Total Degree Bounds for Artin L-functions and Partial Zeta Functions

Lei Fu

Institute of Mathematics, Nankai University, Tianjin, P. R. China leifu@nankai.edu.cn

Daqing Wan

Department of Mathematics, University of California, Irvine, CA 92697 dwan@math.uci.edu

0. Introduction

Let F_{q} be a finite field of characteristic p with q elements. Choose an algebraic closure F of F_{q}. Throughout this paper, schemes, morphisms and sheaves defined over the base field F_{q} are denoted by letters with subscripts 0. We indicate the base extension from F_{q} to F by dropping the subscripts 0. Schemes and morphisms are separated and of finite type.

Let X_{0} be a scheme over F_{q} and let F : X → X be the geometric Frobenius correspondence. For any endomorphism f : X → X, denote by Λ(f) the number of fixed points of f, that is,

Λ(f) = #{x ∈ X(F)|f(x) = x}.

Let G be a finite group acting on X_{0} on the right and let ρ : G → GL(V) be a finite dimensional \overline{Q}_{l}-representation of G, where l is a prime number different from p. For each positive integer k, set

v_{k} = \frac{1}{#G} ∑_{g∈G} Tr(ρ(g)^{−1})Λ(gF^{k}).

The Artin L-function L(X_{0}, ρ, t) is defined by

L(X_{0}, ρ, t) = exp(∑_{k=1}^{∞} v_{k} \frac{t^{k}}{k} ).

By a well known theorem of Grothendieck ([G]), this is a rational function. Define its total degree totdegL(X_{0}, ρ, t) to be the sum of the number of its zeros and the number of its poles counted with multiplicities. In [BS], under the assumption that X_{0} is a closed subscheme of the n-dimensional affine space A_{0}^{n} defined by the vanishing of polynomials of degrees at most d and G acts linearly on X_{0} in the sense that G ⊂ GL(A_{0}^{n}), Bombieri and Sperber prove that

totdegL(X_{0}, ρ, t) ≤ (dimρ)^{2}(4d + 9)^{4n}.

If the action of G on X_{0} is not linear, an explicit total degree bound can also be derived from the remark in [BS], but the resulting bound would depend on the order of the group G. As indicated in [BS], in some special cases, better bounds are known and these improved bounds are of considerable importance in estimating character sums, because they lead to estimates independent of the order of G.

In this paper, we give a bound for totdegL(X_{0}, ρ, t) without assuming G acts linearly on X_{0}. Our bound is independent of the order of G and hence is important for estimating character sums. Precisely, we have

Theorem 0.1. Let X_{0} be a closed subscheme of A_{0}^{n} defined by the vanishing of r polynomials of degrees at most d. Let G be a finite group acting on X_{0} on the right and let ρ : G → GL(V) be a finite dimensional \overline{Q}_{l}-representation. Let V = n_{1}V_{1} ⊕ ⋯⊕n_{k}V_{k} be the decomposition of V into a direct sum of irreducible representations of G, where each n_{j} is the multiplicity of the irreducible representation V_{j} appeared in V. Then we have

totdegL(X_{0}, ρ, t) ≤ 3 · 2^{r+1} · (3 + rd)^{n+1} · (sup_{j}\{\frac{n_{j}}{dimV_{j}} \}).

Obviously we have sup_{j}\{\frac{n_{j}}{dimV_{j}} \} ≤ dimρ. So we have Corollary. Notation as in Theorem 0.1. We have

totdegL(X_{0}, ρ, t) ≤ 3 · 2^{r+1} · (3 + rd)^{n+1} · (dimρ).

The main idea of our proof is to use Katz’s bound on the sum of Betti numbers ([K1]) and a fixed point formula. Originally we only get the estimate in the Corollary. The estimate in Theorem 0.1 is suggested by the referee.

We now describe our second result. Let X_{0} be a closed subscheme of A_{0}^{n} defined by the vanishing of some polynomials. Let d_{1}, …, d_{n} be n positive integers and let N_{d_{1}...d_{n}}(k, X_{0}) be the number of points (x_{1}, …, x_{n}) in X(F) such that x_{1} ∈ F_{q^{kd_{1}}}, …, x_{n} ∈ F_{q^{kdn}}. In [W1], Wan defines the partial zeta function Z_{d_{1}...d_{n}}(X_{0}, t) to be

Z_{d_{1}...d_{n}}(X_{0}, t) = exp(∑_{k=1}^{∞} N_{d_{1},...d_{n}}(k, X_{0})\frac{t^{k}}{k} ).

He proves that Z_{d_{1}...d_{n}}(X_{0}, t) is rational if the integers d_{i}’s can be arranged such that d_{1}|d_{2}| ⋯ |d_{n}, generalizing Dwork’s classical rationality theorem. In general, it is not clear if the partial zeta function is always rational, but Faltings [W1] noted that the partial zeta function is always nearly rational in the sense that there are some roots of unity λ_{k} and some algebraic numbers μ_{k} (k = 1, …, m) such that

Z_{d_{1}...d_{n}}(X_{0}, t) = ∏_{k=1}^{m}(1 − μ_{k}t)^{λ_{k}}.

In the above expression, by collecting similar terms, we may also assume that the μ_{k} (k = 1, …, m) are distinct, but then the λ_{k} may not be roots of unity anymore. In this case, we define the total degree of Z_{d_{1}...d_{n}}(X_{0}, t) to be ∑_{k=1}^{m} |λ_{k}|. Here we fix an archimedean absolute value on the field of algebraic numbers.

Theorem 0.2. Let X_{0} be a closed subscheme of A_{0}^{n} defined by the vanishing of r polynomials of degrees at most d. Let d_{1}, …, d_{n} be n positive integers and let m be their least common multiple. Then we have

totdegZ_{d_{1}...d_{n}}(X_{0}, t) ≤ 3 · 2^{mr+1} · (3 + mrd)^{d_{1}+···+d_{n}+1}.

The main idea of the proof of Theorem 0.2 is again to use Katz’s bound on the sum of Betti numbers, an observation of Faltings, and a fixed point formula. As an immediate application of Theorem 0.2, one obtains an explicit total degree bound for the geometric moment zeta functions [W5] (which are rational functions) attached to a family of algebraic varieties over finite fields. In particular, one obtains explicit estimates for the constants in the higher degree excess theorems in Katz [K2] in his statistical study of the universal family of smooth projective hypersurfaces over finite fields. In some special cases, Theorem 0.2 can be greatly improved, especially about the asymptotic behavior of the bound as m goes to infinity while some of the d_{i}’s are fixed. The universal family of elliptic curve and its relations to dimensions of modular forms of high weights provide such an example. Another interesting case to consider is the higher moment zeta function arising from the universal family of smooth projective hypersurfaces.

As an archimedean analogue of Dwork’s p-adic unit root zeta functions, certain pure weight L-functions arising from a family of algebraic varieties were introduced in [W2]. The rationality of these pure weight L-functions depends on the full strength of Deligne’s main theorem [D] which says that the l-adic higher direct image sheaves with compact support are mixed. We do not yet know how to give an explicit total degree bound for these deeper pure weight L-functions. The “degree bound” problem, that is, the lower bound problem for the p-adic Newton polygon, for the pure (p-adic) slope L-functions in Dwork’s original conjecture is even more difficult, as these pure slope L-functions [W3][W4] are p-adic meromorphic functions and in general not rational any more. In the so-called rank one case, an explicit lower bound for the p-adic Newton polygon has been worked out in [W4]. No non-trivial bound is known in the higher rank case.

Although our principal motivation of this paper comes from theoretical considerations, we have also been influenced by practical algorithmic computations of zeta functions and L-functions. Our explicit bounds are useful in explicitly computing the related zeta functions and L-functions. This is an emerging new subject, which we shall not discuss it here, but see [W6] for an expository introduction.

Acknowledgements: We thank the referee for suggesting the estimate in Theorem 0.1, and for his many other suggestions. The research of Lei Fu is supported by Qiushi Science & Technologies Foundation and Project 973. The research of Daqing Wan is partially supported by the NSF (DMS 9970417) and the NNSF of China (10128103). We also thank the Morningside Center of Mathematics in Beijing, where the present work was carried out.

1. Proof of the Theorems

We need the following result of Katz ([K1], Corollary of Theorem 1):

Proposition 1.1. (Katz) Let X be a closed subscheme of A^{n} over F defined by the vanishing of r polynomials of degrees at most d. We have

∑_{i=0}^{2dimX} dimH_{c}^{i}(X, \overline{Q}_{l}) ≤ 3 · 2^{r+1} · (3 + rd)^{n+1}.

We also need the following fixed point formula:

Proposition 1.2. Let X_{0} be a quasi-projective scheme over F_{q}. For any F_{q}- automorphism σ_{0} : X_{0} → X_{0} of finite order and for any positive integer k, we have

Λ(σF^{k}) = ∑_{i=0}^{2dimX}(−1)^{i}Tr(σF^{k}, H_{c}^{i}(X, \overline{Q}_{l})),

where F : X → X is the geometric Frobenius correspondence and Λ(σF^{k}) is the number of fixed points of σF^{k} : X → X.

To prove this fixed point formula, we use Galois decent theory to show that there exists another scheme X′_{0} over F_{q} such that X′ is isomorphic to X over F and the geometric Frobenius correspondence on X′ coincides with σF on X. Proposition 1.2 for the case k = 1 then follows from the fixed point formula for the geometric Frobenius correspondence on X′. The general case can be proved by working over the base F_{q^{k}}.

Another result we need is the following well-known fact:

Lemma 1.3. Let G be a finite group, ρ : G → GL(V) a finite dimensional \overline{Q}_{l}- representation, and F : V → V a G-equivariant endomorphism of V. Then we have

Tr(F, V^{G}) = \frac{1}{#G} ∑_{g∈G} Tr(gF, V).

Proof. Note that since F is G-equivariant, the subspace V^{G} is invariant under the action of F. So we may talk about Tr(F, V^{G}). Consider the homomorphism

π : V → V, x ↦ \frac{1}{#G} ∑_{g∈G} gx.

Then we have im(π) = V^{G} and π|_{V^{G}} = id. So we have

Tr(F, V^{G}) = Tr(πF, V^{G}) = Tr(πF, V) = \frac{1}{#G} ∑_{g∈G} Tr(gF, V).

Let’s prove Theorem 0.1. Let G be a finite group acting on the right of a quasi-projective scheme X_{0} over F_{q}, let ρ : G → GL(V) be a finite dimensional \overline{Q}_{l}-representation of G and let

L(X_{0}, ρ, t) = exp(∑_{k=1}^{∞} v_{k} \frac{t^{k}}{k} )

be the Artin L-function, where

v_{k} = \frac{1}{#G} ∑_{g∈G} Tr(g^{−1}, V)Λ(gF^{k}).

By Proposition 1.2, we have

v_{k} = \frac{1}{#G} ∑_{g∈G} Tr(g^{−1}, V)Λ(gF^{k})
= \frac{1}{#G} ∑_{g∈G} Tr(g^{−1}, V) ∑_{i=0}^{2dimX}(−1)^{i}Tr(gF^{k}, H_{c}^{i}(X, \overline{Q}_{l}))
= ∑_{i=0}^{2dimX}(−1)^{i} \frac{1}{#G} ∑_{g∈G} Tr(gF^{k}, H_{c}^{i}(X, \overline{Q}_{l}) ⊗_{\overline{Q}_{l}} V^{∗}),

where V^{∗} denotes the representation of G dual to V and F acts trivially on V^{∗}. Combining with Lemma 1.3 applied to the representation H_{c}^{i}(X, \overline{Q}_{l}) ⊗_{\overline{Q}_{l}} V^{∗} of G and the equivariant endomorphism F^{k}, we get

v_{k} = ∑_{i=0}^{2dimX}(−1)^{i}Tr(F^{k}, (H_{c}^{i}(X, \overline{Q}_{l}) ⊗_{\overline{Q}_{l}} V^{∗})^{G}).

So we have

L(X_{0}, ρ, t) = exp(∑_{k=1}^{∞} v_{k} \frac{t^{k}}{k} ) = ∏_{i=0}^{2dimX} det(I − F t, (H_{c}^{i}(X, \overline{Q}_{l}) ⊗_{\overline{Q}_{l}} V^{∗})^{G})^{(−1)^{i+1}}.

Therefore

totdegL(X_{0}, ρ, t) ≤ ∑_{i=0}^{2dimX} dim(H_{c}^{i}(X, \overline{Q}_{l}) ⊗_{\overline{Q}_{l}} V^{∗})^{G}.

Let V = n_{1}V_{1} ⊕ ⋯ ⊕ n_{k}V_{k} be the decomposition of V into a direct sum of irreducible representations of G, where each n_{j} is the multiplicity of the irreducible representation V_{j} appeared in V. Then we have

dim(H_{c}^{i}(X, \overline{Q}_{l}) ⊗_{\overline{Q}_{l}} V^{∗})^{G} = ∑_{j=1}^{k} n_{j}dim(H_{c}^{i}(X, \overline{Q}_{l}) ⊗_{\overline{Q}_{l}} V_{j}^{∗})^{G} = ∑_{j=1}^{k} n_{j}m_{j},

where m_{j} = dim(H_{c}^{i}(X, \overline{Q}_{l}) ⊗_{\overline{Q}_{l}} V_{j}^{∗})^{G}. Note that m_{j} equals the multiplicity of V_{j} appeared in the representation H_{c}^{i}(X, \overline{Q}_{l}) of G. So we have ∑_{j=1}^{k} m_{j}dimV_{j} ≤ dimH_{c}^{i}(X, \overline{Q}_{l}). Therefore

dim(H_{c}^{i}(X, \overline{Q}_{l}) ⊗_{\overline{Q}_{l}} V^{∗})^{G} = ∑_{j=1}^{k} \frac{n_{j}}{dimV_{j}} m_{j}dimV_{j}
≤ (sup_{j}\{\frac{n_{j}}{dimV_{j}} \})(∑_{j=1}^{k} m_{j}dimV_{j})
≤ (sup_{j}\{\frac{n_{j}}{dimV_{j}} \})(dimH_{c}^{i}(X, \overline{Q}_{l})),

and hence

totdegL(X_{0}, ρ, t) ≤ (sup_{j}\{\frac{n_{j}}{dimV_{j}} \})( ∑_{i=0}^{2dimX} dimH_{c}^{i}(X, \overline{Q}_{l})).

Theorem 0.1 then follows from Katz’s bound in Proposition 1.1.

Now we turn to the proof of Theorem 0.2. Let X_{0} be a closed subscheme of A_{0}^{n} over F_{q} defined by the vanishing of r polynomials of degrees at most d. Let d_{1}, …, d_{n} be n positive integers and let m be their least common multiple. Let Y_{0} be the subvariety of the m-fold product X_{0}^{m} defined as follows: For any point (x_{1}, …, x_{m}) in X^{m}(F), where each x_{j} is a point in X(F) ⊂ A^{n}(F), write x_{j} = (x_{1j}, …, x_{nj}). We define Y_{0} to be the closed subvariety of X_{0}^{m} ⊂ A_{0}^{mn} defined by the equations

x_{ij} = x_{ij′} whenever j ≡ j′ mod d_{i} (i ∈ {1, …, n}, j, j′ ∈ {1, …, m}).

Keeping only the variables x_{ij} with 1 ≤ j ≤ d_{i} (1 ≤ i ≤ n), one obtains a closed subvariety Y′_{0} ⊂ A_{0}^{d_{1}+···+d_{n}} defined by mr equations of degrees at most d, and Y′_{0} is isomorphic to Y_{0}. Let σ_{0} : X_{0}^{d} → X_{0}^{d} be the automorphism defined by

(x_{1}, …, x_{d}) → (x_{d}, x_{1}, …, x_{d−1}).

Then Y_{0} is invariant under the action of σ_{0}. For each positive integer k, as Faltings observed, one can show that

N_{d_{1}...d_{n}}(k, X_{0}) = Λ(σF^{k}),

where the left-hand side is the number of points (x_{1}, …, x_{n}) in X(F) such that x_{1} ∈ F_{q^{kd_{1}}}, …, x_{n} ∈ F_{q^{kdn}}, and the right-hand side is the number of fixed points of the endomorphism σF^{k} : Y → Y. (Confer §3 in [W1]). For each i, let λ_{i1}, …, λ_{ik_{i}} be all the distinct eigenvalues of σ acting on H_{c}^{i}(Y, \overline{Q}_{l}) and let H_{i1}, …, H_{ik_{i}} be the corresponding eigenvector spaces. Since σ has finite order, each λ_{ij} is a root of unity and

H_{c}^{i}(Y, \overline{Q}_{l}) = H_{i1} ⊕ ⋯ ⊕ H_{ik_{i}}.

Since F commutes with σ, each H_{ij} is invariant under the action of F. Let μ_{ij1}, …, μ_{ijk_{ij}} be all the eigenvalues of F acting on H_{ij}, where k_{ij} = dimH_{ij}. Each μ_{ijl} is an algebraic number. We have

Tr(σF^{k}, H_{c}^{i}(Y, \overline{Q}_{l})) = ∑_{j=1}^{k_{i}} ∑_{l=1}^{k_{ij}} λ_{ij}μ_{ijl}^{k}.

By Proposition 1.2, we have

N_{d_{1}...d_{n}}(k, X) = Λ(σF^{k}) = ∑_{i=0}^{2dimY} (−1)^{i}Tr(σF^{k}, H_{c}^{i}(Y, \overline{Q}_{l})) = ∑_{i=0}^{2dimY} ∑_{j=1}^{k_{i}} ∑_{l=1}^{k_{ij}}(−1)^{i}λ_{ij}μ_{ijl}^{k}.

So

∑_{k=1}^{∞} N_{d_{1}...d_{n}}(k, X)t^{k} = ∑_{i=0}^{2dimY} ∑_{j=1}^{k_{i}} ∑_{l=1}^{k_{ij}}(−1)^{i}λ_{ij}t\frac{d}{dt} ln(1 − μ_{ijl}t)^{−1},

that is,

t\frac{d}{dt} ln Z_{d_{1}...d_{n}}(X_{0}, t) = ∑_{i=0}^{2dimY} ∑_{j=1}^{k_{i}} ∑_{l=1}^{k_{ij}}(−1)^{i}λ_{ij}t\frac{d}{dt} ln(1 − μ_{ijl}t)^{−1},

where Z_{d_{1}...d_{n}}(X_{0}, t) is the partial zeta function. This implies that

Z_{d_{1}...d_{n}}(X_{0}, t) = ∏_{i=0}^{2dimY} ∏_{j=1}^{k_{i}} ∏_{l=1}^{k_{ij}}(1 − μ_{ijl}t)^{(−1)^{i+1}λ_{ij}},

and

totdegZ_{d_{1}...d_{n}}(X_{0}, t) ≤ ∑_{i=0}^{2dimY} ∑_{j=1}^{k_{i}} ∑_{l=1}^{k_{ij}} |(−1)^{i+1}λ_{ij}|
= ∑_{i=0}^{2dimY} ∑_{j=1}^{k_{i}} ∑_{l=1}^{k_{ij}} 1
= ∑_{i=0}^{2dimY} dimH_{c}^{i}(Y, \overline{Q}_{l})
= ∑_{i=0}^{2dimY} dimH_{c}^{i}(Y′, \overline{Q}_{l}).

Theorem 0.2 follows from this last inequality, Proposition 1.1, and the fact that Y′_{0} is a closed subscheme of A^{d_{1}+···+d_{n}} defined by the vanishing of mr polynomials of degrees at most d.

References

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[G] A. Grothendieck, Formule de Lefschetz et Rationalité des Fonctions L, Séminarie Bourbaki (1964/65), No. 279.

[K1] N. Katz, Sums of Betti numbers in arbitrary characteristic, Finite Fields & Appl., 7(2001), 29-44.

[K2] N. Katz, Frobenius-Schur indicator and the ubiquity of Brock-Granville quadratic excess, Finite Fields & Appl., 7(2001), 45-69.

[W1] D. Wan, Partial zeta functions of algebraic varieties over finite fields, Finite Fields & Appl. 7(2001), 238-251.

[W2] D. Wan, Pure L-functions from algebraic geometry over finite fields, Finite fields and Applications (Augsburg, 1999), 437-461, Springer, Berlin, 2001.

[W3] D. Wan, Higher rank case of Dwork’s conjecture, J. Amer. Math. Soc., Volume 13, 4(2000), 807-852.

[W4] D. Wan, Rank one case of Dwork’s conjecture, J. Amer. Math. Soc., Volume 13, 4(2000), 853-908.

[W5] D. Wan, Moment zeta functions in arithmetic geometry, preprint.

[W6] D. Wan, Algorithmic zeta functions over finite fields, MSRI Algorithmic Number Theory Proceedings, to appear.