficiently smooth complete intersection [AS2]. However, to obtain weaker result stated in Theorem 1.2b, we shall show that it is sufficient to restrict to a subcomplex and this subcomplex is acyclic in positive dimensions for X satisfying the assumption of Theorem 1.2b. By keeping track of the p-adic norm of the Frobenius operator acting on various pieces of the total complex, one then obtains Theorem 1.3

Let X be an affine algebraic set in A^{N}, defined by the vanishing of r polynomials f_{i}(x_{1}, ⋯, x_{N}) (1 ≤ i ≤ N) over F_{q}. We do not assume that X is a complete intersection at this point. To use Dwork’s trace formula, we need to express the zeta function of X in terms of the L-function of the exponential sums associated to the polynomial

g = x_{N+1}f_{1}(x_{1}, ⋯, x_{N}) + ⋯ + x_{N+r}f_{N+r}(x_{1}, ⋯, x_{N}).

Fix a non-trivial additive character Ψ of F_{q} with complex values, let Ψ_{k} = Ψ ∘ tr_{k} be the induced additive character of F_{q^{k}}, where tr_{k} denotes the trace from F_{q^{k}} to F_{q} and k denotes a positive integer. For an F_{q}-regular function h on an F_{q}-variety V, define

S_{k}(V, h) = ∑_{x∈V (F_{q^{k}})} Ψ_{k}(h(x)),

L(V, h, T) = exp(∑_{k=1}^{∞} S_{k}(V, h)\frac{T^{k}}{k}).

It is easy to see that

S_{k}(A^{N+r}, g, T) = q^{r}#X(F_{q^{k}}).

Thus,

Z(X, q^{r}T) = L(A^{N+r}, g, T).  (3.1)

Let Δ_{1} be the convex closure in R^{N+r} of the origin and the exponents of non-zero monomials in g. Normally, it would be natural to work with Δ_{1}. However, we find it easier later to work with a larger polyhedron Δ. For this purpose, we let Δ be the convex closure in R^{N+r} of Δ_{1} and the unit coordinate vectors e_{i} (i = 1, ⋯, N + r), where e_{i} is the unit coordinate vector on the x_{i}-axis. Let C(Δ) be the real cone generated by Δ, i.e., the collection of all non-negative real multiples of points of Δ. It is clear that C(Δ) is just the cone in R^{N+r} with all coordinates non-negative (the first quadrant). The cone C(Δ_{1}) generated by Δ_{1} is a subcone of C(Δ). For a point u = (u_{1}, ⋯, u_{N+r}) ∈ C(Δ), define its weight w(u) to be the smallest real number δ such that the dilation δΔ contains u. In the special case that u = (u_{1}, ⋯, u_{N+r}) is in the smaller cone C(Δ_{1}), we have w(u) = u_{N+1} + ⋯ + u_{N+r}.

Let Ω_{0} = Q_{p}(ζ_{p}, ζ_{q−1}), where Q_{p} denotes the p-adic rational numbers and ζ_{p} (resp. ζ_{q−1}) denotes a primitive p-th (resp. (q − 1)-th) root of unity. Let O_{0} be the ring of integers in Ω_{0} and let π ∈ O_{0} be a uniformizing parameter, so ord_{p}π = 1/(p − 1). Define two p-adic Banach spaces

B = { ∑_{u∈Z^{N+r}∩C(Δ)} A_{u}π^{w(u)}x^{u}|A_{u} ∈ Ω_{0}, A_{u} → 0 as u → ∞},

B_{1} = { ∑_{u∈Z^{N+r}∩C(Δ_{1})} A_{u}π^{w(u)}x^{u}|A_{u} ∈ Ω_{0}, A_{u} → 0 as u → ∞}.

The reason that we choose to work with B instead of B_{1} in this section is that the “reduction” modulo π of B is the full polynomial ring F_{q}[x_{1}, ⋯, x_{N+r}] instead of a smaller conical ring, and thus we will be in a more familiar situation.

Dwork’s trace formula for the torus G_{m}^{N+r} shows that there is a completely continuous Ω_{0}-linear operator α of B such that

L(G_{m}^{N+r}, g, T)^{(−1)^{N+r−1}} = ∏_{l=0}^{N+r} det(I − T q^{l}α|B)^{(−1)^{l}\binom{N+r}{l}}.  (3.2)

The subspace B_{1} is α-stable. Furthermore, the operator α is easily seen to be topologically nilpotent on the quotient space B/B_{1}. Thus, the Fredholm determinants of α acting on both B and B_{1} are the same. In particular, one could use B_{1} in place of B in (3.2). That is, we have

L(G_{m}^{N+r}, g, T)^{(−1)^{N+r−1}} = ∏_{l=0}^{N+r} det(I − T q^{l}α|B_{1})^{(−1)^{l}\binom{N+r}{l}}.  (3.3)

We shall work with the larger space B in this section and restrict to B_{1} in next section.

Formula (3.2) has a homological version. Let S = {1, ⋯, N+r}. There are commuting differential operators D_{i} (i ∈ S) of B such that

α ∘ D_{i} = qD_{i} ∘ α.  (3.4)

Let K_{.} be the Koszul complex

0 → K_{N+r} → K_{N+r−1} → ⋯ → K_{1} → K_{0} → 0  (3.5)

on B defined by D_{1}, ⋯, D_{N+r}. Its component of degree l is given by

K_{l} = ⨁_{|J|=l} Be_{J},

where the sum is over all subsets J of S of cardinality l and e_{J} is a formal symbol. Define an endomorphism α_{l} : K_{l} → K_{l} by

α_{l} = ⨁_{|J|=l} q^{l}α.

By (3.4), these α_{l} induce a chain map α_{.} on the complex in (3.5). By (3.2), one gets the chain level form of the Dwork trace formula for the torus G_{m}^{N+r}:

L(G_{m}^{N+r}, g, T)^{(−1)^{N+r−1}} = ∏_{l=0}^{N+r} det(I − T α_{l}|K_{l})^{(−1)^{l}}.  (3.6)

What we really need is the slightly more subtle trace formula for the affine space A^{N+r}. To describe it, recall that S = {1, ⋯, N + r}. For a subset J of S, let B^{J} denote the space of those power series in B which are divisible by all x_{j} with j ∈ J. Each B^{J} is stable under the operator α. Let K_{.}(S) be the subcomplex of K_{.} defined by

K_{l}(S) = ⨁_{J⊆S,|J|=l} B^{S−J}e_{J}.  (3.7)

Then, the trace formula for the affine (N + r)-space is given by

L(A^{N+r}, g, T)^{(−1)^{N+r−1}} = ∏_{l=0}^{N+r} det(I − T α_{l}|K_{l}(S))^{(−1)^{l}}.  (3.8)

Combining (3.1), (3.8) and (3.7), we deduce with the change of symbol S − J → J that

Z(X, q^{r}T)^{(−1)^{N+r−1}} = ∏_{J∈S} det(I − T q^{N+r−|J|}α|B^{J})^{(−1)^{N+r−|J|}}.  (3.9)

Let S_{1} = {1, ⋯, N} and S_{2} = {N + 1, ⋯, N + r}. The variables in S_{1} and S_{2} play somewhat different role. We need to treat them separately. For J ⊆ S, we can uniquely write J = {J_{1}, J_{2}} with J_{1} ⊆ S_{1} and J_{2} ⊆ S_{2}. In this notation, we also write B^{J} as B^{J_{1},J_{2}}. Equation (3.9) can be rewritten as

Z(X, q^{r}T)^{(−1)^{N+r−1}} = ∏_{J_{1},J_{2}} det(I − T q^{N+r−|J_{1}|−|J_{2}|}α|B^{J_{1},J_{2}})^{(−1)^{N+r−|J_{1}|−|J_{2}|}}. (3.10)

To prove Theorem 1.2b, the first step is to eliminate those factors in (3.10) with |J_{1}| < N. For this purpose, we need the following result. Lemma 3.1. The eigenvalues of α acting on B^{J_{1},J_{2}} are algebraic integers, and are divisible by q^{|J_{2}|} as algebraic integers.

Proof. Since B^{J_{1},J_{2}} is an α-stable subspace of B^{J_{2}}, it suffices to prove the lemma for B^{J_{2}}. We use induction on |J_{2}|.

If |J_{2}| = 0, Möbius inversion of equation (3.2) shows that the eigenvalues of α acting on B are algebraic integers since the reciprocal zeroes and poles of L-functions are algebraic integers. This proves the lemma for the case |J_{2}| = 0.

Assume now that |J_{2}| > 0. Let X^{J_{2}} be the affine variety in (N + r − |J_{2}|)-dimensional space G_{m}^{N}×G_{m}^{r−|J_{2}|} defined by the vanishing of the |J_{2}| polynomials f_{j}(x_{1}, ⋯, x_{N}) (N +j ∈ J_{2}), where G_{m}^{r−|J_{2}|} corresponds to the free toric variables x_{i} with i ∈ S_{2} − J_{2}. Let

g_{J_{2}} = ∑_{N+j∈J_{2}} x_{N+j}f_{j}(x_{1}, ⋯, x_{N}).

Similar to (3.1), one finds that

Z(X^{J_{2}}, q^{|J_{2}|}T) = L(G_{m}^{N+r−|J_{2}|} × A^{|J_{2}|}, g_{J_{2}}, T),  (3.11)

where A^{|J_{2}|} corresponds to the variables x_{N+j} (N + j ∈ J_{2}) in g_{J_{2}}. Let J_{2}^{c} be the total compliment S − J_{2} of J_{2} in S. Applying a slightly more general trace formula to the right side of (3.11), one obtains the formula

Z(X^{J_{2}}, q^{|J_{2}|}T)^{(−1)^{N+r−1}} = ∏_{I_{1}⊆J_{2}^{c},I_{2}⊆J_{2}} det(I − T q^{N+r−|I_{1}|−|I_{2}|}α|B^{I_{2}})^{(−1)^{N+r−|I_{1}|−|I_{2}|}}.

(3.12)

Note that the space B^{I_{2}} in the above formula is independent of I_{1}. By induction, for |I_{2}| < |J_{2}|, the eigenvalues of q^{N+r−|I_{1}|−|I_{2}|}α acting on B^{I_{2}} are algebraic integers, and divisible as algebraic integers by q^{N+r−|I_{1}|}. Since N + r − |I_{1}| ≥ |J_{2}|, it follows from (3.12) that for the p-adic meromorphic function (the part in (3.12) with I_{2} = J_{2}) defined by

F (J_{2}, T) = ∏_{I_{1}⊆J_{2}^{c}} det(I − T q^{N+r−|I_{1}|−|J_{2}|}α|B^{J_{2}})^{(−1)^{N+r−|I_{1}|−|J_{2}|}},  (3.13)

its reciprocal zeroes and poles are algebraic integers and divisible as algebraic integers by q^{|J_{2}|}. Möbius inversion of (3.13) shows that the eigenvalues of α acting on B^{J_{2}} are algebraic integers, and divisible as algebraic integers by q^{|J_{2}|}. The lemma is proved.

By (3.10) and Lemma 3.1, we deduce that the eigenvalues of q^{N+r−|J_{1}|−|J_{2}|}α acting on B^{J_{1},J_{2}} are divisible as algebraic integers by q^{N+r−|J_{1}|}. Since |J_{1}| ≤ N, to prove Theorem 1.2b, it suffices to restrict to those factors in (3.10) with J_{1} = S_{1}. Namely, Theorem 1.2b is reduced to proving the following result.

Lemma 3.2. If the sequence {f_{1}, ⋯, f_{r}} forms a regular sequence of F_{q}[x_{1}, ⋯, x_{N}], then the p-adic meromorphic function defined by

F (X, T) = ∏_{J_{2}∈S_{2}} det(I − T q^{r−|J_{2}|}α|B^{S_{1},J_{2}})^{(−1)^{r−|J_{2}|}}  (3.14)

is a p-adic entire function in T.

Proof. This lemma should be equivalent to a vanishing theorem on cohomology. Thus, we need to find a cohomological formula for F (X, T). For 0 ≤ l ≤ r, define

C_{l}(S) = ⨁_{J_{2}∈S_{2},|J_{2}|=l} B^{S_{1},S_{2}−J_{2}}e_{J_{2}}.

Then C_{l}(S) is a subspace of K_{l}(S), stable under the operator α. We use α_{l} again to denote the operator

α_{l} = ⨁_{J_{2}∈S_{2},|J_{2}|=l} q^{l}α,

acting on C_{l}(S). This should not cause confusion since the space will be specified. Define the boundary operator ∂_{l} : C_{l}(S) → C_{l−1}(S) as follows: If η ∈ B^{S_{1},S_{2}−J_{2}}e_{J_{2}} with J_{2} = {i_{1}, ⋯, i_{l}} and i_{1} < ⋯ < i_{l}, then

∂_{l}(ηe_{J_{2}}) = ∑_{k=1}^{l}(−1)^{k−1}D_{i_{k}}(η)e_{J_{2}−i_{k}},

where D_{N+1}, ⋯, D_{N+r} are the second part of the differential operators {D_{1}, ⋯, D_{N+r}} mentioned in (3.4). Note that D_{j} sends B^{S−j} to B^{S}. In this way, the operator α_{.} becomes a chain map of the complex C.(S):

0 → C_{r}(S) → C_{r−1}(S) → ⋯ → C_{1}(S) → C_{0}(S) → 0.

And we have the formula

\begin{aligned}F (X, T) & = ∏_{l=0}^{r} det(I − T α_{l}|C_{l}(S))^{(−1)^{l}} \\ & = ∏_{l=0}^{r} det (I − T α_{l}|H_{l}(C.(S)))^{(−1)^{l}} , \end{aligned}

where the H_{l} denote the homology groups of the complex C_{.}(S). Lemma 3.2 is reduced to proving the following vanishing result.

Lemma 3.3. If the sequence {f_{1}, ⋯, f_{r}} forms a regular sequence of F_{q}[x_{1}, ⋯, x_{N}], then

H_{l}(C_{.}(S)) = 0, for all l > 0.

Proof. Let B(O_{0}) be the unit ball in B:

B(O_{0}) = { ∑_{u∈Z^{N+r}∩C(Δ)} A_{u}π^{w(u)}x^{u}|A_{u} ∈ O_{0}, A_{u} → 0 as u → ∞}.

The unit ball B(O_{0}) is a flat, complete and separated O_{0}-module. Sending an element ∑ A_{u}π^{w(u)}x^{u} to ∑ A_{u}x^{u} modulo π is a reduction homomorphism from B(O_{0}) into the polynomial ring P = F_{q}[x_{1}, ⋯, x_{N+r}]. Since C(Δ) consists of all points in R^{N+r} with non-negative coordinates, the reduction homomorphism from B(O_{0}) to the polynomial ring P is surjective.

By working with B(O_{0}) instead of B, one finds in a standard manner [Mo, Theorem 8.5] that it suffices to prove that the reduction C̄_{.}(S) modulo π of the complex C_{.}(S) is acyclic in positive dimensions. The reduction of the differential operator D_{N+i} is well known to be given by

D̄_{N+i} = x_{N+i} \frac{∂}{∂x_{N+i}} + x_{N+i}f_{i}.

We want to identify the complex C̄_{.}(S) with a simpler Koszul complex. For this purpose, let

E_{N+i} = \frac{∂}{∂x_{N+i}} + f_{i}.

These are r commuting differential operators acting on the polynomial ring P as ∂/∂x_{N+i} kills f_{j}. Let K_{.}(P, E) be the Koszul complex on P defined by the differential operators E_{N+1}, ⋯, E_{N+r}. Thus, for 0 ≤ l ≤ r, we have

K_{l}(P, E) = ⨁_{|J_{2}|=l,J_{2}∈S_{2}} P e_{J_{2}}.

Define a map from K_{l}(P, E) to C̄_{l}(S) by the rule

ψ : ηe_{J_{2}} ⟶ ( ∏_{i∈S−J_{2}} x_{i})ηe_{J_{2}}.

It is easy to see that ψ defines an F_{q}-vector space isomorphism between K_{l}(P, E) and C̄_{l}(S), where

C̄_{l}(S) = ⨁_{J_{2}∈S_{2},|J_{2}|=l} P^{S_{1},S_{2}−J_{2}}e_{J_{2}}

and P^{J} denotes the space of those polynomials in P which are divisible by all the x_{j} with j ∈ J.

One further checks that ψ is actually a chain map. This follows from the commutative diagram for j ∈ J_{2}:

P e_{J_{2}}  ⟶^{ψ}  P^{S_{1},S_{2}−J_{2}}e_{J_{2}}
↓E_{j}  ↓D̄_{j}
P e_{J_{2}−j} ⟶^{ψ} P^{S_{1},S_{2}−{J_{2}−j}}e_{J_{2}−j}.

Thus, it suffices to prove that the Koszul complex K_{.}(P, E) is acyclic in positive dimensions. This results from the following lemma. Lemma 3.4. If the sequence {f_{1}, ⋯, f_{r}} forms a regular sequence of F_{q}[x_{1}, ⋯, x_{N}], then the sequence {E_{N+1}, ⋯, E_{N+r}} forms a regular sequence for P = F_{q}[x_{1}, ⋯, x_{N+r}].

Proof. Denote F_{q}[x_{1}, ⋯, x_{N}] by P_{0}. Then

P = P_{0}[x_{N+1}, ⋯, x_{N+r}].

We need to prove the claim that if ϕ_{1}, ⋯, ϕ_{r} are in P such that ∑_{i} E_{N+i}ϕ_{i} = 0, then there is a skew-symmetric set a_{ij} ∈ P (1 ≤ i, j ≤ r) such that ϕ_{i} = ∑_{j} E_{N+j}a_{ij}. This is done by induction on the maximal degree k = max_{i} deg(ϕ_{i}), where deg(ϕ_{i}) means the degree of ϕ_{i} in the last r variables {x_{N+1}, ⋯, x_{N+r}}. We now prove the claim and assume ∑_{i} E_{N+i}ϕ_{i} = 0.

If k ≤ 0 (the constant 0 has degree −∞), then all ϕ_{i} are in the subring P_{0}. Since the differential operators ∂/∂x_{N+i} kill P_{0}, we have ∑_{i} f_{i}ϕ_{i} = ∑_{i} E_{N+i}ϕ_{i} = 0. By our assumption, the sequence {f_{1}, ⋯, f_{r}} forms a regular sequence of P_{0}. Thus, there is a skew-symmetric set a_{ij} ∈ P_{0} (1 ≤ i, j ≤ r) such that ϕ_{i} = ∑_{j} f_{j}a_{ij} = ∑_{j} E_{N+j}a_{ij}. The claim is proved for k ≤ 0.

Assume now that k is a positive integer. Let ϕ_{i}^{(k)} be the degree k part of ϕ_{i} with respect to the variables {x_{N+1}, ⋯, x_{N+r}}. The differential operators ∂/∂x_{N+i} reduce the degree by 1 and the multiplication operators f_{i} keep the degree. Comparing the highest degree parts, one finds that ∑_{i} f_{i}ϕ_{i}^{(k)} = 0. Since P is free over P_{0}, the sequence {f_{1}, ⋯, f_{r}} also forms a regular sequence for the larger ring P. Thus, there is a skew-symmetric set b_{ij} ∈ P (1 ≤ i, j ≤ r) of degree k such that ϕ_{i}^{(k)} = ∑_{j} f_{j}b_{ij}. Now,

∑_{i=1}^{r} E_{N+i} (ϕ_{i} − ∑_{j=1}^{r} E_{N+j}(b_{ij})) = 0

and for all 1 ≤ i ≤ r,

deg (ϕ_{i} − ∑_{j=1}^{r} E_{N+j}(b_{ij})) < k.

By induction, the claim is proved.

4. The polar part of the Ax-Katz theorem

In this section, we prove a slight generalization of Theorem 1.3 by taking into account the terms actually appearing in the polynomials f_{i}. Recall that Δ_{1} is the convex closure in R^{N+r} of the origin and the exponents of the non-zero terms in the polynomial g and C(Δ_{1}) is the cone generated by Δ_{1}. For u ∈ C(Δ_{1}), let w(u) be the weight

w(u) = u_{N+1} + ⋯ + u_{N+r}.

For I ∈ S, define

w_{I}(g) = min{w(u) | u ∈ Z^{N+r} ∩ C(Δ_{1}), u_{i} > 0 for all i ∈ I}.

For 1 ≤ i ≤ N, let

ν_{i} = w_{S−i}(g).

By definition, it is clear that ν_{i} − r is a non negative integer for each 1 ≤ i ≤ N as S_{2} ⊆ S − i. Theorem 4.1. Let X be a complete intersection defined by the vanishing of a regular sequence of polynomials {f_{1}, ⋯, f_{r}} over F_{q} in N variables of degrees d_{1}, ⋯, d_{r}. Then, all the reciprocal poles β_{j} of Z(X, T)^{(−1)^{N−r−1}} are divisible by q^{min_{i}(1+ν_{i}−r)} as algebraic integers.

Proof. We now restrict the operator α from B to B_{1}. Since the Fredholm determinants of α acting on both B and B_{1} are the same, equation (3.10) and Lemmas (3.1)-(3.2) remain valid if we replace B by B_{1}. By (3.10) and Lemma 3.2, it suffices to prove that for all J_{1} ⊂ S_{1} (J_{1} ≠ S_{1}) and J_{2} ⊆ S_{2}, the first non-trivial slope of the Newton polygon computed with respect to ord_{q} of the p-adic entire function defined by

Z(J_{1}, J_{2}) = det(I − T q^{N+r−|J_{1}|−|J_{2}|}α|B_{1}^{J_{1},J_{2}}),  (4.1)

is at least min_{i}(1 + ν_{i}). By corollary 4.3 in [AS1], this first non-trivial slope is at least

N + r − |J_{1}| − |J_{2}| + w_{J_{1},J_{2}}(g).  (4.2)

By Lemma 4.5 in [AS1], the minimun of the numbers in (4.2) occurs when |J_{1}| + |J_{2}| is as large as possible, namely, J_{2} = S_{2} and J_{1} = S_{1} − i for some i since we assumed that J_{1} ≠ S_{1}. In the last case, the number in (4.2) reduces to 1 + ν_{i}. By (3.10), this proves that all reciprocal poles β_{j} of Z(X, T)^{(−1)^{N−r−1}} are divisible by q^{min_{i}(1+ν_{i}−r)} as p-adic integers. This divisibility is actually in the stronger sense of algebraic integers as ∏_{j}(1 − β_{j}T), being the denominator of Z(X, T)^{(−1)^{N−r−1}}, is a polynomial with integer coefficients. One simply makes the change of variable T ⟶ q^{− min_{i}(ν_{i}+1−r)}T. A standard argument using various extension fields of F_{q} shows that the resulting denominator is still a polynomial with integer coefficients, see [Ka1]. Theorem 4.1 is proved.

Following [AS1, (5.7)], one checks that

ν_{j} − r ≥ max\{0, \frac{(N − 1 − ∑_{i} d_{i})}{max_{i} d_{i}}\}.

Since ν_{j} − r is a non-negative integer for each 1 ≤ j ≤ N, Theorem 4.1 implies Theorem 1.3.

Finally, we show that some of Ax’s optimal hypersurface examples for his theorem are also optimal hypersurface examples for Theorem 1.3. Let d ≥ 3 and write N = bd + h, where 0 < h ≤ d. We assume that d does not divide (N − 1)(N − 2), i.e., h > 2. One checks that μ = b and ν = b.

Let X be the hypersurface of degree d in A^{N} defined by the polynomial

f(x_{1}, ⋯, x_{N}) = (∑_{i=0}^{b−1} x_{id+1} ⋯ x_{(i+1)d}) + x_{bd+1} ⋯ x_{N}.

Let N(f) be the number of F_{q}-rational points on the hypersurface X. It is shown in [Ax] that q^{b}∥N(f). Using Ax’s recursive formula for N(f), it is straightforward to show more precisely that

N(f) = q^{b}(−1)^{bd+h−1} (1 − (bd + h − b)q + ⋯) ,

where the omitted terms are sums and differences of the terms of the form q^{k} with k ≥ 2. This formula shows that

Z(X, T)^{(−1)^{bd+h−2}} = \frac{(1 − q^{b}T)}{(1 − q^{b+1}T)^{bd+h−b}} R(T),

where R(T) is a rational function whose reciprocal zeroes and poles are divisible by q^{b+2}.

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